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 * Copyright (c) 2011, 2013, Oracle and/or its affiliates. All rights reserved.
 * ORACLE PROPRIETARY/CONFIDENTIAL. Use is subject to license terms.
 *
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 */

package javafx.scene.transform;


import com.sun.javafx.geom.transform.Affine3D;
import com.sun.javafx.geom.transform.BaseTransform;
import javafx.beans.property.DoubleProperty;
import javafx.beans.property.SimpleDoubleProperty;
import javafx.geometry.Point2D;
import javafx.geometry.Point3D;

// PENDING_DOC_REVIEW of this whole class
/**
 * <p>
 * The {@code Affine} class represents a general affine transform. An affine
 * transform performs a linear mapping from 2D/3D coordinates to other 2D/3D
 * coordinates while preserving the "straightness" and "parallelness"
 * of lines.
 * Affine transformations can be constructed using sequence rotations,
 * translations, scales, and shears.</p>
 *
 * <p>
 * For simple transformations application developers should use the
 * specific {@code Translate}, {@code Scale}, {@code Rotate}, or {@code Shear}
 * transforms, which are more lightweight and thus more optimal for this simple
 * purpose. The {@code Affine} class, on the other hand, has the advantage
 * of being able to represent a general affine transform and perform matrix
 * operations on it in place, so it fits better for more complex transformation
 * usages.</p>

 * <p>
 * Such a coordinate transformation can be represented by a 3 row by
 * 4 column matrix. This matrix transforms source coordinates {@code (x,y,z)}
 * into destination coordinates {@code (x',y',z')} by considering
 * them to be a column vector and multiplying the coordinate vector
 * by the matrix according to the following process:</p>
 *
 * <pre>
 *      [ x']   [  mxx  mxy  mxz  tx  ] [ x ]   [ mxx * x + mxy * y + mxz * z + tx ]
 *      [ y'] = [  myx  myy  myz  ty  ] [ y ] = [ myx * x + myy * y + myz * z + ty ]
 *      [ z']   [  mzx  mzy  mzz  tz  ] [ z ]   [ mzx * x + mzy * y + mzz * z + tz ]
 *                                      [ 1 ]
 * </pre>
 * @since JavaFX 2.0
 */
public class Affine extends Transform {

    /**
     * Tracks atomic changes of more elements.
     */
    AffineAtomicChange atomicChange = new AffineAtomicChange();

    /**
     * This constant is used for the internal state2d variable to indicate
     * that no calculations need to be performed and that the source
     * coordinates only need to be copied to their destinations to
     * complete the transformation equation of this transform.
     * @see #state2d
     */
    private static final int APPLY_IDENTITY = 0;

    /**
     * This constant is used for the internal state2d and state3d variables
     * that the translation components of the matrix need to be added
     * to complete the transformation equation of this transform.
     * @see #state2d
     * @see #state3d
     */
    private static final int APPLY_TRANSLATE = 1;

    /**
     * This constant is used for the internal state2d and state3d variables
     * to indicate that the scaling components of the matrix need
     * to be factored in to complete the transformation equation of
     * this transform. If the APPLY_SHEAR bit is also set then it
     * indicates that the scaling components are 0.0.  If the
     * APPLY_SHEAR bit is not also set then it indicates that the
     * scaling components are not 1.0. If neither the APPLY_SHEAR
     * nor the APPLY_SCALE bits are set then the scaling components
     * are 1.0, which means that the x and y components contribute
     * to the transformed coordinate, but they are not multiplied by
     * any scaling factor.
     * @see #state2d
     * @see #state3d
     */
    private static final int APPLY_SCALE = 2;

    /**
     * This constant is used for the internal state2d variable to indicate
     * that the shearing components of the matrix (mxy and myx) need
     * to be factored in to complete the transformation equation of this
     * transform.  The presence of this bit in the state variable changes
     * the interpretation of the APPLY_SCALE bit as indicated in its
     * documentation.
     * @see #state2d
     */
    private static final int APPLY_SHEAR = 4;

    /**
     * This constant is used for the internal state3d variable to indicate
     * that the matrix represents a 2D-only transform.
     */
    private static final int APPLY_NON_3D = 0;

    /**
     * This constant is used for the internal state3d variable to indicate
     * that the matrix is not in any of the recognized simple states
     * and therefore needs a full usage of all elements to complete
     * the transformation equation of this transform.
     */
    private static final int APPLY_3D_COMPLEX = 4;

    /**
     * If this is a 2D transform, this field keeps track of which components
     * of the matrix need to be applied when performing a transformation.
     * If this is a 3D transform, its state is store in the state3d variable
     * and value of state2d is undefined.
     * @see #APPLY_IDENTITY
     * @see #APPLY_TRANSLATE
     * @see #APPLY_SCALE
     * @see #APPLY_SHEAR
     * @see #state3d
     * @see #updateState()
     */
    private transient int state2d;

    /**
     * This field keeps track of whether or not this transform is 3D and if so
     * it tracks several simple states that can be treated faster. If the state
     * is equal to APPLY_NON_3D, this is a 2D transform with its state stored
     * in the state2d variable. If the state is equal to APPLY_3D_COMPLEX,
     * the matrix is not in any of the simple states and needs to be fully
     * processed. Otherwise we recognize scale (mxx, myy and mzz
     * are not all equal to 1.0), translation (tx, ty and tz are not all
     * equal to 0.0) and their combination. In one of the simple states
     * all of the other elements of the matrix are equal to 0.0 (not even
     * shear is allowed).
     * @see #APPLY_NON_3D
     * @see #APPLY_TRANSLATE
     * @see #APPLY_SCALE
     * @see #APPLY_3D_COMPLEX
     * @see #state2d
     * @see #updateState()
     */
    private transient int state3d;

    // Variables used for the elements until user requests creation
    // of the heavy-weight properties
    private double xx;
    private double xy;
    private double xz;
    private double yx;
    private double yy;
    private double yz;
    private double zx;
    private double zy;
    private double zz;
    private double xt;
    private double yt;
    private double zt;

    /**
     * Creates a new instance of {@code Affine} containing an identity transform.
     */
    public Affine() {
        xx = yy = zz = 1.0;
    }

    /**
     * Creates a new instance of {@code Affine} filled with the values from
     * the specified transform.
     * @param transform transform whose matrix is to be filled to the new
     *        instance
     * @throws NullPointerException if the specified {@code transform} is null
     * @since JavaFX 8.0
     */
    public Affine(Transform transform) {
        this(transform.getMxx(), transform.getMxy(), transform.getMxz(),
                                                             transform.getTx(),
             transform.getMyx(), transform.getMyy(), transform.getMyz(),
                                                             transform.getTy(),
             transform.getMzx(), transform.getMzy(), transform.getMzz(),
                                                             transform.getTz());
    }

    /**
     * Creates a new instance of {@code Affine} with a 2D transform specified
     * by the element values.
     * @param mxx the X coordinate scaling element
     * @param mxy the XY coordinate element
     * @param tx the X coordinate translation element
     * @param myx the YX coordinate element
     * @param myy the Y coordinate scaling element
     * @param ty the Y coordinate translation element
     * @since JavaFX 8.0
     */
    public Affine(double mxx, double mxy, double tx,
                  double myx, double myy, double ty) {
        xx = mxx;
        xy = mxy;
        xt = tx;

        yx = myx;
        yy = myy;
        yt = ty;

        zz = 1.0;

        updateState2D();
    }

    /**
     * Creates a new instance of {@code Affine} with a transform specified
     * by the element values.
     * @param mxx the X coordinate scaling element
     * @param mxy the XY coordinate element
     * @param mxz the XZ coordinate element
     * @param tx the X coordinate translation element
     * @param myx the YX coordinate element
     * @param myy the Y coordinate scaling element
     * @param myz the YZ coordinate element
     * @param ty the Y coordinate translation element
     * @param mzx the ZX coordinate element
     * @param mzy the ZY coordinate element
     * @param mzz the Z coordinate scaling element
     * @param tz the Z coordinate translation element
     * @since JavaFX 8.0
     */
    public Affine(double mxx, double mxy, double mxz, double tx,
                  double myx, double myy, double myz, double ty,
                  double mzx, double mzy, double mzz, double tz) {
        xx = mxx;
        xy = mxy;
        xz = mxz;
        xt = tx;

        yx = myx;
        yy = myy;
        yz = myz;
        yt = ty;

        zx = mzx;
        zy = mzy;
        zz = mzz;
        zt = tz;

        updateState();
    }

    /**
     * Creates a new instance of {@code Affine} with a transformation matrix
     * specified by an array.
     * @param matrix array containing the flattened transformation matrix
     * @param type type of matrix contained in the array
     * @param offset offset of the first element in the array
     * @throws IndexOutOfBoundsException if the array is too short for
     *         the specified {@code type} and {@code offset}
     * @throws IllegalArgumentException if the specified matrix is not affine
     *         (the last line of a 2D 3x3 matrix is not {@code [0, 0, 1]} or
     *          the last line of a 3D 4x4 matrix is not {@code [0, 0, 0, 1]}.
     * @throws NullPointerException if the specified {@code matrix}
     *         or {@code type} is null
     * @since JavaFX 8.0
     */
    public Affine(double[] matrix, MatrixType type, int offset) {
        if (matrix.length < offset + type.elements()) {
            throw new IndexOutOfBoundsException("The array is too short.");
        }

        switch(type) {
            default:
                stateError();
                // cannot reach
            case MT_2D_3x3:
                if (matrix[offset + 6] != 0.0 ||
                        matrix[offset + 7] != 0.0 ||
                        matrix[offset + 8] != 1.0) {
                    throw new IllegalArgumentException("The matrix is "
                            + "not affine");
                }
                // fall-through
            case MT_2D_2x3:
                xx = matrix[offset++];
                xy = matrix[offset++];
                xt = matrix[offset++];
                yx = matrix[offset++];
                yy = matrix[offset++];
                yt = matrix[offset];
                zz = 1.0;
                updateState2D();
                return;
            case MT_3D_4x4:
                if (matrix[offset + 12] != 0.0 ||
                        matrix[offset + 13] != 0.0 ||
                        matrix[offset + 14] != 0.0 ||
                        matrix[offset + 15] != 1.0) {
                    throw new IllegalArgumentException("The matrix is "
                            + "not affine");
                }
                // fall-through
            case MT_3D_3x4:
                xx = matrix[offset++];
                xy = matrix[offset++];
                xz = matrix[offset++];
                xt = matrix[offset++];
                yx = matrix[offset++];
                yy = matrix[offset++];
                yz = matrix[offset++];
                yt = matrix[offset++];
                zx = matrix[offset++];
                zy = matrix[offset++];
                zz = matrix[offset++];
                zt = matrix[offset];
                updateState();
                return;
        }
    }

    /**
     * Defines the X coordinate scaling element of the 3x4 matrix.
     */
    private AffineElementProperty mxx;


    public final void setMxx(double value) {
        if (mxx == null) {
            if (xx != value) {
                xx = value;
                postProcessChange();
            }
        } else {
            mxxProperty().set(value);
        }
    }

    @Override
    public final double getMxx() {
        return mxx == null ? xx : mxx.get();
    }

    public final DoubleProperty mxxProperty() {
        if (mxx == null) {
            mxx = new AffineElementProperty(xx) {
                @Override
                public Object getBean() {
                    return Affine.this;
                }

                @Override
                public String getName() {
                    return "mxx";
                }
            };
        }
        return mxx;
    }

    /**
     * Defines the XY coordinate element of the 3x4 matrix.
     */
    private AffineElementProperty mxy;


    public final void setMxy(double value) {
        if (mxy == null) {
            if (xy != value) {
                xy = value;
                postProcessChange();
            }
        } else {
            mxyProperty().set(value);
        }
    }

    @Override
    public final double getMxy() {
        return mxy == null ? xy : mxy.get();
    }

    public final DoubleProperty mxyProperty() {
        if (mxy == null) {
            mxy = new AffineElementProperty(xy) {
                @Override
                public Object getBean() {
                    return Affine.this;
                }

                @Override
                public String getName() {
                    return "mxy";
                }
            };
        }
        return mxy;
    }

    /**
     * Defines the XZ coordinate element of the 3x4 matrix.
     */
    private AffineElementProperty mxz;


    public final void setMxz(double value) {
        if (mxz == null) {
            if (xz != value) {
                xz = value;
                postProcessChange();
            }
        } else {
            mxzProperty().set(value);
        }
    }

    @Override
    public final double getMxz() {
        return mxz == null ? xz : mxz.get();
    }

    public final DoubleProperty mxzProperty() {
        if (mxz == null) {
            mxz = new AffineElementProperty(xz) {
                @Override
                public Object getBean() {
                    return Affine.this;
                }

                @Override
                public String getName() {
                    return "mxz";
                }
            };
        }
        return mxz;
    }

    /**
     * Defines the X coordinate translation element of the 3x4 matrix.
     */
    private AffineElementProperty tx;


    public final void setTx(double value) {
        if (tx == null) {
            if (xt != value) {
                xt = value;
                postProcessChange();
            }
        } else {
            txProperty().set(value);
        }
    }

    @Override
    public final double getTx() {
        return tx == null ? xt : tx.get();
    }

    public final DoubleProperty txProperty() {
        if (tx == null) {
            tx = new AffineElementProperty(xt) {
                @Override
                public Object getBean() {
                    return Affine.this;
                }

                @Override
                public String getName() {
                    return "tx";
                }
            };
        }
        return tx;
    }

    /**
     * Defines the YX coordinate element of the 3x4 matrix.
     */
    private AffineElementProperty myx;


    public final void setMyx(double value) {
        if (myx == null) {
            if (yx != value) {
                yx = value;
                postProcessChange();
            }
        } else {
            myxProperty().set(value);
        }
    }

    @Override
    public final double getMyx() {
        return myx == null ? yx : myx.get();
    }

    public final DoubleProperty myxProperty() {
        if (myx == null) {
            myx = new AffineElementProperty(yx) {
                @Override
                public Object getBean() {
                    return Affine.this;
                }

                @Override
                public String getName() {
                    return "myx";
                }
            };
        }
        return myx;
    }

    /**
     * Defines the Y coordinate scaling element of the 3x4 matrix.
     */
    private AffineElementProperty myy;


    public final void setMyy(double value) {
        if (myy == null) {
            if (yy != value) {
                yy = value;
                postProcessChange();
            }
        } else{
            myyProperty().set(value);
        }
    }

    @Override
    public final double getMyy() {
        return myy == null ? yy : myy.get();
    }

    public final DoubleProperty myyProperty() {
        if (myy == null) {
            myy = new AffineElementProperty(yy) {
                @Override
                public Object getBean() {
                    return Affine.this;
                }

                @Override
                public String getName() {
                    return "myy";
                }
            };
        }
        return myy;
    }

    /**
     * Defines the YZ coordinate element of the 3x4 matrix.
     */
    private AffineElementProperty myz;


    public final void setMyz(double value) {
        if (myz == null) {
            if (yz != value) {
                yz = value;
                postProcessChange();
            }
        } else {
            myzProperty().set(value);
        }
    }

    @Override
    public final double getMyz() {
        return myz == null ? yz : myz.get();
    }

    public final DoubleProperty myzProperty() {
        if (myz == null) {
            myz = new AffineElementProperty(yz) {
                @Override
                public Object getBean() {
                    return Affine.this;
                }

                @Override
                public String getName() {
                    return "myz";
                }
            };
        }
        return myz;
    }

    /**
     * Defines the Y coordinate translation element of the 3x4 matrix.
     */
    private AffineElementProperty ty;


    public final void setTy(double value) {
        if (ty == null) {
            if (yt != value) {
                yt = value;
                postProcessChange();
            }
        } else {
            tyProperty().set(value);
        }
    }

    @Override
    public final double getTy() {
        return ty == null ? yt : ty.get();
    }

    public final DoubleProperty tyProperty() {
        if (ty == null) {
            ty = new AffineElementProperty(yt) {
                @Override
                public Object getBean() {
                    return Affine.this;
                }

                @Override
                public String getName() {
                    return "ty";
                }
            };
        }
        return ty;
    }

    /**
     * Defines the ZX coordinate element of the 3x4 matrix.
     */
    private AffineElementProperty mzx;


    public final void setMzx(double value) {
        if (mzx == null) {
            if (zx != value) {
                zx = value;
                postProcessChange();
            }
        } else {
            mzxProperty().set(value);
        }
    }

    @Override
    public final double getMzx() {
        return mzx == null ? zx : mzx.get();
    }

    public final DoubleProperty mzxProperty() {
        if (mzx == null) {
            mzx = new AffineElementProperty(zx) {
                @Override
                public Object getBean() {
                    return Affine.this;
                }

                @Override
                public String getName() {
                    return "mzx";
                }
            };
        }
        return mzx;
    }

    /**
     * Defines the ZY coordinate element of the 3x4 matrix.
     */
    private AffineElementProperty mzy;


    public final void setMzy(double value) {
        if (mzy == null) {
            if (zy != value) {
                zy = value;
                postProcessChange();
            }
        } else {
            mzyProperty().set(value);
        }
    }

    @Override
    public final double getMzy() {
        return mzy == null ? zy : mzy.get();
    }

    public final DoubleProperty mzyProperty() {
        if (mzy == null) {
            mzy = new AffineElementProperty(zy) {
                @Override
                public Object getBean() {
                    return Affine.this;
                }

                @Override
                public String getName() {
                    return "mzy";
                }
            };
        }
        return mzy;
    }

    /**
     * Defines the Z coordinate scaling element of the 3x4 matrix.
     */
    private AffineElementProperty mzz;


    public final void setMzz(double value) {
        if (mzz == null) {
            if (zz != value) {
                zz = value;
                postProcessChange();
            }
        } else {
            mzzProperty().set(value);
        }
    }

    @Override
    public final double getMzz() {
        return mzz == null ? zz : mzz.get();
    }

    public final DoubleProperty mzzProperty() {
        if (mzz == null) {
            mzz = new AffineElementProperty(zz) {
                @Override
                public Object getBean() {
                    return Affine.this;
                }

                @Override
                public String getName() {
                    return "mzz";
                }
            };
        }
        return mzz;
    }

    /**
     * Defines the Z coordinate translation element of the 3x4 matrix.
     */
    private AffineElementProperty tz;


    public final void setTz(double value) {
        if (tz == null) {
            if (zt != value) {
                zt = value;
                postProcessChange();
            }
        } else {
            tzProperty().set(value);
        }
    }

    @Override
    public final double getTz() {
        return tz == null ? zt : tz.get();
    }

    public final DoubleProperty tzProperty() {
        if (tz == null) {
            tz = new AffineElementProperty(zt) {
                @Override
                public Object getBean() {
                    return Affine.this;
                }

                @Override
                public String getName() {
                    return "tz";
                }
            };
        }
        return tz;
    }

    /**
     * Sets the specified element of the transformation matrix.
     * @param type type of matrix to work with
     * @param row zero-based row number
     * @param column zero-based column number
     * @param value new value of the specified transformation matrix element
     * @throws IndexOutOfBoundsException if the indices are not within
     *         the specified matrix type
     * @throws IllegalArgumentException if setting the value would break
     *         transform's affinity (for convenience the method allows to set
     *         the elements of the last line of a 2D 3x3 matrix to
     *         {@code [0, 0, 1]} and the elements of the last line
     *         of a 3D 4x4 matrix to {@code [0, 0, 0, 1]}).
     * @throws NullPointerException if the specified {@code type} is null
     * @since JavaFX 8.0
     */
    public void setElement(MatrixType type, int row, int column, double value) {
        if (row < 0 || row >= type.rows() ||
                column < 0 || column >= type.columns()) {
            throw new IndexOutOfBoundsException("Index outside of affine "
                    + "matrix " + type + ": [" + row + ", " + column + "]");
        }
        switch(type) {
            default:
                stateError();
                // cannot reach
            case MT_2D_2x3:
                // fall-through
            case MT_2D_3x3:
                if (!isType2D()) {
                    throw new IllegalArgumentException("Cannot access 2D matrix "
                            + "of a 3D transform");
                }
                switch(row) {
                    case 0:
                        switch(column) {
                            case 0: setMxx(value); return;
                            case 1: setMxy(value); return;
                            case 2: setTx(value); return;
                        }
                    case 1:
                        switch(column) {
                            case 0: setMyx(value); return;
                            case 1: setMyy(value); return;
                            case 2: setTy(value); return;
                        }
                    case 2:
                        switch(column) {
                            case 0: if (value == 0.0) return; else break;
                            case 1: if (value == 0.0) return; else break;
                            case 2: if (value == 1.0) return; else break;
                        }
                }
                break;
            case MT_3D_3x4:
                // fall-through
            case MT_3D_4x4:
                switch(row) {
                    case 0:
                        switch(column) {
                            case 0: setMxx(value); return;
                            case 1: setMxy(value); return;
                            case 2: setMxz(value); return;
                            case 3: setTx(value); return;
                        }
                    case 1:
                        switch(column) {
                            case 0: setMyx(value); return;
                            case 1: setMyy(value); return;
                            case 2: setMyz(value); return;
                            case 3: setTy(value); return;
                        }
                    case 2:
                        switch(column) {
                            case 0: setMzx(value); return;
                            case 1: setMzy(value); return;
                            case 2: setMzz(value); return;
                            case 3: setTz(value); return;
                        }
                    case 3:
                        switch(column) {
                            case 0: if (value == 0.0) return; else break;
                            case 1: if (value == 0.0) return; else break;
                            case 2: if (value == 0.0) return; else break;
                            case 3: if (value == 1.0) return; else break;
                        }
                }
                break;
        }
        // reaches here when last line is set to something else than 0 .. 0 1
        throw new IllegalArgumentException("Cannot set affine matrix " + type +
                " element " + "[" + row + ", " + column + "] to " + value);
    }

    /**
     * Affine element property which handles the atomic changes of more
     * properties.
     */
    private class AffineElementProperty extends SimpleDoubleProperty {

        private boolean needsValueChangedEvent = false;
        private double oldValue;

        public AffineElementProperty(double initialValue) {
            super(initialValue);
        }

        @Override
        public void invalidated() {
            // if an atomic change runs, postpone the change notifications
            if (!atomicChange.runs()) {
                updateState();
                transformChanged();
            }
        }

        @Override
        protected void fireValueChangedEvent() {
            // if an atomic change runs, postpone the change notifications
            if (!atomicChange.runs()) {
                super.fireValueChangedEvent();
            } else {
                needsValueChangedEvent = true;
            }
        }

        /**
         * Called before an atomic change
         */
        private void preProcessAtomicChange() {
            // remember the value before an atomic change
            oldValue = get();
        }

        /**
         * Called after an atomic change
         */
        private void postProcessAtomicChange() {
            // if there was a change notification  during the atomic change,
            // fire it now
            if (needsValueChangedEvent) {
                needsValueChangedEvent = false;
                // the value might have change back an forth
                // (happens quite commonly for transforms with pivot)
                if (oldValue != get()) {
                    super.fireValueChangedEvent();
                }
            }
        }
    }

    /**
     * Called by each element property after value change to
     * update the state variables and call transform change notifications.
     * If an atomic change runs, this is a NOP and the work is done
     * in the end of the entire atomic change.
     */
    private void postProcessChange() {
        if (!atomicChange.runs()) {
            updateState();
            transformChanged();
        }
    }

    /* *************************************************************************
     *                                                                         *
     *                           State getters                                 *
     *                                                                         *
     **************************************************************************/

    @Override
    boolean computeIs2D() {
        return (state3d == APPLY_NON_3D);
    }

    @Override
    boolean computeIsIdentity() {
        return state3d == APPLY_NON_3D && state2d == APPLY_IDENTITY;
    }

    @Override
    public double determinant() {
        if (state3d == APPLY_NON_3D) {
            return getDeterminant2D();
        } else {
            return getDeterminant3D();
        }
    }

    /**
     * 2D implementation of {@code determinant()}.
     * The behavior is undefined if this is a 3D transform.
     * @return determinant
     */
    private double getDeterminant2D() {
        // assert(state3d == APPLY_NON_3D)
        switch (state2d) {
            default:
                stateError();
                // cannot reach
            case APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE:
            case APPLY_SHEAR | APPLY_SCALE:
                return getMxx() * getMyy() - getMxy() * getMyx();
            case APPLY_SHEAR | APPLY_TRANSLATE:
            case APPLY_SHEAR:
                return -(getMxy() * getMyx());
            case APPLY_SCALE | APPLY_TRANSLATE:
            case APPLY_SCALE:
                return getMxx() * getMyy();
            case APPLY_TRANSLATE:
            case APPLY_IDENTITY:
                return 1.0;
        }
    }

    /**
     * 3D implementation of {@code determinant()}.
     * The behavior is undefined if this is a 2D transform.
     * @return determinant
     */
    private double getDeterminant3D() {
        // assert(state3d != APPLY_NON_3D)
        switch(state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_TRANSLATE:
                return 1.0;
            case APPLY_SCALE:
            case APPLY_SCALE | APPLY_TRANSLATE:
                return getMxx() * getMyy() * getMzz();
            case APPLY_3D_COMPLEX:
                final double myx = getMyx();
                final double myy = getMyy();
                final double myz = getMyz();
                final double mzx = getMzx();
                final double mzy = getMzy();
                final double mzz = getMzz();

                return (getMxx() * (myy * mzz - mzy * myz) +
                        getMxy() * (myz * mzx - mzz * myx) +
                        getMxz() * (myx * mzy - mzx * myy));
        }
    }

    /* *************************************************************************
     *                                                                         *
     *                         Transform creators                              *
     *                                                                         *
     **************************************************************************/

    @Override
    public Transform createConcatenation(Transform transform) {
        Affine a = clone();
        a.append(transform);
        return a;
    }

    @Override
    public Affine createInverse() throws NonInvertibleTransformException {
        Affine t = clone();
        t.invert();
        return t;
    }

    @Override
    public Affine clone() {
        return new Affine(this);
    }

    /* *************************************************************************
     *                                                                         *
     *                           Matrix setters                                *
     *                                                                         *
     **************************************************************************/

    /**
     * Sets the values of this instance to the values provided by the specified
     * transform.
     * @param transform transform whose matrix is to be filled to this instance
     * @throws NullPointerException if the specified {@code transform} is null
     * @since JavaFX 8.0
     */
    public void setToTransform(Transform transform) {
        setToTransform(
                transform.getMxx(), transform.getMxy(),
                                         transform.getMxz(), transform.getTx(),
                transform.getMyx(), transform.getMyy(),
                                         transform.getMyz(), transform.getTy(),
                transform.getMzx(), transform.getMzy(),
                                         transform.getMzz(), transform.getTz());
    }

    /**
     * Sets the values of this instance to the 2D transform specified
     * by the element values.
     * @param mxx the X coordinate scaling element
     * @param mxy the XY coordinate element
     * @param tx the X coordinate translation element
     * @param myx the YX coordinate element
     * @param myy the Y coordinate scaling element
     * @param ty the Y coordinate translation element
     * @since JavaFX 8.0
     */
    public void setToTransform(double mxx, double mxy, double tx,
                               double myx, double myy, double ty) {
        setToTransform(mxx, mxy, 0.0, tx,
                myx, myy, 0.0, ty,
                0.0, 0.0, 1.0, 0.0);
    }

    /**
     * Sets the values of this instance to the transform specified
     * by the element values.
     * @param mxx the X coordinate scaling element
     * @param mxy the XY coordinate element
     * @param mxz the XZ coordinate element
     * @param tx the X coordinate translation element
     * @param myx the YX coordinate element
     * @param myy the Y coordinate scaling element
     * @param myz the YZ coordinate element
     * @param ty the Y coordinate translation element
     * @param mzx the ZX coordinate element
     * @param mzy the ZY coordinate element
     * @param mzz the Z coordinate scaling element
     * @param tz the Z coordinate translation element
     * @since JavaFX 8.0
     */
    public void setToTransform(double mxx, double mxy, double mxz, double tx,
                               double myx, double myy, double myz, double ty,
                               double mzx, double mzy, double mzz, double tz)
    {
        atomicChange.start();

        setMxx(mxx);
        setMxy(mxy);
        setMxz(mxz);
        setTx(tx);

        setMyx(myx);
        setMyy(myy);
        setMyz(myz);
        setTy(ty);

        setMzx(mzx);
        setMzy(mzy);
        setMzz(mzz);
        setTz(tz);

        updateState();
        atomicChange.end();
    }

    /**
     * Sets the values of this instance to the transformation matrix
     * specified by an array.
     * @param matrix array containing the flattened transformation matrix
     * @param type type of matrix contained in the array
     * @param offset offset of the first element in the array
     * @throws IndexOutOfBoundsException if the array is too short for
     *         the specified {@code type} and {@code offset}
     * @throws IllegalArgumentException if the specified matrix is not affine
     *         (the last line of a 2D 3x3 matrix is not {@code [0, 0, 1]} or
     *          the last line of a 3D 4x4 matrix is not {@code [0, 0, 0, 1]}.
     * @throws NullPointerException if the specified {@code matrix}
     *         or {@code type} is null
     * @since JavaFX 8.0
     */
    public void setToTransform(double[] matrix, MatrixType type, int offset) {
        if (matrix.length < offset + type.elements()) {
            throw new IndexOutOfBoundsException("The array is too short.");
        }

        switch(type) {
            default:
                stateError();
                // cannot reach
            case MT_2D_3x3:
                if (matrix[offset + 6] != 0.0 ||
                        matrix[offset + 7] != 0.0 ||
                        matrix[offset + 8] != 1.0) {
                    throw new IllegalArgumentException("The matrix is "
                            + "not affine");
                }
                // fall-through
            case MT_2D_2x3:
                setToTransform(matrix[offset++], matrix[offset++],
                        matrix[offset++], matrix[offset++],
                        matrix[offset++], matrix[offset++]);
                return;
            case MT_3D_4x4:
                if (matrix[offset + 12] != 0.0 ||
                        matrix[offset + 13] != 0.0 ||
                        matrix[offset + 14] != 0.0 ||
                        matrix[offset + 15] != 1.0) {
                    throw new IllegalArgumentException("The matrix is "
                            + "not affine");
                }
                // fall-through
            case MT_3D_3x4:
                setToTransform(matrix[offset++], matrix[offset++],
                        matrix[offset++], matrix[offset++], matrix[offset++],
                        matrix[offset++], matrix[offset++], matrix[offset++],
                        matrix[offset++], matrix[offset++], matrix[offset++],
                        matrix[offset++]);
                return;
        }
    }

    /**
     * Resets this transform to the identity transform.
     * @since JavaFX 8.0
     */
    public void setToIdentity() {
        atomicChange.start();

        if (state3d != APPLY_NON_3D) {
            setMxx(1.0); setMxy(0.0); setMxz(0.0); setTx(0.0);
            setMyx(0.0); setMyy(1.0); setMyz(0.0); setTy(0.0);
            setMzx(0.0); setMzy(0.0); setMzz(1.0); setTz(0.0);
            state3d = APPLY_NON_3D;
            state2d = APPLY_IDENTITY;
        } else if (state2d != APPLY_IDENTITY) {
            setMxx(1.0); setMxy(0.0); setTx(0.0);
            setMyx(0.0); setMyy(1.0); setTy(0.0);
            state2d = APPLY_IDENTITY;
        }

        atomicChange.end();
    }

    /* *************************************************************************
     *                                                                         *
     *                           Matrix operations                             *
     *                                                                         *
     **************************************************************************/


               /* *************************************************
                *                   Inversion                     *
                **************************************************/

    /**
     * Inverts this transform in place.
     * @throws NonInvertibleTransformException if this transform
     *         cannot be inverted
     * @since JavaFX 8.0
     */
    public void invert() throws NonInvertibleTransformException {
        atomicChange.start();

        if (state3d == APPLY_NON_3D) {
            invert2D();
            updateState2D();
        } else {
            invert3D();
            updateState();
        }

        atomicChange.end();
    }

    /**
     * 2D implementation of {@code invert()}.
     * The behavior is undefined for a 3D transform.
     */
    private void invert2D() throws NonInvertibleTransformException {
        double Mxx, Mxy, Mxt;
        double Myx, Myy, Myt;
        double det;
        // assert(state3d == APPLY_NON_3D)

        switch (state2d) {
            default:
                stateError();
                // cannot reach
            case APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE:
                Mxx = getMxx(); Mxy = getMxy(); Mxt = getTx();
                Myx = getMyx(); Myy = getMyy(); Myt = getTy();
                det = getDeterminant2D();
                if (det == 0.0) {
                    atomicChange.cancel();
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                setMxx(Myy / det);
                setMyx(-Myx / det);
                setMxy(-Mxy / det);
                setMyy(Mxx / det);
                setTx((Mxy * Myt - Myy * Mxt) / det);
                setTy((Myx * Mxt - Mxx * Myt) / det);
                return;
            case APPLY_SHEAR | APPLY_SCALE:
                Mxx = getMxx(); Mxy = getMxy();
                Myx = getMyx(); Myy = getMyy();
                det = getDeterminant2D();
                if (det == 0.0) {
                    atomicChange.cancel();
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                setMxx(Myy / det);
                setMyx(-Myx / det);
                setMxy(-Mxy / det);
                setMyy(Mxx / det);
                return;
            case APPLY_SHEAR | APPLY_TRANSLATE:
                Mxy = getMxy(); Mxt = getTx();
                Myx = getMyx(); Myt = getTy();
                if (Mxy == 0.0 || Myx == 0.0) {
                    atomicChange.cancel();
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                setMyx(1.0 / Mxy);
                setMxy(1.0 / Myx);
                setTx(-Myt / Myx);
                setTy(-Mxt / Mxy);
                return;
            case APPLY_SHEAR:
                Mxy = getMxy();
                Myx = getMyx();
                if (Mxy == 0.0 || Myx == 0.0) {
                    atomicChange.cancel();
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                setMyx(1.0 / Mxy);
                setMxy(1.0 / Myx);
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
                Mxx = getMxx(); Mxt = getTx();
                Myy = getMyy(); Myt = getTy();
                if (Mxx == 0.0 || Myy == 0.0) {
                    atomicChange.cancel();
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                setMxx(1.0 / Mxx);
                setMyy(1.0 / Myy);
                setTx(-Mxt / Mxx);
                setTy(-Myt / Myy);
                return;
            case APPLY_SCALE:
                Mxx = getMxx();
                Myy = getMyy();
                if (Mxx == 0.0 || Myy == 0.0) {
                    atomicChange.cancel();
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                setMxx(1.0 / Mxx);
                setMyy(1.0 / Myy);
                return;
            case APPLY_TRANSLATE:
                setTx(-getTx());
                setTy(-getTy());
                return;
            case APPLY_IDENTITY:
                return;
        }
    }

    /**
     * 3D implementation of {@code invert()}.
     * The behavior is undefined if this is a 2D transform.
     */
    private void invert3D() throws NonInvertibleTransformException {

        switch(state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_TRANSLATE:
                setTx(-getTx());
                setTy(-getTy());
                setTz(-getTz());
                return;
            case APPLY_SCALE:
                final double mxx_s = getMxx();
                final double myy_s = getMyy();
                final double mzz_s = getMzz();
                if (mxx_s == 0.0 || myy_s == 0.0 || mzz_s == 0.0) {
                    atomicChange.cancel();
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                setMxx(1.0 / mxx_s);
                setMyy(1.0 / myy_s);
                setMzz(1.0 / mzz_s);
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
                final double mxx_st = getMxx();
                final double tx_st = getTx();
                final double myy_st = getMyy();
                final double ty_st = getTy();
                final double mzz_st = getMzz();
                final double tz_st = getTz();
                if (mxx_st == 0.0 || myy_st == 0.0 || mzz_st == 0.0) {
                    atomicChange.cancel();
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                setMxx(1.0 / mxx_st);
                setMyy(1.0 / myy_st);
                setMzz(1.0 / mzz_st);
                setTx(-tx_st / mxx_st);
                setTy(-ty_st / myy_st);
                setTz(-tz_st / mzz_st);
                return;
            case APPLY_3D_COMPLEX:

                // InvM = Transpose(Cofactor(M)) / det(M)
                // Cofactor(M) = matrix of cofactors(0..3,0..3)
                // cofactor(r,c) = (-1 if r+c is odd) * minor(r,c)
                // minor(r,c) = det(M with row r and col c removed)
                // For an Affine3D matrix, minor(r, 3) is {0, 0, 0, det}
                // which generates {0, 0, 0, 1} and so can be ignored.

                final double mxx = getMxx();
                final double mxy = getMxy();
                final double mxz = getMxz();
                final double tx = getTx();
                final double myx = getMyx();
                final double myy = getMyy();
                final double myz = getMyz();
                final double ty = getTy();
                final double mzy = getMzy();
                final double mzx = getMzx();
                final double mzz = getMzz();
                final double tz = getTz();

                final double det =
                        mxx * (myy * mzz - mzy * myz) +
                        mxy * (myz * mzx - mzz * myx) +
                        mxz * (myx * mzy - mzx * myy);

                if (det == 0.0) {
                    atomicChange.cancel();
                    throw new NonInvertibleTransformException("Determinant is 0");
                }

                final double cxx =   myy * mzz - myz * mzy;
                final double cyx = - myx * mzz + myz * mzx;
                final double czx =   myx * mzy - myy * mzx;
                final double cxt = - mxy * (myz * tz - mzz  * ty)
                                   - mxz * (ty  * mzy - tz  * myy)
                                   - tx  * (myy * mzz - mzy * myz);
                final double cxy = - mxy * mzz + mxz * mzy;
                final double cyy =   mxx * mzz - mxz * mzx;
                final double czy = - mxx * mzy + mxy * mzx;
                final double cyt =   mxx * (myz * tz  - mzz * ty)
                                   + mxz * (ty  * mzx - tz  * myx)
                                   + tx  * (myx * mzz - mzx * myz);
                final double cxz =   mxy * myz - mxz * myy;
                final double cyz = - mxx * myz + mxz * myx;
                final double czz =   mxx * myy - mxy * myx;
                final double czt = - mxx * (myy * tz - mzy  * ty)
                                   - mxy * (ty  * mzx - tz  * myx)
                                   - tx  * (myx * mzy - mzx * myy);

                setMxx(cxx / det);
                setMxy(cxy / det);
                setMxz(cxz / det);
                setTx(cxt / det);
                setMyx(cyx / det);
                setMyy(cyy / det);
                setMyz(cyz / det);
                setTy(cyt / det);
                setMzx(czx / det);
                setMzy(czy / det);
                setMzz(czz / det);
                setTz(czt / det);
                return;
        }
    }

               /* *************************************************
                *             General concatenations              *
                **************************************************/

    /**
     * <p>
     * Appends the specified transform to this instance.
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding the two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * {@code transform} second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified transform.
     * </p>
     *
     * @param transform transform to be appended to this instance
     * @throws NullPointerException if the specified {@code transform} is null
     * @since JavaFX 8.0
     */
    public void append(Transform transform) {
        transform.appendTo(this);
    }

    /**
     * <p>
     * Appends the 2D transform specified by the element values to this instance.
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding the two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * {@code transform} second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified transform.
     * </p>
     * @param mxx the X coordinate scaling element of the transform to be
     *            appended
     * @param mxy the XY coordinate element of the transform to be appended
     * @param tx the X coordinate translation element of the transform to be
     *            appended
     * @param myx the YX coordinate element of the transform to be appended
     * @param myy the Y coordinate scaling element of the transform to be
     *            appended
     * @param ty the Y coordinate translation element of the transform to be
     *            appended
     * @since JavaFX 8.0
     */
    public void append(double mxx, double mxy, double tx,
                        double myx, double myy, double ty) {

        if (state3d == APPLY_NON_3D) {

            atomicChange.start();

            final double m_xx = getMxx();
            final double m_xy = getMxy();
            final double m_yx = getMyx();
            final double m_yy = getMyy();

            setMxx(m_xx * mxx + m_xy * myx);
            setMxy(m_xx * mxy + m_xy * myy);
            setTx(m_xx * tx + m_xy * ty + getTx());
            setMyx(m_yx * mxx + m_yy * myx);
            setMyy(m_yx * mxy + m_yy * myy);
            setTy(m_yx * tx + m_yy * ty + getTy());

            updateState();
            atomicChange.end();
        } else {
            append(mxx, mxy, 0.0, tx,
                    myx, myy, 0.0, ty,
                    0.0, 0.0, 1.0, 0.0);
        }
    }

    /**
     * <p>
     * Appends the transform specified by the element values to this instance.
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding the two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * {@code transform} second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified transform.
     * </p>
     *
     * @param mxx the X coordinate scaling element of the transform to be
     *            appended
     * @param mxy the XY coordinate element of the transform to be appended
     * @param mxz the XZ coordinate element of the transform to be appended
     * @param tx the X coordinate translation element of the transform to be
     *            appended
     * @param myx the YX coordinate element of the transform to be appended
     * @param myy the Y coordinate scaling element of the transform to be
     *            appended
     * @param myz the YZ coordinate element of the transform to be appended
     * @param ty the Y coordinate translation element of the transform to be
     *            appended
     * @param mzx the ZX coordinate element of the transform to be appended
     * @param mzy the ZY coordinate element of the transform to be appended
     * @param mzz the Z coordinate scaling element of the transform to be
     *            appended
     * @param tz the Z coordinate translation element of the transform to be
     *            appended
     * @since JavaFX 8.0
     */
    public void append(double mxx, double mxy, double mxz, double tx,
                       double myx, double myy, double myz, double ty,
                       double mzx, double mzy, double mzz, double tz)
    {
        atomicChange.start();

        final double m_xx = getMxx();
        final double m_xy = getMxy();
        final double m_xz = getMxz();
        final double t_x = getTx();
        final double m_yx = getMyx();
        final double m_yy = getMyy();
        final double m_yz = getMyz();
        final double t_y = getTy();
        final double m_zx = getMzx();
        final double m_zy = getMzy();
        final double m_zz = getMzz();
        final double t_z = getTz();

        setMxx(m_xx * mxx + m_xy * myx + m_xz * mzx);
        setMxy(m_xx * mxy + m_xy * myy + m_xz * mzy);
        setMxz(m_xx * mxz + m_xy * myz + m_xz * mzz);
        setTx( m_xx * tx + m_xy * ty + m_xz * tz + t_x);
        setMyx(m_yx * mxx + m_yy * myx + m_yz * mzx);
        setMyy(m_yx * mxy + m_yy * myy + m_yz * mzy);
        setMyz(m_yx * mxz + m_yy * myz + m_yz * mzz);
        setTy( m_yx * tx + m_yy * ty + m_yz * tz + t_y);
        setMzx(m_zx * mxx + m_zy * myx + m_zz * mzx);
        setMzy(m_zx * mxy + m_zy * myy + m_zz * mzy);
        setMzz(m_zx * mxz + m_zy * myz + m_zz * mzz);
        setTz( m_zx * tx + m_zy * ty + m_zz * tz + t_z);

        updateState();
        atomicChange.end();
    }

    /**
     * <p>
     * Appends the transform specified by the array to this instance.
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding the two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * {@code transform} second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified transform.
     * </p>
     *
     * @param matrix array containing the flattened transformation matrix
     *               to be appended
     * @param type type of matrix contained in the array
     * @param offset offset of the first matrix element in the array
     * @throws IndexOutOfBoundsException if the array is too short for
     *         the specified {@code type} and {@code offset}
     * @throws IllegalArgumentException if the specified matrix is not affine
     *         (the last line of a 2D 3x3 matrix is not {@code [0, 0, 1]} or
     *          the last line of a 3D 4x4 matrix is not {@code [0, 0, 0, 1]}.
     * @throws NullPointerException if the specified {@code matrix}
     *         or {@code type} is null
     * @since JavaFX 8.0
     */
    public void append(double[] matrix, MatrixType type, int offset) {
        if (matrix.length < offset + type.elements()) {
            throw new IndexOutOfBoundsException("The array is too short.");
        }

        switch(type) {
            default:
                stateError();
                // cannot reach
            case MT_2D_3x3:
                if (matrix[offset + 6] != 0.0 ||
                        matrix[offset + 7] != 0.0 ||
                        matrix[offset + 8] != 1.0) {
                    throw new IllegalArgumentException("The matrix is "
                            + "not affine");
                }
                // fall-through
            case MT_2D_2x3:
                append(matrix[offset++], matrix[offset++],
                        matrix[offset++], matrix[offset++],
                        matrix[offset++], matrix[offset++]);
                return;
            case MT_3D_4x4:
                if (matrix[offset + 12] != 0.0 ||
                        matrix[offset + 13] != 0.0 ||
                        matrix[offset + 14] != 0.0 ||
                        matrix[offset + 15] != 1.0) {
                    throw new IllegalArgumentException("The matrix is "
                            + "not affine");
                }
                // fall-through
            case MT_3D_3x4:
                append(matrix[offset++], matrix[offset++], matrix[offset++],
                        matrix[offset++], matrix[offset++], matrix[offset++],
                        matrix[offset++], matrix[offset++], matrix[offset++],
                        matrix[offset++], matrix[offset++], matrix[offset++]);
                return;
        }
    }

    @Override
    void appendTo(Affine a) {
        switch(state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                switch(state2d) {
                    case APPLY_IDENTITY:
                        return;
                    case APPLY_TRANSLATE:
                        a.appendTranslation(getTx(), getTy());
                        return;
                    case APPLY_SCALE:
                        a.appendScale(getMxx(), getMyy());
                        return;
                    case APPLY_SCALE | APPLY_TRANSLATE:
                        a.appendTranslation(getTx(), getTy());
                        a.appendScale(getMxx(), getMyy());
                        return;
                    default:
                        a.append(getMxx(), getMxy(), getTx(),
                                 getMyx(), getMyy(), getTy());
                        return;
                }
            case APPLY_TRANSLATE:
                a.appendTranslation(getTx(), getTy(), getTz());
                return;
            case APPLY_SCALE:
                a.appendScale(getMxx(), getMyy(), getMzz());
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
                a.appendTranslation(getTx(), getTy(), getTz());
                a.appendScale(getMxx(), getMyy(), getMzz());
                return;
            case APPLY_3D_COMPLEX:
                a.append(getMxx(), getMxy(), getMxz(), getTx(),
                         getMyx(), getMyy(), getMyz(), getTy(),
                         getMzx(), getMzy(), getMzz(), getTz());
                return;
        }
    }

    /**
     * <p>
     * Prepends the specified transform to this instance.
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding the two transforms to its
     * {@code getTransforms()} list, the specified {@code transform} first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified transform.
     * </p>
     *
     * @param transform transform to be prepended to this instance
     * @throws NullPointerException if the specified {@code transform} is null
     * @since JavaFX 8.0
     */
    public void prepend(Transform transform) {
        transform.prependTo(this);
    }

    /**
     * <p>
     * Prepends the 2D transform specified by the element values to this instance.
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding the two transforms to its
     * {@code getTransforms()} list, the specified {@code transform} first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified transform.
     * </p>
     *
     * @param mxx the X coordinate scaling element of the transform to be
     *            prepended
     * @param mxy the XY coordinate element of the transform to be prepended
     * @param tx the X coordinate translation element of the transform to be
     *            prepended
     * @param myx the YX coordinate element of the transform to be prepended
     * @param myy the Y coordinate scaling element of the transform to be
     *            prepended
     * @param ty the Y coordinate translation element of the transform to be
     *            prepended
     * @since JavaFX 8.0
     */
    public void prepend(double mxx, double mxy, double tx,
                        double myx, double myy, double ty) {

        if (state3d == APPLY_NON_3D) {
            atomicChange.start();

            final double m_xx = getMxx();
            final double m_xy = getMxy();
            final double t_x = getTx();
            final double m_yx = getMyx();
            final double m_yy = getMyy();
            final double t_y = getTy();

            setMxx(mxx * m_xx + mxy * m_yx);
            setMxy(mxx * m_xy + mxy * m_yy);
            setTx(mxx * t_x + mxy * t_y + tx);
            setMyx(myx * m_xx + myy * m_yx);
            setMyy(myx * m_xy + myy * m_yy);
            setTy(myx * t_x + myy * t_y + ty);

            updateState2D();
            atomicChange.end();
        } else {
            prepend(mxx, mxy, 0.0, tx,
                   myx, myy, 0.0, ty,
                   0.0, 0.0, 1.0, 0.0);
        }
    }

    /**
     * <p>
     * Prepends the transform specified by the element values to this instance.
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding the two transforms to its
     * {@code getTransforms()} list, the specified {@code transform} first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified transform.
     * </p>
     *
     * @param mxx the X coordinate scaling element of the transform to be
     *            prepended
     * @param mxy the XY coordinate element of the transform to be prepended
     * @param mxz the XZ coordinate element of the transform to be prepended
     * @param tx the X coordinate translation element of the transform to be
     *            prepended
     * @param myx the YX coordinate element of the transform to be prepended
     * @param myy the Y coordinate scaling element of the transform to be
     *            prepended
     * @param myz the YZ coordinate element of the transform to be prepended
     * @param ty the Y coordinate translation element of the transform to be
     *            prepended
     * @param mzx the ZX coordinate element of the transform to be prepended
     * @param mzy the ZY coordinate element of the transform to be prepended
     * @param mzz the Z coordinate scaling element of the transform to be
     *            prepended
     * @param tz the Z coordinate translation element of the transform to be
     *            prepended
     * @since JavaFX 8.0
     */
    public void prepend(double mxx, double mxy, double mxz, double tx,
                        double myx, double myy, double myz, double ty,
                        double mzx, double mzy, double mzz, double tz) {
        atomicChange.start();

        final double m_xx = getMxx();
        final double m_xy = getMxy();
        final double m_xz = getMxz();
        final double t_x = getTx();
        final double m_yx = getMyx();
        final double m_yy = getMyy();
        final double m_yz = getMyz();
        final double t_y = getTy();
        final double m_zx = getMzx();
        final double m_zy = getMzy();
        final double m_zz = getMzz();
        final double t_z = getTz();

        setMxx(mxx * m_xx + mxy * m_yx + mxz * m_zx);
        setMxy(mxx * m_xy + mxy * m_yy + mxz * m_zy);
        setMxz(mxx * m_xz + mxy * m_yz + mxz * m_zz);
        setTx( mxx * t_x + mxy * t_y + mxz * t_z + tx);
        setMyx(myx * m_xx + myy * m_yx + myz * m_zx);
        setMyy(myx * m_xy + myy * m_yy + myz * m_zy);
        setMyz(myx * m_xz + myy * m_yz + myz * m_zz);
        setTy( myx * t_x + myy * t_y + myz * t_z + ty);
        setMzx(mzx * m_xx + mzy * m_yx + mzz * m_zx);
        setMzy(mzx * m_xy + mzy * m_yy + mzz * m_zy);
        setMzz(mzx * m_xz + mzy * m_yz + mzz * m_zz);
        setTz( mzx * t_x + mzy * t_y + mzz * t_z + tz);

        updateState();
        atomicChange.end();
    }

    /**
     * <p>
     * Prepends the transform specified by the array to this instance.
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding the two transforms to its
     * {@code getTransforms()} list, the specified {@code transform} first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified transform.
     * </p>
     *
     * @param matrix array containing the flattened transformation matrix
     *               to be prepended
     * @param type type of matrix contained in the array
     * @param offset offset of the first matrix element in the array
     * @throws IndexOutOfBoundsException if the array is too short for
     *         the specified {@code type} and {@code offset}
     * @throws IllegalArgumentException if the specified matrix is not affine
     *         (the last line of a 2D 3x3 matrix is not {@code [0, 0, 1]} or
     *          the last line of a 3D 4x4 matrix is not {@code [0, 0, 0, 1]}.
     * @throws NullPointerException if the specified {@code matrix}
     *         or {@code type} is null
     * @since JavaFX 8.0
     */
    public void prepend(double[] matrix, MatrixType type, int offset) {
        if (matrix.length < offset + type.elements()) {
            throw new IndexOutOfBoundsException("The array is too short.");
        }

        switch(type) {
            default:
                stateError();
                // cannot reach
            case MT_2D_3x3:
                if (matrix[offset + 6] != 0.0 ||
                        matrix[offset + 7] != 0.0 ||
                        matrix[offset + 8] != 1.0) {
                    throw new IllegalArgumentException("The matrix is "
                            + "not affine");
                }
                // fall-through
            case MT_2D_2x3:
                prepend(matrix[offset++], matrix[offset++],
                        matrix[offset++], matrix[offset++],
                        matrix[offset++], matrix[offset++]);
                return;
            case MT_3D_4x4:
                if (matrix[offset + 12] != 0.0 ||
                        matrix[offset + 13] != 0.0 ||
                        matrix[offset + 14] != 0.0 ||
                        matrix[offset + 15] != 1.0) {
                    throw new IllegalArgumentException("The matrix is "
                            + "not affine");
                }
                // fall-through
            case MT_3D_3x4:
                prepend(matrix[offset++], matrix[offset++], matrix[offset++],
                        matrix[offset++], matrix[offset++], matrix[offset++],
                        matrix[offset++], matrix[offset++], matrix[offset++],
                        matrix[offset++], matrix[offset++], matrix[offset++]);
                return;
        }
    }

    @Override
    void prependTo(Affine a) {
        switch(state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                switch(state2d) {
                    case APPLY_IDENTITY:
                        return;
                    case APPLY_TRANSLATE:
                        a.prependTranslation(getTx(), getTy());
                        return;
                    case APPLY_SCALE:
                        a.prependScale(getMxx(), getMyy());
                        return;
                    case APPLY_SCALE | APPLY_TRANSLATE:
                        a.prependScale(getMxx(), getMyy());
                        a.prependTranslation(getTx(), getTy());
                        return;
                    default:
                        a.prepend(getMxx(), getMxy(), getTx(),
                                  getMyx(), getMyy(), getTy());
                        return;
                }
            case APPLY_TRANSLATE:
                a.prependTranslation(getTx(), getTy(), getTz());
                return;
            case APPLY_SCALE:
                a.prependScale(getMxx(), getMyy(), getMzz());
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
                a.prependScale(getMxx(), getMyy(), getMzz());
                a.prependTranslation(getTx(), getTy(), getTz());
                return;
            case APPLY_3D_COMPLEX:
                a.prepend(getMxx(), getMxy(), getMxz(), getTx(),
                          getMyx(), getMyy(), getMyz(), getTy(),
                          getMzx(), getMzy(), getMzz(), getTz());
                return;
        }
    }


               /* *************************************************
                *                    Translate                    *
                **************************************************/

    /**
     * <p>
     * Appends the 2D translation to this instance.
     * It is equivalent to {@code append(new Translate(tx, ty))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * translation second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified translation.
     * </p>
     * @param tx the X coordinate translation
     * @param ty the Y coordinate translation
     * @since JavaFX 8.0
     */
    public void appendTranslation(double tx, double ty) {
        atomicChange.start();
        translate2D(tx, ty);
        atomicChange.end();
    }

    /**
     * <p>
     * Appends the translation to this instance.
     * It is equivalent to {@code append(new Translate(tx, ty, tz))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * translation second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified translation.
     * </p>
     * @param tx the X coordinate translation
     * @param ty the Y coordinate translation
     * @param tz the Z coordinate translation
     * @since JavaFX 8.0
     */
    public void appendTranslation(double tx, double ty, double tz) {
        atomicChange.start();
        translate3D(tx, ty, tz);
        atomicChange.end();
    }

    /**
     * 2D implementation of {@code appendTranslation()}.
     * If this is a 3D transform, the call is redirected to {@code transalte3D()}.
     */
    private void translate2D(double tx, double ty) {
        if (state3d != APPLY_NON_3D) {
            translate3D(tx, ty, 0.0);
            return;
        }

        switch (state2d) {
            default:
                stateError();
                // cannot reach
            case APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE:
                setTx(tx * getMxx() + ty * getMxy() + getTx());
                setTy(tx * getMyx() + ty * getMyy() + getTy());
                if (getTx() == 0.0 && getTy() == 0.0) {
                    state2d = APPLY_SHEAR | APPLY_SCALE;
                }
                return;
            case APPLY_SHEAR | APPLY_SCALE:
                setTx(tx * getMxx() + ty * getMxy());
                setTy(tx * getMyx() + ty * getMyy());
                if (getTx() != 0.0 || getTy() != 0.0) {
                    state2d = APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE;
                }
                return;
            case APPLY_SHEAR | APPLY_TRANSLATE:
                setTx(ty * getMxy() + getTx());
                setTy(tx * getMyx() + getTy());
                if (getTx() == 0.0 && getTy() == 0.0) {
                    state2d = APPLY_SHEAR;
                }
                return;
            case APPLY_SHEAR:
                setTx(ty * getMxy());
                setTy(tx * getMyx());
                if (getTx() != 0.0 || getTy() != 0.0) {
                    state2d = APPLY_SHEAR | APPLY_TRANSLATE;
                }
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
                setTx(tx * getMxx() + getTx());
                setTy(ty * getMyy() + getTy());
                if (getTx() == 0.0 && getTy() == 0.0) {
                    state2d = APPLY_SCALE;
                }
                return;
            case APPLY_SCALE:
                setTx(tx * getMxx());
                setTy(ty * getMyy());
                if (getTx() != 0.0 || getTy() != 0.0) {
                    state2d = APPLY_SCALE | APPLY_TRANSLATE;
                }
                return;
            case APPLY_TRANSLATE:
                setTx(tx + getTx());
                setTy(ty + getTy());
                if (getTx() == 0.0 && getTy() == 0.0) {
                    state2d = APPLY_IDENTITY;
                }
                return;
            case APPLY_IDENTITY:
                setTx(tx);
                setTy(ty);
                if (tx != 0.0 || ty != 0.0) {
                    state2d = APPLY_TRANSLATE;
                }
                return;
        }
    }

    /**
     * 3D implementation of {@code appendTranslation()}.
     * Works fine if this is a 2D transform.
     */
    private void translate3D(double tx, double ty, double tz) {
        switch(state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                translate2D(tx, ty);
                if (tz != 0.0) {
                    setTz(tz);
                    if ((state2d & APPLY_SHEAR) == 0) {
                        state3d = (state2d & APPLY_SCALE) | APPLY_TRANSLATE;
                    } else {
                        state3d = APPLY_3D_COMPLEX;
                    }
                }
                return;
            case APPLY_TRANSLATE:
                setTx(tx + getTx());
                setTy(ty + getTy());
                setTz(tz + getTz());
                if (getTz() == 0.0) {
                    state3d = APPLY_NON_3D;
                    if (getTx() == 0.0 && getTy() == 0.0) {
                        state2d = APPLY_IDENTITY;
                    } else {
                        state2d = APPLY_TRANSLATE;
                    }
                }
                return;
            case APPLY_SCALE:
                setTx(tx * getMxx());
                setTy(ty * getMyy());
                setTz(tz * getMzz());
                if (getTx() != 0.0 || getTy() != 0.0 || getTz() != 0.0) {
                    state3d |= APPLY_TRANSLATE;
                }
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
                setTx(tx * getMxx() + getTx());
                setTy(ty * getMyy() + getTy());
                setTz(tz * getMzz() + getTz());
                if (getTz() == 0.0) {
                    if (getTx() == 0.0 && getTy() == 0.0) {
                        state3d = APPLY_SCALE;
                    }
                    if (getMzz() == 1.0) {
                        state2d = state3d;
                        state3d = APPLY_NON_3D;
                    }
                }
                return;
            case APPLY_3D_COMPLEX:
                setTx(tx * getMxx() + ty * getMxy() + tz * getMxz() + getTx());
                setTy(tx * getMyx() + ty * getMyy() + tz * getMyz() + getTy());
                setTz(tx * getMzx() + ty * getMzy() + tz * getMzz() + getTz());
                updateState();
                return;
        }
    }

    /**
     * <p>
     * Prepends the translation to this instance.
     * It is equivalent to {@code prepend(new Translate(tx, ty, tz))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified translation first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified translation.
     * </p>
     * @param tx the X coordinate translation
     * @param ty the Y coordinate translation
     * @param tz the Z coordinate translation
     * @since JavaFX 8.0
     */
    public void prependTranslation(double tx, double ty, double tz) {
        atomicChange.start();
        preTranslate3D(tx, ty, tz);
        atomicChange.end();
    }

    /**
     * <p>
     * Prepends the 2D translation to this instance.
     * It is equivalent to {@code prepend(new Translate(tx, ty))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified translation first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified translation.
     * </p>
     * @param tx the X coordinate translation
     * @param ty the Y coordinate translation
     * @since JavaFX 8.0
     */
    public void prependTranslation(double tx, double ty) {
        atomicChange.start();
        preTranslate2D(tx, ty);
        atomicChange.end();
    }

    /**
     * 2D implementation of {@code prependTranslation()}.
     * If this is a 3D transform, the call is redirected to
     * {@code preTransalte3D}.
     * @since JavaFX 8.0
     */
    private void preTranslate2D(double tx, double ty) {
        if (state3d != APPLY_NON_3D) {
            preTranslate3D(tx, ty, 0.0);
            return;
        }

        setTx(getTx() + tx);
        setTy(getTy() + ty);

        if (getTx() == 0.0 && getTy() == 0.0) {
            state2d &= ~APPLY_TRANSLATE;
        } else {
            state2d |= APPLY_TRANSLATE;
        }
    }

    /**
     * 3D implementation of {@code prependTranslation()}.
     * Works fine if this is a 2D transform.
     */
    private void preTranslate3D(double tx, double ty, double tz) {
        switch(state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                preTranslate2D(tx, ty);

                if (tz != 0.0) {
                    setTz(tz);

                    if ((state2d & APPLY_SHEAR) == 0) {
                        state3d = (state2d & APPLY_SCALE) | APPLY_TRANSLATE;
                    } else {
                        state3d = APPLY_3D_COMPLEX;
                    }
                }
                return;
            case APPLY_TRANSLATE:
                setTx(getTx() + tx);
                setTy(getTy() + ty);
                setTz(getTz() + tz);
                if (getTz() == 0.0) {
                    state3d = APPLY_NON_3D;
                    if (getTx() == 0.0 && getTy() == 0.0) {
                        state2d = APPLY_IDENTITY;
                    } else {
                        state2d = APPLY_TRANSLATE;
                    }
                }
                return;
            case APPLY_SCALE:
                setTx(tx);
                setTy(ty);
                setTz(tz);
                if (tx != 0.0 || ty != 0.0 || tz != 0.0) {
                    state3d |= APPLY_TRANSLATE;
                }
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
                setTx(getTx() + tx);
                setTy(getTy() + ty);
                setTz(getTz() + tz);

                if (getTz() == 0.0) {
                    if (getTx() == 0.0 && getTy() == 0.0) {
                        state3d = APPLY_SCALE;
                    }
                    if (getMzz() == 1.0) {
                        state2d = state3d;
                        state3d = APPLY_NON_3D;
                    }
                }
                return;
            case APPLY_3D_COMPLEX:
                setTx(getTx() + tx);
                setTy(getTy() + ty);
                setTz(getTz() + tz);
                if (getTz() == 0.0 && getMxz() == 0.0 && getMyz() == 0.0 &&
                        getMzx() == 0.0 && getMzy() == 0.0 && getMzz() == 1.0) {
                    state3d = APPLY_NON_3D;
                    updateState2D();
                } // otherwise state remains COMPLEX
                return;
        }
    }

               /* *************************************************
                *                      Scale                      *
                **************************************************/

    /**
     * <p>
     * Appends the 2D scale to this instance.
     * It is equivalent to {@code append(new Scale(sx, sy))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * scale second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified scale.
     * </p>
     * @param sx the X coordinate scale factor
     * @param sy the Y coordinate scale factor
     * @since JavaFX 8.0
     */
    public void appendScale(double sx, double sy) {
        atomicChange.start();
        scale2D(sx, sy);
        atomicChange.end();
    }

    /**
     * <p>
     * Appends the 2D scale with pivot to this instance.
     * It is equivalent to {@code append(new Scale(sx, sy, pivotX, pivotY))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * scale second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified scale.
     * </p>
     * @param sx the X coordinate scale factor
     * @param sy the Y coordinate scale factor
     * @param pivotX the X coordinate of the point about which the scale occurs
     * @param pivotY the Y coordinate of the point about which the scale occurs
     * @since JavaFX 8.0
     */
    public void appendScale(double sx, double sy,
            double pivotX, double pivotY) {
        atomicChange.start();
        if (pivotX != 0.0 || pivotY != 0.0) {
            translate2D(pivotX, pivotY);
            scale2D(sx, sy);
            translate2D(-pivotX, -pivotY);
        } else {
            scale2D(sx, sy);
        }
        atomicChange.end();
    }

    /**
     * <p>
     * Appends the 2D scale with pivot to this instance.
     * It is equivalent to
     * {@code append(new Scale(sx, sy, pivot.getX(), pivot.getY())}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * scale second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified scale.
     * </p>
     * @param sx the X coordinate scale factor
     * @param sy the Y coordinate scale factor
     * @param pivot the point about which the scale occurs
     * @throws NullPointerException if the specified {@code pivot} is null
     * @since JavaFX 8.0
     */
    public void appendScale(double sx, double sy, Point2D pivot) {
        appendScale(sx, sy, pivot.getX(), pivot.getY());
    }

    /**
     * <p>
     * Appends the scale to this instance.
     * It is equivalent to {@code append(new Scale(sx, sy, sz))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * scale second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified scale.
     * </p>
     * @param sx the X coordinate scale factor
     * @param sy the Y coordinate scale factor
     * @param sz the Z coordinate scale factor
     * @since JavaFX 8.0
     */
    public void appendScale(double sx, double sy, double sz) {
        atomicChange.start();
        scale3D(sx, sy, sz);
        atomicChange.end();
    }

    /**
     * <p>
     * Appends the scale with pivot to this instance.
     * It is equivalent to {@code append(new Scale(sx, sy, sz, pivotX,
     * pivotY, pivotZ))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * scale second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified scale.
     * </p>
     * @param sx the X coordinate scale factor
     * @param sy the Y coordinate scale factor
     * @param sz the Z coordinate scale factor
     * @param pivotX the X coordinate of the point about which the scale occurs
     * @param pivotY the Y coordinate of the point about which the scale occurs
     * @param pivotZ the Z coordinate of the point about which the scale occurs
     * @since JavaFX 8.0
     */
    public void appendScale(double sx, double sy, double sz,
            double pivotX, double pivotY, double pivotZ) {
        atomicChange.start();
        if (pivotX != 0.0 || pivotY != 0.0 || pivotZ != 0.0) {
            translate3D(pivotX, pivotY, pivotZ);
            scale3D(sx, sy, sz);
            translate3D(-pivotX, -pivotY, -pivotZ);
        } else {
            scale3D(sx, sy, sz);
        }
        atomicChange.end();
    }

    /**
     * <p>
     * Appends the scale with pivot to this instance.
     * It is equivalent to {@code append(new Scale(sx, sy, sz, pivot.getX(),
     * pivot.getY(), pivot.getZ()))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * scale second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified scale.
     * </p>
     * @param sx the X coordinate scale factor
     * @param sy the Y coordinate scale factor
     * @param sz the Z coordinate scale factor
     * @param pivot the point about which the scale occurs
     * @throws NullPointerException if the specified {@code pivot} is null
     * @since JavaFX 8.0
     */
    public void appendScale(double sx, double sy, double sz, Point3D pivot) {
        appendScale(sx, sy, sz, pivot.getX(), pivot.getY(), pivot.getZ());
    }

    /**
     * 2D implementation of {@code appendScale()}.
     * If this is a 3D transform, the call is redirected to {@code scale3D()}.
     */
    private void scale2D(double sx, double sy) {
        if (state3d != APPLY_NON_3D) {
            scale3D(sx, sy, 1.0);
            return;
        }

        int mystate = state2d;
        switch (mystate) {
            default:
                stateError();
                // cannot reach
            case APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE:
            case APPLY_SHEAR | APPLY_SCALE:
                setMxx(getMxx() * sx);
                setMyy(getMyy() * sy);
                // fall-through
            case APPLY_SHEAR | APPLY_TRANSLATE:
            case APPLY_SHEAR:
                setMxy(getMxy() * sy);
                setMyx(getMyx() * sx);
                if (getMxy() == 0.0 && getMyx() == 0.0) {
                    mystate &= APPLY_TRANSLATE;
                    if (getMxx() != 1.0 || getMyy() != 1.0) {
                        mystate |= APPLY_SCALE;
                    }
                    state2d = mystate;
                } else if (getMxx() == 0.0 && getMyy() == 0.0) {
                    state2d &= ~APPLY_SCALE;
                }
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
            case APPLY_SCALE:
                setMxx(getMxx() * sx);
                setMyy(getMyy() * sy);
                if (getMxx() == 1.0 && getMyy() == 1.0) {
                    state2d = (mystate &= APPLY_TRANSLATE);
                }
                return;
            case APPLY_TRANSLATE:
            case APPLY_IDENTITY:
                setMxx(sx);
                setMyy(sy);
                if (sx != 1.0 || sy != 1.0) {
                    state2d = (mystate | APPLY_SCALE);
                }
                return;
        }
    }

    /**
     * 3D implementation of {@code appendScale()}.
     * Works fine if this is a 2D transform.
     */
    private void scale3D(double sx, double sy, double sz) {
        switch (state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                scale2D(sx, sy);
                if (sz != 1.0) {
                    setMzz(sz);
                    if ((state2d & APPLY_SHEAR) == 0) {
                        state3d = (state2d & APPLY_TRANSLATE) | APPLY_SCALE;
                    } else {
                        state3d = APPLY_3D_COMPLEX;
                    }
                }
                return;
            case APPLY_TRANSLATE:
                setMxx(sx);
                setMyy(sy);
                setMzz(sz);
                if (sx != 1.0 || sy != 1.0 || sz != 1.0) {
                    state3d |= APPLY_SCALE;
                }
                return;
            case APPLY_SCALE:
                setMxx(getMxx() * sx);
                setMyy(getMyy() * sy);
                setMzz(getMzz() * sz);
                if (getMzz() == 1.0) {
                    state3d = APPLY_NON_3D;
                    if (getMxx() == 1.0 && getMyy() == 1.0) {
                        state2d = APPLY_IDENTITY;
                    } else {
                        state2d = APPLY_SCALE;
                    }
                }
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
                setMxx(getMxx() * sx);
                setMyy(getMyy() * sy);
                setMzz(getMzz() * sz);

                if (getMxx() == 1.0 && getMyy() == 1.0 && getMzz() == 1.0) {
                    state3d &= ~APPLY_SCALE;
                }
                if (getTz() == 0.0 && getMzz() == 1.0) {
                    state2d = state3d;
                    state3d = APPLY_NON_3D;
                }
                return;
            case APPLY_3D_COMPLEX:
                setMxx(getMxx() * sx);
                setMxy(getMxy() * sy);
                setMxz(getMxz() * sz);

                setMyx(getMyx() * sx);
                setMyy(getMyy() * sy);
                setMyz(getMyz() * sz);

                setMzx(getMzx() * sx);
                setMzy(getMzy() * sy);
                setMzz(getMzz() * sz);

                if (sx == 0.0 || sy == 0.0 || sz == 0.0) {
                    updateState();
                } // otherwise state remains COMPLEX
                return;
        }
    }

    /**
     * <p>
     * Prepends the 2D scale to this instance.
     * It is equivalent to {@code prepend(new Scale(sx, sy))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified scale first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified scale.
     * </p>
     * @param sx the X coordinate scale factor
     * @param sy the Y coordinate scale factor
     * @since JavaFX 8.0
     */
    public void prependScale(double sx, double sy) {

        atomicChange.start();
        preScale2D(sx, sy);
        atomicChange.end();
    }

    /**
     * <p>
     * Prepends the 2D scale with pivot to this instance.
     * It is equivalent to {@code prepend(new Scale(sx, sy, pivotX, pivotY))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified scale first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified scale.
     * </p>
     * @param sx the X coordinate scale factor
     * @param sy the Y coordinate scale factor
     * @param pivotX the X coordinate of the point about which the scale occurs
     * @param pivotY the Y coordinate of the point about which the scale occurs
     * @since JavaFX 8.0
     */
    public void prependScale(double sx, double sy,
            double pivotX, double pivotY) {
        atomicChange.start();
        if (pivotX != 0.0 || pivotY != 0.0) {
            preTranslate2D(-pivotX, -pivotY);
            preScale2D(sx, sy);
            preTranslate2D(pivotX, pivotY);
        } else {
            preScale2D(sx, sy);
        }
        atomicChange.end();
    }

    /**
     * <p>
     * Prepends the 2D scale with pivot to this instance.
     * It is equivalent to {@code prepend(new Scale(sx, sy, pivot.getX(),
     * </p>pivot.getY()))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified scale first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified scale.
     * </p>
     * @param sx the X coordinate scale factor
     * @param sy the Y coordinate scale factor
     * @param pivot the point about which the scale occurs
     * @throws NullPointerException if the specified {@code pivot} is null
     * @since JavaFX 8.0
     */
    public void prependScale(double sx, double sy, Point2D pivot) {
        prependScale(sx, sy, pivot.getX(), pivot.getY());
    }

    /**
     * <p>
     * Prepends the scale to this instance.
     * It is equivalent to {@code prepend(new Scale(sx, sy, sz))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified scale first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified scale.
     * </p>
     * @param sx the X coordinate scale factor
     * @param sy the Y coordinate scale factor
     * @param sz the Z coordinate scale factor
     * @since JavaFX 8.0
     */
    public void prependScale(double sx, double sy, double sz) {
        atomicChange.start();
        preScale3D(sx, sy, sz);
        atomicChange.end();
    }

    /**
     * <p>
     * Prepends the scale with pivot to this instance.
     * It is equivalent to
     * {@code prepend(new Scale(sx, sy, sz, pivotX, pivotY, pivotZ))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified scale first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified scale.
     * </p>
     * @param sx the X coordinate scale factor
     * @param sy the Y coordinate scale factor
     * @param sz the Z coordinate scale factor
     * @param pivotX the X coordinate of the point about which the scale occurs
     * @param pivotY the Y coordinate of the point about which the scale occurs
     * @param pivotZ the Z coordinate of the point about which the scale occurs
     * @since JavaFX 8.0
     */
    public void prependScale(double sx, double sy, double sz,
            double pivotX, double pivotY, double pivotZ) {

        atomicChange.start();
        if (pivotX != 0.0 || pivotY != 0.0 || pivotZ != 0.0) {
            preTranslate3D(-pivotX, -pivotY, -pivotZ);
            preScale3D(sx, sy, sz);
            preTranslate3D(pivotX, pivotY, pivotZ);
        } else {
            preScale3D(sx, sy, sz);
        }
        atomicChange.end();
    }

    /**
     * <p>
     * Prepends the scale with pivot to this instance.
     * It is equivalent to {@code prepend(new Scale(sx, sy, sz, pivot.getX(),
     * pivot.getY(), pivot.getZ()))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified scale first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified scale.
     * </p>
     * @param sx the X coordinate scale factor
     * @param sy the Y coordinate scale factor
     * @param sz the Z coordinate scale factor
     * @param pivot the point about which the scale occurs
     * @throws NullPointerException if the specified {@code pivot} is null
     * @since JavaFX 8.0
     */
    public void prependScale(double sx, double sy, double sz, Point3D pivot) {
        prependScale(sx, sy, sz, pivot.getX(), pivot.getY(), pivot.getZ());
    }

    /**
     * 2D implementation of {@code prependScale()}.
     * If this is a 3D transform, the call is redirected to {@code preScale3D()}.
     */
    private void preScale2D(double sx, double sy) {

        if (state3d != APPLY_NON_3D) {
            preScale3D(sx, sy, 1.0);
            return;
        }

        int mystate = state2d;
        switch (mystate) {
            default:
                stateError();
                // cannot reach
            case APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE:
                setTx(getTx() * sx);
                setTy(getTy() * sy);
                if (getTx() == 0.0 && getTy() == 0.0) {
                    mystate = mystate & ~APPLY_TRANSLATE;
                    state2d = mystate;
                }
                // fall-through
            case APPLY_SHEAR | APPLY_SCALE:
                setMxx(getMxx() * sx);
                setMyy(getMyy() * sy);
                // fall-through
            case APPLY_SHEAR:
                setMxy(getMxy() * sx);
                setMyx(getMyx() * sy);
                if (getMxy() == 0.0 && getMyx() == 0.0) {
                    mystate &= APPLY_TRANSLATE;
                    if (getMxx() != 1.0 || getMyy() != 1.0) {
                        mystate |= APPLY_SCALE;
                    }
                    state2d = mystate;
                }
                return;
            case APPLY_SHEAR | APPLY_TRANSLATE:
                setTx(getTx() * sx);
                setTy(getTy() * sy);
                setMxy(getMxy() * sx);
                setMyx(getMyx() * sy);
                if (getMxy() == 0.0 && getMyx() == 0.0) {
                    if (getTx() == 0.0 && getTy() == 0.0) {
                        state2d = APPLY_SCALE;
                    } else {
                        state2d = APPLY_SCALE | APPLY_TRANSLATE;
                    }
                } else if (getTx() ==0.0 && getTy() == 0.0) {
                    state2d = APPLY_SHEAR;
                }
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
                setTx(getTx() * sx);
                setTy(getTy() * sy);
                if (getTx() == 0.0 && getTy() == 0.0) {
                    mystate = mystate & ~APPLY_TRANSLATE;
                    state2d = mystate;
                }
                // fall-through
            case APPLY_SCALE:
                setMxx(getMxx() * sx);
                setMyy(getMyy() * sy);
                if (getMxx() == 1.0 && getMyy() == 1.0) {
                    state2d = (mystate &= APPLY_TRANSLATE);
                }
                return;
            case APPLY_TRANSLATE:
                setTx(getTx() * sx);
                setTy(getTy() * sy);
                if (getTx() == 0.0 && getTy() == 0.0) {
                    mystate = mystate & ~APPLY_TRANSLATE;
                    state2d = mystate;
                }
                // fall-through
            case APPLY_IDENTITY:
                setMxx(sx);
                setMyy(sy);
                if (sx != 1.0 || sy != 1.0) {
                    state2d = mystate | APPLY_SCALE;
                }
                return;
        }
    }

    /**
     * 3D implementation of {@code prependScale()}.
     * Works fine if this is a 2D transform.
     */
    private void preScale3D(double sx, double sy, double sz) {
        switch (state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                preScale2D(sx, sy);
                if (sz != 1.0) {
                    setMzz(sz);
                    if ((state2d & APPLY_SHEAR) == 0) {
                        state3d = (state2d & APPLY_TRANSLATE) | APPLY_SCALE;
                    } else {
                        state3d = APPLY_3D_COMPLEX;
                    }
                }
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
                setTx(getTx() * sx);
                setTy(getTy() * sy);
                setTz(getTz() * sz);
                setMxx(getMxx() * sx);
                setMyy(getMyy() * sy);
                setMzz(getMzz() * sz);
                if (getTx() == 0.0 && getTy() == 0.0 && getTz() == 0.0) {
                    state3d &= ~APPLY_TRANSLATE;
                }
                if (getMxx() == 1.0 && getMyy() == 1.0 && getMzz() == 1.0) {
                    state3d &= ~APPLY_SCALE;
                }
                if (getTz() == 0.0 && getMzz() == 1.0) {
                    state2d = state3d;
                    state3d = APPLY_NON_3D;
                }
                return;
            case APPLY_SCALE:
                setMxx(getMxx() * sx);
                setMyy(getMyy() * sy);
                setMzz(getMzz() * sz);
                if (getMzz() == 1.0) {
                    state3d = APPLY_NON_3D;
                    if (getMxx() == 1.0 && getMyy() == 1.0) {
                        state2d = APPLY_IDENTITY;
                    } else {
                        state2d = APPLY_SCALE;
                    }
                }
                return;
            case APPLY_TRANSLATE:
                setTx(getTx() * sx);
                setTy(getTy() * sy);
                setTz(getTz() * sz);
                setMxx(sx);
                setMyy(sy);
                setMzz(sz);
                if (getTx() == 0.0 && getTy() == 0.0 && getTz() == 0.0) {
                    state3d &= ~APPLY_TRANSLATE;
                }
                if (sx != 1.0 || sy != 1.0 || sz != 1.0) {
                    state3d |= APPLY_SCALE;
                }
                return;
            case APPLY_3D_COMPLEX:
                setMxx(getMxx() * sx);
                setMxy(getMxy() * sx);
                setMxz(getMxz() * sx);
                setTx(getTx() * sx);

                setMyx(getMyx() * sy);
                setMyy(getMyy() * sy);
                setMyz(getMyz() * sy);
                setTy(getTy() * sy);

                setMzx(getMzx() * sz);
                setMzy(getMzy() * sz);
                setMzz(getMzz() * sz);
                setTz(getTz() * sz);

                if (sx == 0.0 || sy == 0.0 || sz == 0.0) {
                    updateState();
                } // otherwise state remains COMPLEX
                return;
        }
    }

               /* *************************************************
                *                     Shear                       *
                **************************************************/

    /**
     * <p>
     * Appends the shear to this instance.
     * It is equivalent to {@code append(new Shear(sx, sy))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * shear second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified shear.
     * </p>
     * @param shx the XY coordinate element
     * @param shy the YX coordinate element
     * @since JavaFX 8.0
     */
    public void appendShear(double shx, double shy) {
        atomicChange.start();
        shear2D(shx, shy);
        atomicChange.end();
    }

    /**
     * <p>
     * Appends the shear with pivot to this instance.
     * It is equivalent to {@code append(new Shear(sx, sy, pivotX, pivotY))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * shear second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified shear.
     * </p>
     * @param shx the XY coordinate element
     * @param shy the YX coordinate element
     * @param pivotX the X coordinate of the shear pivot point
     * @param pivotY the Y coordinate of the shear pivot point
     * @since JavaFX 8.0
     */
    public void appendShear(double shx, double shy,
            double pivotX, double pivotY) {
        atomicChange.start();
        if (pivotX != 0.0 || pivotY != 0.0) {
            translate2D(pivotX, pivotY);
            shear2D(shx, shy);
            translate2D(-pivotX, -pivotY);
        } else {
            shear2D(shx, shy);
        }
        atomicChange.end();
    }

    /**
     * <p>
     * Appends the shear with pivot to this instance.
     * It is equivalent to {@code append(new Shear(sx, sy,
     * pivot.getX(), pivot.getY()))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * shear second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified shear.
     * </p>
     * @param shx the XY coordinate element
     * @param shy the YX coordinate element
     * @param pivot the shear pivot point
     * @throws NullPointerException if the specified {@code pivot} is null
     * @since JavaFX 8.0
     */
    public void appendShear(double shx, double shy, Point2D pivot) {
        appendShear(shx, shy, pivot.getX(), pivot.getY());
    }

    /**
     * 2D implementation of {@code appendShear()}.
     * If this is a 3D transform, the call is redirected to {@code shear3D()}.
     */
    private void shear2D(double shx, double shy) {

        if (state3d != APPLY_NON_3D) {
            shear3D(shx, shy);
            return;
        }

        int mystate = state2d;
        switch (mystate) {
            default:
                stateError();
                // cannot reach
            case APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE:
            case APPLY_SHEAR | APPLY_SCALE:
                double M0, M1;
                M0 = getMxx();
                M1 = getMxy();
                setMxx(M0 + M1 * shy);
                setMxy(M0 * shx + M1);

                M0 = getMyx();
                M1 = getMyy();
                setMyx(M0 + M1 * shy);
                setMyy(M0 * shx + M1);
                updateState2D();
                return;
            case APPLY_SHEAR | APPLY_TRANSLATE:
            case APPLY_SHEAR:
                setMxx(getMxy() * shy);
                setMyy(getMyx() * shx);
                if (getMxx() != 0.0 || getMyy() != 0.0) {
                    state2d = mystate | APPLY_SCALE;
                }
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
            case APPLY_SCALE:
                setMxy(getMxx() * shx);
                setMyx(getMyy() * shy);
                if (getMxy() != 0.0 || getMyx() != 0.0) {
                    state2d = mystate | APPLY_SHEAR;
                }
                return;
            case APPLY_TRANSLATE:
            case APPLY_IDENTITY:
                setMxy(shx);
                setMyx(shy);
                if (getMxy() != 0.0 || getMyx() != 0.0) {
                    state2d = mystate | APPLY_SCALE | APPLY_SHEAR;
                }
                return;
        }
    }

    /**
     * 3D implementation of {@code appendShear()}.
     * Works fine if this is a 2D transform.
     */
    private void shear3D(double shx, double shy) {
        switch (state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                // cannot happen because currently there is no 3D appendShear
                // that would call this method directly
                shear2D(shx, shy);
                return;
            case APPLY_TRANSLATE:
                setMxy(shx);
                setMyx(shy);
                if (shx != 0.0 || shy != 0.0) {
                    state3d = APPLY_3D_COMPLEX;
                }
                return;
            case APPLY_SCALE:
            case APPLY_SCALE | APPLY_TRANSLATE:
                setMxy(getMxx() * shx);
                setMyx(getMyy() * shy);
                if (getMxy() != 0.0 || getMyx() != 0.0) {
                    state3d = APPLY_3D_COMPLEX;
                }
                return;
            case APPLY_3D_COMPLEX:
                final double m_xx = getMxx();
                final double m_xy = getMxy();
                final double m_yx = getMyx();
                final double m_yy = getMyy();
                final double m_zx = getMzx();
                final double m_zy = getMzy();

                setMxx(m_xx + m_xy * shy);
                setMxy(m_xy + m_xx * shx);
                setMyx(m_yx + m_yy * shy);
                setMyy(m_yy + m_yx * shx);
                setMzx(m_zx + m_zy * shy);
                setMzy(m_zy + m_zx * shx);
                updateState();
                return;
        }
    }

    /**
     * <p>
     * Prepends the shear to this instance.
     * It is equivalent to {@code prepend(new Shear(sx, sy))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified shear first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified shear.
     * </p>
     * @param shx the XY coordinate element
     * @param shy the YX coordinate element
     * @since JavaFX 8.0
     */
    public void prependShear(double shx, double shy) {
        atomicChange.start();
        preShear2D(shx, shy);
        atomicChange.end();
    }

    /**
     * <p>
     * Prepends the shear with pivot to this instance.
     * It is equivalent to {@code prepend(new Shear(sx, sy, pivotX, pivotY))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified shear first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified shear.
     * </p>
     * @param shx the XY coordinate element
     * @param shy the YX coordinate element
     * @param pivotX the X coordinate of the shear pivot point
     * @param pivotY the Y coordinate of the shear pivot point
     * @since JavaFX 8.0
     */
    public void prependShear(double shx, double shy,
            double pivotX, double pivotY) {
        atomicChange.start();
        if (pivotX != 0.0 || pivotY != 0.0) {
            preTranslate2D(-pivotX, -pivotY);
            preShear2D(shx, shy);
            preTranslate2D(pivotX, pivotY);
        } else {
            preShear2D(shx, shy);
        }
        atomicChange.end();
    }

    /**
     * <p>
     * Prepends the shear with pivot to this instance.
     * It is equivalent to {@code prepend(new Shear(sx, sy, pivot.getX(),
     * pivot.getY()))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified shear first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified shear.
     * </p>
     * @param shx the XY coordinate element
     * @param shy the YX coordinate element
     * @param pivot the shear pivot point
     * @throws NullPointerException if the specified {@code pivot} is null
     * @since JavaFX 8.0
     */
    public void prependShear(double shx, double shy, Point2D pivot) {
        prependShear(shx, shy, pivot.getX(), pivot.getY());
    }

    /**
     * 2D implementation of {@code prependShear()}.
     * If this is a 3D transform, the call is redirected to {@code preShear3D()}.
     */
    private void preShear2D(double shx, double shy) {

        if (state3d != APPLY_NON_3D) {
            preShear3D(shx, shy);
            return;
        }

        int mystate = state2d;

        switch (mystate) {
            default:
                stateError();
                // cannot reach
            case APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE:
            case APPLY_SHEAR | APPLY_TRANSLATE:
                final double t_x_1 = getTx();
                final double t_y_1 = getTy();
                setTx(t_x_1 + shx * t_y_1);
                setTy(t_y_1 + shy * t_x_1);
                // fall-through
            case APPLY_SHEAR | APPLY_SCALE:
            case APPLY_SHEAR:
                final double m_xx = getMxx();
                final double m_xy = getMxy();
                final double m_yx = getMyx();
                final double m_yy = getMyy();

                setMxx(m_xx + shx * m_yx);
                setMxy(m_xy + shx * m_yy);
                setMyx(shy * m_xx + m_yx);
                setMyy(shy * m_xy + m_yy);
                updateState2D();
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
                final double t_x_2 = getTx();
                final double t_y_2 = getTy();
                setTx(t_x_2 + shx * t_y_2);
                setTy(t_y_2 + shy * t_x_2);
                if (getTx() == 0.0 && getTy() == 0.0) {
                    mystate = mystate & ~APPLY_TRANSLATE;
                    state2d = mystate;
                }
                // fall-through
            case APPLY_SCALE:
                setMxy(shx * getMyy());
                setMyx(shy * getMxx());
                if (getMxy() != 0.0 || getMyx() != 0.0) {
                    state2d = mystate | APPLY_SHEAR;
                }
                return;
            case APPLY_TRANSLATE:
                final double t_x_3 = getTx();
                final double t_y_3 = getTy();
                setTx(t_x_3 + shx * t_y_3);
                setTy(t_y_3 + shy * t_x_3);
                if (getTx() == 0.0 && getTy() == 0.0) {
                    mystate = mystate & ~APPLY_TRANSLATE;
                    state2d = mystate;
                }
                // fall-through
            case APPLY_IDENTITY:
                setMxy(shx);
                setMyx(shy);
                if (getMxy() != 0.0 || getMyx() != 0.0) {
                    state2d = mystate | APPLY_SCALE | APPLY_SHEAR;
                }
                return;
        }
    }

    /**
     * 3D implementation of {@code prependShear()}.
     * Works fine if this is a 2D transform.
     */
    private void preShear3D(double shx, double shy) {

        switch (state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                // cannot happen because currently there is no 3D prependShear
                // that would call this method directly
                preShear2D(shx, shy);
                return;
            case APPLY_TRANSLATE:
                final double tx_t = getTx();
                setMxy(shx);
                setTx(tx_t + getTy() * shx);
                setMyx(shy);
                setTy(tx_t * shy + getTy());

                if (shx != 0.0 || shy != 0.0) {
                    state3d = APPLY_3D_COMPLEX;
                }
                return;
            case APPLY_SCALE:
                setMxy(getMyy() * shx);
                setMyx(getMxx() * shy);

                if (getMxy() != 0.0 || getMyx() != 0.0) {
                    state3d = APPLY_3D_COMPLEX;
                }
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
                final double tx_st = getTx();
                setMxy(getMyy() * shx);
                setTx(tx_st + getTy() * shx);
                setMyx(getMxx() * shy);
                setTy(tx_st * shy + getTy());

                if (getMxy() != 0.0 || getMyx() != 0.0) {
                    state3d = APPLY_3D_COMPLEX;
                }
                return;
            case APPLY_3D_COMPLEX:

                final double m_xx = getMxx();
                final double m_xy = getMxy();
                final double m_yx = getMyx();
                final double t_x = getTx();
                final double m_yy = getMyy();
                final double m_xz = getMxz();
                final double m_yz = getMyz();
                final double t_y = getTy();

                setMxx(m_xx + m_yx * shx);
                setMxy(m_xy + m_yy * shx);
                setMxz(m_xz + m_yz * shx);
                setTx(t_x + t_y * shx);
                setMyx(m_xx * shy + m_yx);
                setMyy(m_xy * shy + m_yy);
                setMyz(m_xz * shy + m_yz);
                setTy(t_x * shy + t_y);

                updateState();
                return;
        }
    }

               /* *************************************************
                *                     Rotate                      *
                **************************************************/

    /**
     * <p>
     * Appends the 2D rotation to this instance.
     * It is equivalent to {@code append(new Rotate(angle))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * rotation second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified rotation.
     * </p>
     * @param angle the angle of the rotation in degrees
     * @since JavaFX 8.0
     */
    public void appendRotation(double angle) {
        atomicChange.start();
        rotate2D(angle);
        atomicChange.end();
    }

    /**
     * <p>
     * Appends the 2D rotation with pivot to this instance.
     * It is equivalent to {@code append(new Rotate(angle, pivotX, pivotY))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * rotation second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified rotation.
     * </p>
     * @param angle the angle of the rotation in degrees
     * @param pivotX the X coordinate of the rotation pivot point
     * @param pivotY the Y coordinate of the rotation pivot point
     * @since JavaFX 8.0
     */
    public void appendRotation(double angle,
            double pivotX, double pivotY) {
        atomicChange.start();
        if (pivotX != 0.0 || pivotY != 0.0) {
            translate2D(pivotX, pivotY);
            rotate2D(angle);
            translate2D(-pivotX, -pivotY);
        } else {
            rotate2D(angle);
        }
        atomicChange.end();
    }

    /**
     * <p>
     * Appends the 2D rotation with pivot to this instance.
     * It is equivalent to {@code append(new Rotate(angle, pivot.getX(),
     * pivot.getY()))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * rotation second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified rotation.
     * </p>
     * @param angle the angle of the rotation in degrees
     * @param pivot the rotation pivot point
     * @throws NullPointerException if the specified {@code pivot} is null
     * @since JavaFX 8.0
     */
    public void appendRotation(double angle, Point2D pivot) {
        appendRotation(angle, pivot.getX(), pivot.getY());
    }

    /**
     * <p>
     * Appends the rotation to this instance.
     * It is equivalent to {@code append(new Rotate(angle, pivotX, pivotY,
     * pivotZ, new Point3D(axisX, axisY, axisZ)))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * rotation second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified rotation.
     * </p>
     * @param angle the angle of the rotation in degrees
     * @param pivotX the X coordinate of the rotation pivot point
     * @param pivotY the Y coordinate of the rotation pivot point
     * @param pivotZ the Z coordinate of the rotation pivot point
     * @param axisX the X coordinate magnitude of the rotation axis
     * @param axisY the Y coordinate magnitude of the rotation axis
     * @param axisZ the Z coordinate magnitude of the rotation axis
     * @since JavaFX 8.0
     */
    public void appendRotation(double angle,
            double pivotX, double pivotY, double pivotZ,
            double axisX, double axisY, double axisZ) {
        atomicChange.start();
        if (pivotX != 0.0 || pivotY != 0.0 || pivotZ != 0.0) {
            translate3D(pivotX, pivotY, pivotZ);
            rotate3D(angle, axisX, axisY, axisZ);
            translate3D(-pivotX, -pivotY, -pivotZ);
        } else {
            rotate3D(angle, axisX, axisY, axisZ);
        }
        atomicChange.end();
    }

    /**
     * <p>
     * Appends the rotation to this instance.
     * It is equivalent to {@code append(new Rotate(angle, pivotX, pivotY,
     * pivotZ, axis))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * rotation second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified rotation.
     * </p>
     * @param angle the angle of the rotation in degrees
     * @param pivotX the X coordinate of the rotation pivot point
     * @param pivotY the Y coordinate of the rotation pivot point
     * @param pivotZ the Z coordinate of the rotation pivot point
     * @param axis the rotation axis
     * @throws NullPointerException if the specified {@code axis} is null
     * @since JavaFX 8.0
     */
    public void appendRotation(double angle,
            double pivotX, double pivotY, double pivotZ,
            Point3D axis) {
        appendRotation(angle, pivotX, pivotY, pivotZ,
                axis.getX(), axis.getY(), axis.getZ());
    }

    /**
     * <p>
     * Appends the rotation to this instance.
     * It is equivalent to {@code append(new Rotate(angle, pivot.getX(),
     * pivot.getY(), pivot.getZ(), axis))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, {@code this} transform first and the specified
     * rotation second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the right by the transformation matrix of
     * the specified rotation.
     * </p>
     * @param angle the angle of the rotation in degrees
     * @param pivot the rotation pivot point
     * @param axis the rotation axis
     * @throws NullPointerException if the specified {@code pivot}
     *         or {@code axis} is null
     * @since JavaFX 8.0
     */
    public void appendRotation(double angle, Point3D pivot, Point3D axis) {
        appendRotation(angle, pivot.getX(), pivot.getY(), pivot.getZ(),
                axis.getX(), axis.getY(), axis.getZ());
    }

    /**
     * Implementation of the {@code appendRotation()} around an arbitrary axis.
     */
    private void rotate3D(double angle, double axisX, double axisY, double axisZ) {
        if (axisX == 0.0 && axisY == 0.0) {
            if (axisZ > 0.0) {
                rotate3D(angle);
            } else if (axisZ < 0.0) {
                rotate3D(-angle);
            } // else rotating about zero vector - NOP
            return;
        }

        double mag = Math.sqrt(axisX * axisX + axisY * axisY + axisZ * axisZ);

        if (mag == 0.0) {
            return;
        }

        mag = 1.0 / mag;
        final double ax = axisX * mag;
        final double ay = axisY * mag;
        final double az = axisZ * mag;

        final double sinTheta = Math.sin(Math.toRadians(angle));
        final double cosTheta = Math.cos(Math.toRadians(angle));
        final double t = 1.0 - cosTheta;

        final double xz = ax * az;
        final double xy = ax * ay;
        final double yz = ay * az;

        final double Txx = t * ax * ax + cosTheta;
        final double Txy = t * xy - sinTheta * az;
        final double Txz = t * xz + sinTheta * ay;

        final double Tyx = t * xy + sinTheta * az;
        final double Tyy = t * ay * ay + cosTheta;
        final double Tyz = t * yz - sinTheta * ax;

        final double Tzx = t * xz - sinTheta * ay;
        final double Tzy = t * yz + sinTheta * ax;
        final double Tzz = t * az * az + cosTheta;

        switch (state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                switch (state2d) {
                    default:
                        stateError();
                        // cannot reach
                    case APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE:
                    case APPLY_SHEAR | APPLY_SCALE:
                        final double xx_sst = getMxx();
                        final double xy_sst = getMxy();
                        final double yx_sst = getMyx();
                        final double yy_sst = getMyy();
                        setMxx(xx_sst * Txx + xy_sst * Tyx);
                        setMxy(xx_sst * Txy + xy_sst * Tyy);
                        setMxz(xx_sst * Txz + xy_sst * Tyz);
                        setMyx(yx_sst * Txx + yy_sst * Tyx);
                        setMyy(yx_sst * Txy + yy_sst * Tyy);
                        setMyz(yx_sst * Txz + yy_sst * Tyz);
                        setMzx(Tzx);
                        setMzy(Tzy);
                        setMzz(Tzz);
                        break;
                    case APPLY_SHEAR | APPLY_TRANSLATE:
                    case APPLY_SHEAR:
                        final double xy_sht = getMxy();
                        final double yx_sht = getMyx();
                        setMxx(xy_sht * Tyx);
                        setMxy(xy_sht * Tyy);
                        setMxz(xy_sht * Tyz);
                        setMyx(yx_sht * Txx);
                        setMyy(yx_sht * Txy);
                        setMyz(yx_sht * Txz);
                        setMzx(Tzx);
                        setMzy(Tzy);
                        setMzz(Tzz);
                        break;
                    case APPLY_SCALE | APPLY_TRANSLATE:
                    case APPLY_SCALE:
                        final double xx_s = getMxx();
                        final double yy_s = getMyy();
                        setMxx(xx_s * Txx);
                        setMxy(xx_s * Txy);
                        setMxz(xx_s * Txz);
                        setMyx(yy_s * Tyx);
                        setMyy(yy_s * Tyy);
                        setMyz(yy_s * Tyz);
                        setMzx(Tzx);
                        setMzy(Tzy);
                        setMzz(Tzz);
                        break;
                    case APPLY_TRANSLATE:
                    case APPLY_IDENTITY:
                        setMxx(Txx);
                        setMxy(Txy);
                        setMxz(Txz);
                        setMyx(Tyx);
                        setMyy(Tyy);
                        setMyz(Tyz);
                        setMzx(Tzx);
                        setMzy(Tzy);
                        setMzz(Tzz);
                        break;
                }
                break;
            case APPLY_TRANSLATE:
                setMxx(Txx);
                setMxy(Txy);
                setMxz(Txz);
                setMyx(Tyx);
                setMyy(Tyy);
                setMyz(Tyz);
                setMzx(Tzx);
                setMzy(Tzy);
                setMzz(Tzz);
                break;
            case APPLY_SCALE:
            case APPLY_SCALE | APPLY_TRANSLATE:
                final double xx_st = getMxx();
                final double yy_st = getMyy();
                final double zz_st = getMzz();
                setMxx(xx_st * Txx);
                setMxy(xx_st * Txy);
                setMxz(xx_st * Txz);
                setMyx(yy_st * Tyx);
                setMyy(yy_st * Tyy);
                setMyz(yy_st * Tyz);
                setMzx(zz_st * Tzx);
                setMzy(zz_st * Tzy);
                setMzz(zz_st * Tzz);
                break;
            case APPLY_3D_COMPLEX:
                final double m_xx = getMxx();
                final double m_xy = getMxy();
                final double m_xz = getMxz();
                final double m_yx = getMyx();
                final double m_yy = getMyy();
                final double m_yz = getMyz();
                final double m_zx = getMzx();
                final double m_zy = getMzy();
                final double m_zz = getMzz();
                setMxx(m_xx * Txx + m_xy * Tyx + m_xz * Tzx /* + mxt * 0.0 */);
                setMxy(m_xx * Txy + m_xy * Tyy + m_xz * Tzy /* + mxt * 0.0 */);
                setMxz(m_xx * Txz + m_xy * Tyz + m_xz * Tzz /* + mxt * 0.0 */);
                setMyx(m_yx * Txx + m_yy * Tyx + m_yz * Tzx /* + myt * 0.0 */);
                setMyy(m_yx * Txy + m_yy * Tyy + m_yz * Tzy /* + myt * 0.0 */);
                setMyz(m_yx * Txz + m_yy * Tyz + m_yz * Tzz /* + myt * 0.0 */);
                setMzx(m_zx * Txx + m_zy * Tyx + m_zz * Tzx /* + mzt * 0.0 */);
                setMzy(m_zx * Txy + m_zy * Tyy + m_zz * Tzy /* + mzt * 0.0 */);
                setMzz(m_zx * Txz + m_zy * Tyz + m_zz * Tzz /* + mzt * 0.0 */);
                break;
        }
        updateState();
    }

    /**
     * Table of 2D state changes during predictable quadrant rotations where
     * the shear and scaleAffine values are swapped and negated.
     */
    private static final int rot90conversion[] = {
        /* IDENTITY => */        APPLY_SHEAR,
        /* TRANSLATE (TR) => */  APPLY_SHEAR | APPLY_TRANSLATE,
        /* SCALE (SC) => */      APPLY_SHEAR,
        /* SC | TR => */         APPLY_SHEAR | APPLY_TRANSLATE,
        /* SHEAR (SH) => */      APPLY_SCALE,
        /* SH | TR => */         APPLY_SCALE | APPLY_TRANSLATE,
        /* SH | SC => */         APPLY_SHEAR | APPLY_SCALE,
        /* SH | SC | TR => */    APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE,
    };

    /**
     * 2D implementation of {@code appendRotation}.
     * If this is a 3D transform, the call is redirected to {@code rotate3D()}.
     */
    private void rotate2D(double theta) {
        if (state3d != APPLY_NON_3D) {
            rotate3D(theta);
            return;
        }

        double sin = Math.sin(Math.toRadians(theta));
        if (sin == 1.0) {
            rotate2D_90();
        } else if (sin == -1.0) {
            rotate2D_270();
        } else {
            double cos = Math.cos(Math.toRadians(theta));
            if (cos == -1.0) {
                rotate2D_180();
            } else if (cos != 1.0) {
                double M0, M1;
                M0 = getMxx();
                M1 = getMxy();
                setMxx(cos * M0 + sin * M1);
                setMxy(-sin * M0 + cos * M1);
                M0 = getMyx();
                M1 = getMyy();
                setMyx(cos * M0 + sin * M1);
                setMyy(-sin * M0 + cos * M1);
                updateState2D();
            }
        }
    }

    /**
     * 2D implementation of {@code appendRotation} for 90 degrees rotation
     * around Z axis.
     * Behaves wrong when called for a 3D transform.
     */
    private void rotate2D_90() {
        double M0 = getMxx();
        setMxx(getMxy());
        setMxy(-M0);
        M0 = getMyx();
        setMyx(getMyy());
        setMyy(-M0);
        int newstate = rot90conversion[state2d];
        if ((newstate & (APPLY_SHEAR | APPLY_SCALE)) == APPLY_SCALE &&
                getMxx() == 1.0 && getMyy() == 1.0) {
            newstate -= APPLY_SCALE;
        } else if ((newstate & (APPLY_SHEAR | APPLY_SCALE)) == APPLY_SHEAR &&
                getMxy() == 0.0 && getMyx() == 0.0) {
            newstate = (newstate & ~APPLY_SHEAR | APPLY_SCALE);
        }
        state2d = newstate;
    }

    /**
     * 2D implementation of {@code appendRotation} for 180 degrees rotation
     * around Z axis.
     * Behaves wrong when called for a 3D transform.
     */
    private void rotate2D_180() {
        setMxx(-getMxx());
        setMyy(-getMyy());
        int oldstate = state2d;
        if ((oldstate & (APPLY_SHEAR)) != 0) {
            // If there was a shear, then this rotation has no
            // effect on the state.
            setMxy(-getMxy());
            setMyx(-getMyx());
        } else {
            // No shear means the SCALE state may toggle when
            // m00 and m11 are negated.
            if (getMxx() == 1.0 && getMyy() == 1.0) {
                state2d = oldstate & ~APPLY_SCALE;
            } else {
                state2d = oldstate | APPLY_SCALE;
            }
        }
    }

    /**
     * 2D implementation of {@code appendRotation} for 270 degrees rotation
     * around Z axis.
     * Behaves wrong when called for a 3D transform.
     */
    private void rotate2D_270() {
        double M0 = getMxx();
        setMxx(-getMxy());
        setMxy(M0);
        M0 = getMyx();
        setMyx(-getMyy());
        setMyy(M0);
        int newstate = rot90conversion[state2d];
        if ((newstate & (APPLY_SHEAR | APPLY_SCALE)) == APPLY_SCALE &&
                getMxx() == 1.0 && getMyy() == 1.0) {
            newstate -= APPLY_SCALE;
        } else if ((newstate & (APPLY_SHEAR | APPLY_SCALE)) == APPLY_SHEAR &&
                getMxy() == 0.0 && getMyx() == 0.0) {
            newstate = (newstate & ~APPLY_SHEAR | APPLY_SCALE);
        }
        state2d = newstate;
    }

    /**
     * 3D implementation of {@code appendRotation} around Z axis.
     * If this is a 2D transform, the call is redirected to {@code rotate2D()}.
     */
    private void rotate3D(double theta) {
        if (state3d == APPLY_NON_3D) {
            rotate2D(theta);
            return;
        }

        double sin = Math.sin(Math.toRadians(theta));
        if (sin == 1.0) {
            rotate3D_90();
        } else if (sin == -1.0) {
            rotate3D_270();
        } else {
            double cos = Math.cos(Math.toRadians(theta));
            if (cos == -1.0) {
                rotate3D_180();
            } else if (cos != 1.0) {
                double M0, M1;
                M0 = getMxx();
                M1 = getMxy();
                setMxx(cos * M0 + sin * M1);
                setMxy(-sin * M0 + cos * M1);
                M0 = getMyx();
                M1 = getMyy();
                setMyx(cos * M0 + sin * M1);
                setMyy(-sin * M0 + cos * M1);
                M0 = getMzx();
                M1 = getMzy();
                setMzx(cos * M0 + sin * M1);
                setMzy(-sin * M0 + cos * M1);
                updateState();
            }
        }
    }

    /**
     * 3D implementation of {@code appendRotation} for 90 degrees rotation
     * around Z axis.
     * Behaves wrong when called for a 2D transform.
     */
    private void rotate3D_90() {
        double M0 = getMxx();
        setMxx(getMxy());
        setMxy(-M0);
        M0 = getMyx();
        setMyx(getMyy());
        setMyy(-M0);
        M0 = getMzx();
        setMzx(getMzy());
        setMzy(-M0);
        switch(state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_TRANSLATE:
                state3d = APPLY_3D_COMPLEX;
                return;
            case APPLY_SCALE:
            case APPLY_SCALE | APPLY_TRANSLATE:
                if (getMxy() != 0.0 || getMyx() != 0.0) {
                    state3d = APPLY_3D_COMPLEX;
                }
                return;
            case APPLY_3D_COMPLEX:
                updateState();
                return;
        }
    }

    /**
     * 3D implementation of {@code appendRotation} for 180 degrees rotation
     * around Z axis.
     * Behaves wrong when called for a 2D transform.
     */
    private void rotate3D_180() {
        final double mxx = getMxx();
        final double myy = getMyy();
        setMxx(-mxx);
        setMyy(-myy);
        if (state3d == APPLY_3D_COMPLEX) {
            setMxy(-getMxy());
            setMyx(-getMyx());
            setMzx(-getMzx());
            setMzy(-getMzy());
            updateState();
            return;
        }

        if (mxx == -1.0 && myy == -1.0 && getMzz() == 1.0) {
            // must have been 3d because of translation, which remained
            state3d &= ~APPLY_SCALE;
        } else {
            state3d |= APPLY_SCALE;
        }
    }

    /**
     * 3D implementation of {@code appendRotation} for 270 degrees rotation
     * around Z axis.
     * Behaves wrong when called for a 2D transform.
     */
    private void rotate3D_270() {
        double M0 = getMxx();
        setMxx(-getMxy());
        setMxy(M0);
        M0 = getMyx();
        setMyx(-getMyy());
        setMyy(M0);
        M0 = getMzx();
        setMzx(-getMzy());
        setMzy(M0);
        switch(state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_TRANSLATE:
                state3d = APPLY_3D_COMPLEX;
                return;
            case APPLY_SCALE:
            case APPLY_SCALE | APPLY_TRANSLATE:
                if (getMxy() != 0.0 || getMyx() != 0.0) {
                    state3d = APPLY_3D_COMPLEX;
                }
                return;
            case APPLY_3D_COMPLEX:
                updateState();
                return;
        }
    }

    /**
     * <p>
     * Prepends the 2D rotation to this instance.
     * It is equivalent to {@code prepend(new Rotate(angle))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified rotation first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified rotation.
     * </p>
     * @param angle the angle of the rotation in degrees
     * @since JavaFX 8.0
     */
    public void prependRotation(double angle) {
        atomicChange.start();
        preRotate2D(angle);
        atomicChange.end();
    }

    /**
     * <p>
     * Prepends the 2D rotation with pivot to this instance.
     * It is equivalent to {@code prepend(new Rotate(angle, pivotX, pivotY))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified rotation first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified rotation.
     * </p>
     * @param angle the angle of the rotation in degrees
     * @param pivotX the X coordinate of the rotation pivot point
     * @param pivotY the Y coordinate of the rotation pivot point
     * @since JavaFX 8.0
     */
    public void prependRotation(double angle, double pivotX, double pivotY) {
        atomicChange.start();
        if (pivotX != 0.0 || pivotY != 0.0) {
            preTranslate2D(-pivotX, -pivotY);
            preRotate2D(angle);
            preTranslate2D(pivotX, pivotY);
        } else {
            preRotate2D(angle);
        }
        atomicChange.end();
    }

    /**
     * <p>
     * Prepends the 2D rotation with pivot to this instance.
     * It is equivalent to {@code prepend(new Rotate(angle, pivot.getX(),
     * pivot.getY()))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified rotation first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified rotation.
     * </p>
     * @param angle the angle of the rotation in degrees
     * @param pivot the rotation pivot point
     * @throws NullPointerException if the specified {@code pivot} is null
     * @since JavaFX 8.0
     */
    public void prependRotation(double angle, Point2D pivot) {
        prependRotation(angle, pivot.getX(), pivot.getY());
    }

    /**
     * <p>
     * Prepends the rotation to this instance.
     * It is equivalent to {@code prepend(new Rotate(angle, pivotX, pivotY,
     * pivotZ, new Point3D(axisX, axisY, axisZ)))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified rotation first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified rotation.
     * </p>
     * @param angle the angle of the rotation in degrees
     * @param pivotX the X coordinate of the rotation pivot point
     * @param pivotY the Y coordinate of the rotation pivot point
     * @param pivotZ the Z coordinate of the rotation pivot point
     * @param axisX the X coordinate magnitude of the rotation axis
     * @param axisY the Y coordinate magnitude of the rotation axis
     * @param axisZ the Z coordinate magnitude of the rotation axis
     * @since JavaFX 8.0
     */
    public void prependRotation(double angle,
            double pivotX, double pivotY, double pivotZ,
            double axisX, double axisY, double axisZ) {
        atomicChange.start();
        if (pivotX != 0.0 || pivotY != 0.0 || pivotZ != 0.0) {
            preTranslate3D(-pivotX, -pivotY, -pivotZ);
            preRotate3D(angle, axisX, axisY, axisZ);
            preTranslate3D(pivotX, pivotY, pivotZ);
        } else {
            preRotate3D(angle, axisX, axisY, axisZ);
        }
        atomicChange.end();
    }

    /**
     * <p>
     * Prepends the rotation to this instance.
     * It is equivalent to {@code prepend(new Rotate(angle, pivotX, pivotY,
     * pivotZ, axis))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified rotation first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified rotation.
     * </p>
     * @param angle the angle of the rotation in degrees
     * @param pivotX the X coordinate of the rotation pivot point
     * @param pivotY the Y coordinate of the rotation pivot point
     * @param pivotZ the Z coordinate of the rotation pivot point
     * @param axis the rotation axis
     * @throws NullPointerException if the specified {@code axis} is null
     * @since JavaFX 8.0
     */
    public void prependRotation(double angle,
            double pivotX, double pivotY, double pivotZ,
            Point3D axis) {
        prependRotation(angle, pivotX, pivotY, pivotZ,
                axis.getX(), axis.getY(), axis.getZ());
    }

    /**
     * <p>
     * Prepends the rotation to this instance.
     * It is equivalent to {@code prepend(new Rotate(angle, pivot.getX(),
     * pivot.getY(), pivot.getZ(), axis))}.
     * </p><p>
     * The operation modifies this transform in a way that applying it to a node
     * has the same effect as adding two transforms to its
     * {@code getTransforms()} list, the specified rotation first
     * and {@code this} transform second.
     * </p><p>
     * From the matrix point of view, the transformation matrix of this
     * transform is multiplied on the left by the transformation matrix of
     * the specified rotation.
     * </p>
     * @param angle the angle of the rotation in degrees
     * @param pivot the rotation pivot point
     * @param axis the rotation axis
     * @throws NullPointerException if the specified {@code pivot}
     *         or {@code axis} is null
     * @since JavaFX 8.0
     */
    public void prependRotation(double angle, Point3D pivot, Point3D axis) {
        prependRotation(angle, pivot.getX(), pivot.getY(), pivot.getZ(),
                axis.getX(), axis.getY(), axis.getZ());
    }

    /**
     * Implementation of the {@code prependRotation()} around an arbitrary axis.
     */
    private void preRotate3D(double angle,
            double axisX, double axisY, double axisZ) {

        if (axisX == 0.0 && axisY == 0.0) {
            if (axisZ > 0.0) {
                preRotate3D(angle);
            } else if (axisZ < 0.0) {
                preRotate3D(-angle);
            } // else rotating about zero vector - NOP
            return;
        }

        double mag = Math.sqrt(axisX * axisX + axisY * axisY + axisZ * axisZ);

        if (mag == 0.0) {
            return;
        }

        mag = 1.0 / mag;
        final double ax = axisX * mag;
        final double ay = axisY * mag;
        final double az = axisZ * mag;

        final double sinTheta = Math.sin(Math.toRadians(angle));
        final double cosTheta = Math.cos(Math.toRadians(angle));
        final double t = 1.0 - cosTheta;

        final double xz = ax * az;
        final double xy = ax * ay;
        final double yz = ay * az;

        final double Txx = t * ax * ax + cosTheta;
        final double Txy = t * xy - sinTheta * az;
        final double Txz = t * xz + sinTheta * ay;

        final double Tyx = t * xy + sinTheta * az;
        final double Tyy = t * ay * ay + cosTheta;
        final double Tyz = t * yz - sinTheta * ax;

        final double Tzx = t * xz - sinTheta * ay;
        final double Tzy = t * yz + sinTheta * ax;
        final double Tzz = t * az * az + cosTheta;

        switch (state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                switch (state2d) {
                    default:
                        stateError();
                        // cannot reach
                    case APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE:
                        final double xx_sst = getMxx();
                        final double xy_sst = getMxy();
                        final double tx_sst = getTx();
                        final double yx_sst = getMyx();
                        final double yy_sst = getMyy();
                        final double ty_sst = getTy();
                        setMxx(Txx * xx_sst + Txy * yx_sst);
                        setMxy(Txx * xy_sst + Txy * yy_sst);
                        setMxz(Txz);
                        setTx( Txx * tx_sst  + Txy * ty_sst);
                        setMyx(Tyx * xx_sst + Tyy * yx_sst);
                        setMyy(Tyx * xy_sst + Tyy * yy_sst);
                        setMyz(Tyz);
                        setTy( Tyx * tx_sst  + Tyy * ty_sst);
                        setMzx(Tzx * xx_sst + Tzy * yx_sst);
                        setMzy(Tzx * xy_sst + Tzy * yy_sst);
                        setMzz(Tzz);
                        setTz( Tzx * tx_sst  + Tzy * ty_sst);
                        break;
                    case APPLY_SHEAR | APPLY_SCALE:
                        final double xx_ss = getMxx();
                        final double xy_ss = getMxy();
                        final double yx_ss = getMyx();
                        final double yy_ss = getMyy();
                        setMxx(Txx * xx_ss + Txy * yx_ss);
                        setMxy(Txx * xy_ss + Txy * yy_ss);
                        setMxz(Txz);
                        setMyx(Tyx * xx_ss + Tyy * yx_ss);
                        setMyy(Tyx * xy_ss + Tyy * yy_ss);
                        setMyz(Tyz);
                        setMzx(Tzx * xx_ss + Tzy * yx_ss);
                        setMzy(Tzx * xy_ss + Tzy * yy_ss);
                        setMzz(Tzz);
                        break;
                    case APPLY_SHEAR | APPLY_TRANSLATE:
                        final double xy_sht = getMxy();
                        final double tx_sht = getTx();
                        final double yx_sht = getMyx();
                        final double ty_sht = getTy();
                        setMxx(Txy * yx_sht);
                        setMxy(Txx * xy_sht);
                        setMxz(Txz);
                        setTx( Txx * tx_sht  + Txy * ty_sht);
                        setMyx(Tyy * yx_sht);
                        setMyy(Tyx * xy_sht);
                        setMyz(Tyz);
                        setTy( Tyx * tx_sht  + Tyy * ty_sht);
                        setMzx(Tzy * yx_sht);
                        setMzy(Tzx * xy_sht);
                        setMzz(Tzz);
                        setTz( Tzx * tx_sht  + Tzy * ty_sht);
                        break;
                    case APPLY_SHEAR:
                        final double xy_sh = getMxy();
                        final double yx_sh = getMyx();
                        setMxx(Txy * yx_sh);
                        setMxy(Txx * xy_sh);
                        setMxz(Txz);
                        setMyx(Tyy * yx_sh);
                        setMyy(Tyx * xy_sh);
                        setMyz(Tyz);
                        setMzx(Tzy * yx_sh);
                        setMzy(Tzx * xy_sh);
                        setMzz(Tzz);
                        break;
                    case APPLY_SCALE | APPLY_TRANSLATE:
                        final double xx_st = getMxx();
                        final double tx_st = getTx();
                        final double yy_st = getMyy();
                        final double ty_st = getTy();
                        setMxx(Txx * xx_st);
                        setMxy(Txy * yy_st);
                        setMxz(Txz);
                        setTx( Txx * tx_st  + Txy * ty_st);
                        setMyx(Tyx * xx_st);
                        setMyy(Tyy * yy_st);
                        setMyz(Tyz);
                        setTy( Tyx * tx_st  + Tyy * ty_st);
                        setMzx(Tzx * xx_st);
                        setMzy(Tzy * yy_st);
                        setMzz(Tzz);
                        setTz( Tzx * tx_st  + Tzy * ty_st);
                        break;
                    case APPLY_SCALE:
                        final double xx_s = getMxx();
                        final double yy_s = getMyy();
                        setMxx(Txx * xx_s);
                        setMxy(Txy * yy_s);
                        setMxz(Txz);
                        setMyx(Tyx * xx_s);
                        setMyy(Tyy * yy_s);
                        setMyz(Tyz);
                        setMzx(Tzx * xx_s);
                        setMzy(Tzy * yy_s);
                        setMzz(Tzz);
                        break;
                    case APPLY_TRANSLATE:
                        final double tx_t = getTx();
                        final double ty_t = getTy();
                        setMxx(Txx);
                        setMxy(Txy);
                        setMxz(Txz);
                        setTx( Txx * tx_t  + Txy * ty_t);
                        setMyx(Tyx);
                        setMyy(Tyy);
                        setMyz(Tyz);
                        setTy( Tyx * tx_t  + Tyy * ty_t);
                        setMzx(Tzx);
                        setMzy(Tzy);
                        setMzz(Tzz);
                        setTz( Tzx * tx_t  + Tzy * ty_t);
                        break;
                    case APPLY_IDENTITY:
                        setMxx(Txx);
                        setMxy(Txy);
                        setMxz(Txz);
                        setMyx(Tyx);
                        setMyy(Tyy);
                        setMyz(Tyz);
                        setMzx(Tzx);
                        setMzy(Tzy);
                        setMzz(Tzz);
                        break;
                }
                break;
            case APPLY_TRANSLATE:
                final double tx_t = getTx();
                final double ty_t = getTy();
                final double tz_t = getTz();
                setMxx(Txx);
                setMxy(Txy);
                setMxz(Txz);
                setMyx(Tyx);
                setMyy(Tyy);
                setMyz(Tyz);
                setMzx(Tzx);
                setMzy(Tzy);
                setMzz(Tzz);
                setTx( Txx * tx_t  + Txy * ty_t  + Txz * tz_t);
                setTy( Tyx * tx_t  + Tyy * ty_t  + Tyz * tz_t);
                setTz( Tzx * tx_t  + Tzy * ty_t  + Tzz * tz_t);
                break;
            case APPLY_SCALE:
                final double xx_s = getMxx();
                final double yy_s = getMyy();
                final double zz_s = getMzz();
                setMxx(Txx * xx_s);
                setMxy(Txy * yy_s);
                setMxz(Txz * zz_s);
                setMyx(Tyx * xx_s);
                setMyy(Tyy * yy_s);
                setMyz(Tyz * zz_s);
                setMzx(Tzx * xx_s);
                setMzy(Tzy * yy_s);
                setMzz(Tzz * zz_s);
                break;
            case APPLY_SCALE | APPLY_TRANSLATE:
                final double xx_st = getMxx();
                final double tx_st = getTx();
                final double yy_st = getMyy();
                final double ty_st = getTy();
                final double zz_st = getMzz();
                final double tz_st = getTz();
                setMxx(Txx * xx_st);
                setMxy(Txy * yy_st);
                setMxz(Txz * zz_st);
                setTx( Txx * tx_st  + Txy * ty_st  + Txz * tz_st);
                setMyx(Tyx * xx_st);
                setMyy(Tyy * yy_st);
                setMyz(Tyz * zz_st);
                setTy( Tyx * tx_st  + Tyy * ty_st  + Tyz * tz_st);
                setMzx(Tzx * xx_st);
                setMzy(Tzy * yy_st);
                setMzz(Tzz * zz_st);
                setTz( Tzx * tx_st  + Tzy * ty_st  + Tzz * tz_st);
                break;
            case APPLY_3D_COMPLEX:
                final double m_xx = getMxx();
                final double m_xy = getMxy();
                final double m_xz = getMxz();
                final double t_x = getTx();
                final double m_yx = getMyx();
                final double m_yy = getMyy();
                final double m_yz = getMyz();
                final double t_y = getTy();
                final double m_zx = getMzx();
                final double m_zy = getMzy();
                final double m_zz = getMzz();
                final double t_z = getTz();
                setMxx(Txx * m_xx + Txy * m_yx + Txz * m_zx /* + Ttx * 0.0 */);
                setMxy(Txx * m_xy + Txy * m_yy + Txz * m_zy /* + Ttx * 0.0 */);
                setMxz(Txx * m_xz + Txy * m_yz + Txz * m_zz /* + Ttx * 0.0 */);
                setTx( Txx * t_x  + Txy * t_y  + Txz * t_z  /* + Ttx * 0.0 */);
                setMyx(Tyx * m_xx + Tyy * m_yx + Tyz * m_zx /* + Tty * 0.0 */);
                setMyy(Tyx * m_xy + Tyy * m_yy + Tyz * m_zy /* + Tty * 0.0 */);
                setMyz(Tyx * m_xz + Tyy * m_yz + Tyz * m_zz /* + Tty * 0.0 */);
                setTy( Tyx * t_x  + Tyy * t_y  + Tyz * t_z  /* + Tty * 0.0 */);
                setMzx(Tzx * m_xx + Tzy * m_yx + Tzz * m_zx /* + Ttz * 0.0 */);
                setMzy(Tzx * m_xy + Tzy * m_yy + Tzz * m_zy /* + Ttz * 0.0 */);
                setMzz(Tzx * m_xz + Tzy * m_yz + Tzz * m_zz /* + Ttz * 0.0 */);
                setTz( Tzx * t_x  + Tzy * t_y  + Tzz * t_z  /* + Ttz * 0.0 */);
                break;
        }

        updateState();
    }

    /**
     * 2D implementation of {@code prependRotation}.
     * If this is a 3D transform, the call is redirected to {@code preRotate3D()}.
     */
    private void preRotate2D(double theta) {

        if (state3d != APPLY_NON_3D) {
            preRotate3D(theta);
            return;
        }

        double sin = Math.sin(Math.toRadians(theta));
        if (sin == 1.0) {
            preRotate2D_90();
        } else if (sin == -1.0) {
            preRotate2D_270();
        } else {
            double cos = Math.cos(Math.toRadians(theta));
            if (cos == -1.0) {
                preRotate2D_180();
            } else if (cos != 1.0) {
                double M0, M1;
                M0 = getMxx();
                M1 = getMyx();
                setMxx(cos * M0 - sin * M1);
                setMyx(sin * M0 + cos * M1);
                M0 = getMxy();
                M1 = getMyy();
                setMxy(cos * M0 - sin * M1);
                setMyy(sin * M0 + cos * M1);
                M0 = getTx();
                M1 = getTy();
                setTx(cos * M0 - sin * M1);
                setTy(sin * M0 + cos * M1);
                updateState2D();
            }
        }
    }

    /**
     * 2D implementation of {@code prependRotation} for 90 degrees rotation
     * around Z axis.
     * Behaves wrong when called for a 3D transform.
     */
    private void preRotate2D_90() {
        double M0 = getMxx();
        setMxx(-getMyx());
        setMyx(M0);
        M0 = getMxy();
        setMxy(-getMyy());
        setMyy(M0);
        M0 = getTx();
        setTx(-getTy());
        setTy(M0);

        int newstate = rot90conversion[state2d];
        if ((newstate & (APPLY_SHEAR | APPLY_SCALE)) == APPLY_SCALE &&
                getMxx() == 1.0 && getMyy() == 1.0) {
            newstate -= APPLY_SCALE;
        } else if ((newstate & (APPLY_SHEAR | APPLY_SCALE)) == APPLY_SHEAR &&
                getMxy() == 0.0 && getMyx() == 0.0) {
            newstate = (newstate & ~APPLY_SHEAR | APPLY_SCALE);
        }
        state2d = newstate;
    }

    /**
     * 2D implementation of {@code prependRotation} for 180 degrees rotation
     * around Z axis.
     * Behaves wrong when called for a 3D transform.
     */
    private void preRotate2D_180() {
        setMxx(-getMxx());
        setMxy(-getMxy());
        setTx(-getTx());
        setMyx(-getMyx());
        setMyy(-getMyy());
        setTy(-getTy());

        if ((state2d & APPLY_SHEAR) != 0) {
            if (getMxx() == 0.0 && getMyy() == 0.0) {
                state2d &= ~APPLY_SCALE;
            } else {
                state2d |= APPLY_SCALE;
            }
        } else {
            if (getMxx() == 1.0 && getMyy() == 1.0) {
                state2d &= ~APPLY_SCALE;
            } else {
                state2d |= APPLY_SCALE;
            }
        }
    }

    /**
     * 2D implementation of {@code prependRotation} for 270 degrees rotation
     * around Z axis.
     * Behaves wrong when called for a 3D transform.
     */
    private void preRotate2D_270() {
        double M0 = getMxx();
        setMxx(getMyx());
        setMyx(-M0);
        M0 = getMxy();
        setMxy(getMyy());
        setMyy(-M0);
        M0 = getTx();
        setTx(getTy());
        setTy(-M0);

        int newstate = rot90conversion[state2d];
        if ((newstate & (APPLY_SHEAR | APPLY_SCALE)) == APPLY_SCALE &&
                getMxx() == 1.0 && getMyy() == 1.0) {
            newstate -= APPLY_SCALE;
        } else if ((newstate & (APPLY_SHEAR | APPLY_SCALE)) == APPLY_SHEAR &&
                getMxy() == 0.0 && getMyx() == 0.0) {
            newstate = (newstate & ~APPLY_SHEAR | APPLY_SCALE);
        }
        state2d = newstate;
    }

    /**
     * 3D implementation of {@code prependRotation} around Z axis.
     * If this is a 2D transform, the call is redirected to {@code preRotate2D()}.
     */
    private void preRotate3D(double theta) {
        if (state3d == APPLY_NON_3D) {
            preRotate2D(theta);
            return;
        }

        double sin = Math.sin(Math.toRadians(theta));
        if (sin == 1.0) {
            preRotate3D_90();
        } else if (sin == -1.0) {
            preRotate3D_270();
        } else {
            double cos = Math.cos(Math.toRadians(theta));
            if (cos == -1.0) {
                preRotate3D_180();
            } else if (cos != 1.0) {
                double M0, M1;
                M0 = getMxx();
                M1 = getMyx();
                setMxx(cos * M0 - sin * M1);
                setMyx(sin * M0 + cos * M1);
                M0 = getMxy();
                M1 = getMyy();
                setMxy(cos * M0 - sin * M1);
                setMyy(sin * M0 + cos * M1);
                M0 = getMxz();
                M1 = getMyz();
                setMxz(cos * M0 - sin * M1);
                setMyz(sin * M0 + cos * M1);
                M0 = getTx();
                M1 = getTy();
                setTx(cos * M0 - sin * M1);
                setTy(sin * M0 + cos * M1);
                updateState();
            }
        }
    }

    /**
     * 3D implementation of {@code prependRotation} for 90 degrees rotation
     * around Z axis.
     * Behaves wrong when called for a 2D transform.
     */
    private void preRotate3D_90() {
        double M0 = getMxx();
        setMxx(-getMyx());
        setMyx(M0);
        M0 = getMxy();
        setMxy(-getMyy());
        setMyy(M0);
        M0 = getMxz();
        setMxz(-getMyz());
        setMyz(M0);
        M0 = getTx();
        setTx(-getTy());
        setTy(M0);

        switch(state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_TRANSLATE:
                state3d = APPLY_3D_COMPLEX;
                return;
            case APPLY_SCALE:
            case APPLY_SCALE | APPLY_TRANSLATE:
                if (getMxy() != 0.0 || getMyx() != 0.0) {
                    state3d = APPLY_3D_COMPLEX;
                }
                return;
            case APPLY_3D_COMPLEX:
                updateState();
                return;
        }
    }

    /**
     * 3D implementation of {@code prependRotation} for 180 degrees rotation
     * around Z axis.
     * Behaves wrong when called for a 2D transform.
     */
    private void preRotate3D_180() {
        final double mxx = getMxx();
        final double myy = getMyy();
        setMxx(-mxx);
        setMyy(-myy);
        setTx(-getTx());
        setTy(-getTy());

        if (state3d == APPLY_3D_COMPLEX) {
            setMxy(-getMxy());
            setMxz(-getMxz());
            setMyx(-getMyx());
            setMyz(-getMyz());
            updateState();
            return;
        }

        if (mxx == -1.0 && myy == -1.0 && getMzz() == 1.0) {
            // must have been 3d because of translation, which remained
            state3d &= ~APPLY_SCALE;
        } else {
            state3d |= APPLY_SCALE;
        }
    }

    /**
     * 3D implementation of {@code prependRotation} for 270 degrees rotation
     * around Z axis.
     * Behaves wrong when called for a 2D transform.
     */
    private void preRotate3D_270() {
        double M0 = getMxx();
        setMxx(getMyx());
        setMyx(-M0);
        M0 = getMxy();
        setMxy(getMyy());
        setMyy(-M0);
        M0 = getMxz();
        setMxz(getMyz());
        setMyz(-M0);
        M0 = getTx();
        setTx(getTy());
        setTy(-M0);

        switch(state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_TRANSLATE:
                state3d = APPLY_3D_COMPLEX;
                return;
            case APPLY_SCALE:
            case APPLY_SCALE | APPLY_TRANSLATE:
                if (getMxy() != 0.0 || getMyx() != 0.0) {
                    state3d = APPLY_3D_COMPLEX;
                }
                return;
            case APPLY_3D_COMPLEX:
                updateState();
                return;
        }
    }

    /* *************************************************************************
     *                                                                         *
     *                     Transform, Inverse Transform                        *
     *                                                                         *
     **************************************************************************/

    @Override
    public Point2D transform(double x, double y) {
        ensureCanTransform2DPoint();

        switch (state2d) {
            default:
                stateError();
                // cannot reach
            case APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE:
                return new Point2D(
                    getMxx() * x + getMxy() * y + getTx(),
                    getMyx() * x + getMyy() * y + getTy());
            case APPLY_SHEAR | APPLY_SCALE:
                return new Point2D(
                    getMxx() * x + getMxy() * y,
                    getMyx() * x + getMyy() * y);
            case APPLY_SHEAR | APPLY_TRANSLATE:
                return new Point2D(
                        getMxy() * y + getTx(),
                        getMyx() * x + getTy());
            case APPLY_SHEAR:
                return new Point2D(getMxy() * y, getMyx() * x);
            case APPLY_SCALE | APPLY_TRANSLATE:
                return new Point2D(
                        getMxx() * x + getTx(),
                        getMyy() * y + getTy());
            case APPLY_SCALE:
                return new Point2D(getMxx() * x, getMyy() * y);
            case APPLY_TRANSLATE:
                return new Point2D(x + getTx(), y + getTy());
            case APPLY_IDENTITY:
                return new Point2D(x, y);
        }
    }

    @Override
    public Point3D transform(double x, double y, double z) {
        switch (state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                switch (state2d) {
                    default:
                        stateError();
                        // cannot reach
                    case APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE:
                        return new Point3D(
                            getMxx() * x + getMxy() * y + getTx(),
                            getMyx() * x + getMyy() * y + getTy(), z);
                    case APPLY_SHEAR | APPLY_SCALE:
                        return new Point3D(
                            getMxx() * x + getMxy() * y,
                            getMyx() * x + getMyy() * y, z);
                    case APPLY_SHEAR | APPLY_TRANSLATE:
                        return new Point3D(
                                getMxy() * y + getTx(), getMyx() * x + getTy(),
                                z);
                    case APPLY_SHEAR:
                        return new Point3D(getMxy() * y, getMyx() * x, z);
                    case APPLY_SCALE | APPLY_TRANSLATE:
                        return new Point3D(
                                getMxx() * x + getTx(), getMyy() * y + getTy(),
                                z);
                    case APPLY_SCALE:
                        return new Point3D(getMxx() * x, getMyy() * y, z);
                    case APPLY_TRANSLATE:
                        return new Point3D(x + getTx(), y + getTy(), z);
                    case APPLY_IDENTITY:
                        return new Point3D(x, y, z);
                }
            case APPLY_TRANSLATE:
                return new Point3D(x + getTx(), y + getTy(), z + getTz());
            case APPLY_SCALE:
                return new Point3D(getMxx() * x, getMyy() * y, getMzz() * z);
            case APPLY_SCALE | APPLY_TRANSLATE:
                return new Point3D(
                        getMxx() * x + getTx(),
                        getMyy() * y + getTy(),
                        getMzz() * z + getTz());
            case APPLY_3D_COMPLEX:
                return new Point3D(
                    getMxx() * x + getMxy() * y + getMxz() * z + getTx(),
                    getMyx() * x + getMyy() * y + getMyz() * z + getTy(),
                    getMzx() * x + getMzy() * y + getMzz() * z + getTz());
        }
    }

    @Override
    void transform2DPointsImpl(double[] srcPts, int srcOff,
            double[] dstPts, int dstOff, int numPts) {

        double mxx, mxy, tx, myx, myy, ty;

        switch (state2d) {
            default:
                stateError();
                // cannot reach
            case APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE:
                mxx = getMxx(); mxy = getMxy(); tx = getTx();
                myx = getMyx(); myy = getMyy(); ty = getTy();
                while (--numPts >= 0) {
                    final double x = srcPts[srcOff++];
                    final double y = srcPts[srcOff++];
                    dstPts[dstOff++] = mxx * x + mxy * y + tx;
                    dstPts[dstOff++] = myx * x + myy * y + ty;
                }
                return;
            case APPLY_SHEAR | APPLY_SCALE:
                mxx = getMxx(); mxy = getMxy();
                myx = getMyx(); myy = getMyy();
                while (--numPts >= 0) {
                    final double x = srcPts[srcOff++];
                    final double y = srcPts[srcOff++];
                    dstPts[dstOff++] = mxx * x + mxy * y;
                    dstPts[dstOff++] = myx * x + myy * y;
                }
                return;
            case APPLY_SHEAR | APPLY_TRANSLATE:
                mxy = getMxy(); tx = getTx();
                myx = getMyx(); ty = getTy();
                while (--numPts >= 0) {
                    final double x = srcPts[srcOff++];
                    dstPts[dstOff++] = mxy * srcPts[srcOff++] + tx;
                    dstPts[dstOff++] = myx * x + ty;
                }
                return;
            case APPLY_SHEAR:
                mxy = getMxy();
                myx = getMyx();
                while (--numPts >= 0) {
                    final double x = srcPts[srcOff++];
                    dstPts[dstOff++] = mxy * srcPts[srcOff++];
                    dstPts[dstOff++] = myx * x;
                }
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
                mxx = getMxx(); tx = getTx();
                myy = getMyy(); ty = getTy();
                while (--numPts >= 0) {
                    dstPts[dstOff++] = mxx * srcPts[srcOff++] + tx;
                    dstPts[dstOff++] = myy * srcPts[srcOff++] + ty;
                }
                return;
            case APPLY_SCALE:
                mxx = getMxx();
                myy = getMyy();
                while (--numPts >= 0) {
                    dstPts[dstOff++] = mxx * srcPts[srcOff++];
                    dstPts[dstOff++] = myy * srcPts[srcOff++];
                }
                return;
            case APPLY_TRANSLATE:
                tx = getTx();
                ty = getTy();
                while (--numPts >= 0) {
                    dstPts[dstOff++] = srcPts[srcOff++] + tx;
                    dstPts[dstOff++] = srcPts[srcOff++] + ty;
                }
                return;
            case APPLY_IDENTITY:
                if (srcPts != dstPts || srcOff != dstOff) {
                    System.arraycopy(srcPts, srcOff, dstPts, dstOff,
                                     numPts * 2);
                }
                return;
        }
    }

    @Override
    void transform3DPointsImpl(double[] srcPts, int srcOff,
            double[] dstPts, int dstOff, int numPts) {

        double mxx, mxy, tx, myx, myy, ty, mzz, tz;

        switch(state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                switch (state2d) {
                    default:
                        stateError();
                        // cannot reach
                    case APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE:
                        mxx = getMxx(); mxy = getMxy(); tx = getTx();
                        myx = getMyx(); myy = getMyy(); ty = getTy();
                        while (--numPts >= 0) {
                            final double x = srcPts[srcOff++];
                            final double y = srcPts[srcOff++];
                            dstPts[dstOff++] = mxx * x + mxy * y + tx;
                            dstPts[dstOff++] = myx * x + myy * y + ty;
                            dstPts[dstOff++] = srcPts[srcOff++];
                        }
                        return;
                    case APPLY_SHEAR | APPLY_SCALE:
                        mxx = getMxx(); mxy = getMxy();
                        myx = getMyx(); myy = getMyy();
                        while (--numPts >= 0) {
                            final double x = srcPts[srcOff++];
                            final double y = srcPts[srcOff++];
                            dstPts[dstOff++] = mxx * x + mxy * y;
                            dstPts[dstOff++] = myx * x + myy * y;
                            dstPts[dstOff++] = srcPts[srcOff++];
                        }
                        return;
                    case APPLY_SHEAR | APPLY_TRANSLATE:
                        mxy = getMxy(); tx = getTx();
                        myx = getMyx(); ty = getTy();
                        while (--numPts >= 0) {
                            final double x = srcPts[srcOff++];
                            dstPts[dstOff++] = mxy * srcPts[srcOff++] + tx;
                            dstPts[dstOff++] = myx * x + ty;
                            dstPts[dstOff++] = srcPts[srcOff++];
                        }
                        return;
                    case APPLY_SHEAR:
                        mxy = getMxy();
                        myx = getMyx();
                        while (--numPts >= 0) {
                            final double x = srcPts[srcOff++];
                            dstPts[dstOff++] = mxy * srcPts[srcOff++];
                            dstPts[dstOff++] = myx * x;
                            dstPts[dstOff++] = srcPts[srcOff++];
                        }
                        return;
                    case APPLY_SCALE | APPLY_TRANSLATE:
                        mxx = getMxx(); tx = getTx();
                        myy = getMyy(); ty = getTy();
                        while (--numPts >= 0) {
                            dstPts[dstOff++] = mxx * srcPts[srcOff++] + tx;
                            dstPts[dstOff++] = myy * srcPts[srcOff++] + ty;
                            dstPts[dstOff++] = srcPts[srcOff++];
                        }
                        return;
                    case APPLY_SCALE:
                        mxx = getMxx();
                        myy = getMyy();
                        while (--numPts >= 0) {
                            dstPts[dstOff++] = mxx * srcPts[srcOff++];
                            dstPts[dstOff++] = myy * srcPts[srcOff++];
                            dstPts[dstOff++] = srcPts[srcOff++];
                        }
                        return;
                    case APPLY_TRANSLATE:
                        tx = getTx();
                        ty = getTy();
                        while (--numPts >= 0) {
                            dstPts[dstOff++] = srcPts[srcOff++] + tx;
                            dstPts[dstOff++] = srcPts[srcOff++] + ty;
                            dstPts[dstOff++] = srcPts[srcOff++];
                        }
                        return;
                    case APPLY_IDENTITY:
                        if (srcPts != dstPts || srcOff != dstOff) {
                            System.arraycopy(srcPts, srcOff, dstPts, dstOff,
                                             numPts * 3);
                        }
                        return;
                }
                // cannot reach
            case APPLY_TRANSLATE:
                tx = getTx();
                ty = getTy();
                tz = getTz();
                while (--numPts >= 0) {
                    dstPts[dstOff++] = srcPts[srcOff++] + tx;
                    dstPts[dstOff++] = srcPts[srcOff++] + ty;
                    dstPts[dstOff++] = srcPts[srcOff++] + tz;
                }
                return;
            case APPLY_SCALE:
                mxx = getMxx();
                myy = getMyy();
                mzz = getMzz();
                while (--numPts >= 0) {
                    dstPts[dstOff++] = mxx * srcPts[srcOff++];
                    dstPts[dstOff++] = myy * srcPts[srcOff++];
                    dstPts[dstOff++] = mzz * srcPts[srcOff++];
                }
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
                mxx = getMxx(); tx = getTx();
                myy = getMyy(); ty = getTy();
                mzz = getMzz(); tz = getTz();
                while (--numPts >= 0) {
                    dstPts[dstOff++] = mxx * srcPts[srcOff++] + tx;
                    dstPts[dstOff++] = myy * srcPts[srcOff++] + ty;
                    dstPts[dstOff++] = mzz * srcPts[srcOff++] + tz;
                }
                return;
            case APPLY_3D_COMPLEX:
                mxx = getMxx();
                mxy = getMxy();
                double mxz = getMxz();
                tx = getTx();
                myx = getMyx();
                myy = getMyy();
                double myz = getMyz();
                ty = getTy();
                double mzx = getMzx();
                double mzy = getMzy();
                mzz = getMzz();
                tz = getTz();

                while (--numPts >= 0) {
                    final double x = srcPts[srcOff++];
                    final double y = srcPts[srcOff++];
                    final double z = srcPts[srcOff++];

                    dstPts[dstOff++] = mxx * x + mxy * y + mxz * z + tx;
                    dstPts[dstOff++] = myx * x + myy * y + myz * z + ty;
                    dstPts[dstOff++] = mzx * x + mzy * y + mzz * z + tz;
                }
                return;
        }
    }

    @Override
    public Point2D deltaTransform(double x, double y) {
        ensureCanTransform2DPoint();

        switch (state2d) {
            default:
                stateError();
                // cannot reach
            case APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE:
            case APPLY_SHEAR | APPLY_SCALE:
                return new Point2D(
                    getMxx() * x + getMxy() * y,
                    getMyx() * x + getMyy() * y);
            case APPLY_SHEAR | APPLY_TRANSLATE:
            case APPLY_SHEAR:
                return new Point2D(getMxy() * y, getMyx() * x);
            case APPLY_SCALE | APPLY_TRANSLATE:
            case APPLY_SCALE:
                return new Point2D(getMxx() * x, getMyy() * y);
            case APPLY_TRANSLATE:
            case APPLY_IDENTITY:
                return new Point2D(x, y);
        }
    }

    @Override
    public Point3D deltaTransform(double x, double y, double z) {
        switch (state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                switch (state2d) {
                    default:
                        stateError();
                        // cannot reach
                    case APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE:
                    case APPLY_SHEAR | APPLY_SCALE:
                        return new Point3D(
                            getMxx() * x + getMxy() * y,
                            getMyx() * x + getMyy() * y, z);
                    case APPLY_SHEAR | APPLY_TRANSLATE:
                    case APPLY_SHEAR:
                        return new Point3D(getMxy() * y, getMyx() * x, z);
                    case APPLY_SCALE | APPLY_TRANSLATE:
                    case APPLY_SCALE:
                        return new Point3D(getMxx() * x, getMyy() * y, z);
                    case APPLY_TRANSLATE:
                    case APPLY_IDENTITY:
                        return new Point3D(x, y, z);
                }
            case APPLY_TRANSLATE:
                return new Point3D(x, y, z);
            case APPLY_SCALE:
            case APPLY_SCALE | APPLY_TRANSLATE:
                return new Point3D(getMxx() * x, getMyy() * y, getMzz() * z);
            case APPLY_3D_COMPLEX:
                return new Point3D(
                    getMxx() * x + getMxy() * y + getMxz() * z,
                    getMyx() * x + getMyy() * y + getMyz() * z,
                    getMzx() * x + getMzy() * y + getMzz() * z);
        }
    }

    @Override
    public Point2D inverseTransform(double x, double y)
            throws NonInvertibleTransformException {
        ensureCanTransform2DPoint();

        switch (state2d) {
            default:
                return super.inverseTransform(x, y);
            case APPLY_SHEAR | APPLY_TRANSLATE:
                final double mxy_st = getMxy();
                final double myx_st = getMyx();
                if (mxy_st == 0.0 || myx_st == 0.0) {
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                return new Point2D(
                        (1.0 / myx_st) * y - getTy() / myx_st,
                        (1.0 / mxy_st) * x - getTx() / mxy_st);
            case APPLY_SHEAR:
                final double mxy_s = getMxy();
                final double myx_s = getMyx();
                if (mxy_s == 0.0 || myx_s == 0.0) {
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                return new Point2D((1.0 / myx_s) * y, (1.0 / mxy_s) * x);
            case APPLY_SCALE | APPLY_TRANSLATE:
                final double mxx_st = getMxx();
                final double myy_st = getMyy();
                if (mxx_st == 0.0 || myy_st == 0.0) {
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                return new Point2D(
                        (1.0 / mxx_st) * x - getTx() / mxx_st,
                        (1.0 / myy_st) * y - getTy() / myy_st);
            case APPLY_SCALE:
                final double mxx_s = getMxx();
                final double myy_s = getMyy();
                if (mxx_s == 0.0 || myy_s == 0.0) {
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                return new Point2D((1.0 / mxx_s) * x, (1.0 / myy_s) * y);
            case APPLY_TRANSLATE:
                return new Point2D(x - getTx(), y - getTy());
            case APPLY_IDENTITY:
                return new Point2D(x, y);
        }
    }

    @Override
    public Point3D inverseTransform(double x, double y, double z)
            throws NonInvertibleTransformException {
        switch(state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                switch (state2d) {
                    default:
                        return super.inverseTransform(x, y, z);
                    case APPLY_SHEAR | APPLY_TRANSLATE:
                        final double mxy_st = getMxy();
                        final double myx_st = getMyx();
                        if (mxy_st == 0.0 || myx_st == 0.0) {
                            throw new NonInvertibleTransformException(
                                    "Determinant is 0");
                        }
                        return new Point3D(
                                (1.0 / myx_st) * y - getTy() / myx_st,
                                (1.0 / mxy_st) * x - getTx() / mxy_st, z);
                    case APPLY_SHEAR:
                        final double mxy_s = getMxy();
                        final double myx_s = getMyx();
                        if (mxy_s == 0.0 || myx_s == 0.0) {
                            throw new NonInvertibleTransformException(
                                    "Determinant is 0");
                        }
                        return new Point3D(
                                (1.0 / myx_s) * y,
                                (1.0 / mxy_s) * x, z);
                    case APPLY_SCALE | APPLY_TRANSLATE:
                        final double mxx_st = getMxx();
                        final double myy_st = getMyy();
                        if (mxx_st == 0.0 || myy_st == 0.0) {
                            throw new NonInvertibleTransformException(
                                    "Determinant is 0");
                        }
                        return new Point3D(
                                (1.0 / mxx_st) * x - getTx() / mxx_st,
                                (1.0 / myy_st) * y - getTy() / myy_st, z);
                    case APPLY_SCALE:
                        final double mxx_s = getMxx();
                        final double myy_s = getMyy();
                        if (mxx_s == 0.0 || myy_s == 0.0) {
                            throw new NonInvertibleTransformException(
                                    "Determinant is 0");
                        }
                        return new Point3D((1.0 / mxx_s) * x, (1.0 / myy_s) * y, z);
                    case APPLY_TRANSLATE:
                        return new Point3D(x - getTx(), y - getTy(), z);
                    case APPLY_IDENTITY:
                        return new Point3D(x, y, z);
                }
            case APPLY_TRANSLATE:
                return new Point3D(x - getTx(), y - getTy(), z - getTz());
            case APPLY_SCALE:
                final double mxx_s = getMxx();
                final double myy_s = getMyy();
                final double mzz_s = getMzz();
                if (mxx_s == 0.0 || myy_s == 0.0 || mzz_s == 0.0) {
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                return new Point3D(
                        (1.0 / mxx_s) * x,
                        (1.0 / myy_s) * y,
                        (1.0 / mzz_s) * z);
            case APPLY_SCALE | APPLY_TRANSLATE:
                final double mxx_st = getMxx();
                final double myy_st = getMyy();
                final double mzz_st = getMzz();
                if (mxx_st == 0.0 || myy_st == 0.0 || mzz_st == 0.0) {
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                return new Point3D(
                        (1.0 / mxx_st) * x - getTx() / mxx_st,
                        (1.0 / myy_st) * y - getTy() / myy_st,
                        (1.0 / mzz_st) * z - getTz() / mzz_st);
            case APPLY_3D_COMPLEX:
                return super.inverseTransform(x, y, z);
        }
    }

    @Override
    void inverseTransform2DPointsImpl(double[] srcPts, int srcOff,
            double[] dstPts, int dstOff, int numPts)
            throws NonInvertibleTransformException {

        double mxx, mxy, tx, myx, myy, ty, tmp;

        switch (state2d) {
            default:
                super.inverseTransform2DPointsImpl(srcPts, srcOff,
                        dstPts, dstOff, numPts);
                return;

            case APPLY_SHEAR | APPLY_TRANSLATE:
                mxy = getMxy(); tx = getTx();
                myx = getMyx(); ty = getTy();
                if (mxy == 0.0 || myx == 0.0) {
                    throw new NonInvertibleTransformException("Determinant is 0");
                }

                tmp = tx;
                tx = -ty / myx;
                ty = -tmp / mxy;

                tmp = myx;
                myx = 1.0 / mxy;
                mxy = 1.0 / tmp;

                while (--numPts >= 0) {
                    final double x = srcPts[srcOff++];
                    dstPts[dstOff++] = mxy * srcPts[srcOff++] + tx;
                    dstPts[dstOff++] = myx * x + ty;
                }
                return;
            case APPLY_SHEAR:
                mxy = getMxy();
                myx = getMyx();
                if (mxy == 0.0 || myx == 0.0) {
                    throw new NonInvertibleTransformException("Determinant is 0");
                }

                tmp = myx;
                myx = 1.0 / mxy;
                mxy = 1.0 / tmp;

                while (--numPts >= 0) {
                    final double x = srcPts[srcOff++];
                    dstPts[dstOff++] = mxy * srcPts[srcOff++];
                    dstPts[dstOff++] = myx * x;
                }
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
                mxx = getMxx(); tx = getTx();
                myy = getMyy(); ty = getTy();
                if (mxx == 0.0 || myy == 0.0) {
                    throw new NonInvertibleTransformException("Determinant is 0");
                }

                tx = -tx / mxx;
                ty = -ty / myy;
                mxx = 1.0 / mxx;
                myy = 1.0 / myy;

                while (--numPts >= 0) {
                    dstPts[dstOff++] = mxx * srcPts[srcOff++] + tx;
                    dstPts[dstOff++] = myy * srcPts[srcOff++] + ty;
                }
                return;
            case APPLY_SCALE:
                mxx = getMxx();
                myy = getMyy();
                if (mxx == 0.0 || myy == 0.0) {
                    throw new NonInvertibleTransformException("Determinant is 0");
                }

                mxx = 1.0 / mxx;
                myy = 1.0 / myy;

                while (--numPts >= 0) {
                    dstPts[dstOff++] = mxx * srcPts[srcOff++];
                    dstPts[dstOff++] = myy * srcPts[srcOff++];
                }
                return;
            case APPLY_TRANSLATE:
                tx = getTx();
                ty = getTy();
                while (--numPts >= 0) {
                    dstPts[dstOff++] = srcPts[srcOff++] - tx;
                    dstPts[dstOff++] = srcPts[srcOff++] - ty;
                }
                return;
            case APPLY_IDENTITY:
                if (srcPts != dstPts || srcOff != dstOff) {
                    System.arraycopy(srcPts, srcOff, dstPts, dstOff,
                                     numPts * 2);
                }
                return;
        }
    }

    @Override
    void inverseTransform3DPointsImpl(double[] srcPts, int srcOff,
            double[] dstPts, int dstOff, int numPts)
            throws NonInvertibleTransformException {

        double mxx, mxy, tx, myx, myy, ty, mzz, tz, tmp;

        switch (state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                switch (state2d) {
                    default:
                        super.inverseTransform3DPointsImpl(srcPts, srcOff,
                                dstPts, dstOff, numPts);
                        return;

                    case APPLY_SHEAR | APPLY_TRANSLATE:
                        mxy = getMxy(); tx = getTx();
                        myx = getMyx(); ty = getTy();
                        if (mxy == 0.0 || myx == 0.0) {
                            throw new NonInvertibleTransformException(
                                    "Determinant is 0");
                        }

                        tmp = tx;
                        tx = -ty / myx;
                        ty = -tmp / mxy;

                        tmp = myx;
                        myx = 1.0 / mxy;
                        mxy = 1.0 / tmp;

                        while (--numPts >= 0) {
                            final double x = srcPts[srcOff++];
                            dstPts[dstOff++] = mxy * srcPts[srcOff++] + tx;
                            dstPts[dstOff++] = myx * x + ty;
                            dstPts[dstOff++] = srcPts[srcOff++];
                        }
                        return;
                    case APPLY_SHEAR:
                        mxy = getMxy();
                        myx = getMyx();
                        if (mxy == 0.0 || myx == 0.0) {
                            throw new NonInvertibleTransformException(
                                    "Determinant is 0");
                        }

                        tmp = myx;
                        myx = 1.0 / mxy;
                        mxy = 1.0 / tmp;

                        while (--numPts >= 0) {
                            final double x = srcPts[srcOff++];
                            dstPts[dstOff++] = mxy * srcPts[srcOff++];
                            dstPts[dstOff++] = myx * x;
                            dstPts[dstOff++] = srcPts[srcOff++];
                        }
                        return;
                    case APPLY_SCALE | APPLY_TRANSLATE:
                        mxx = getMxx(); tx = getTx();
                        myy = getMyy(); ty = getTy();
                        if (mxx == 0.0 || myy == 0.0) {
                            throw new NonInvertibleTransformException(
                                    "Determinant is 0");
                        }

                        tx = -tx / mxx;
                        ty = -ty / myy;
                        mxx = 1.0 / mxx;
                        myy = 1.0 / myy;

                        while (--numPts >= 0) {
                            dstPts[dstOff++] = mxx * srcPts[srcOff++] + tx;
                            dstPts[dstOff++] = myy * srcPts[srcOff++] + ty;
                            dstPts[dstOff++] = srcPts[srcOff++];
                        }
                        return;
                    case APPLY_SCALE:
                        mxx = getMxx();
                        myy = getMyy();
                        if (mxx == 0.0 || myy == 0.0) {
                            throw new NonInvertibleTransformException(
                                    "Determinant is 0");
                        }

                        mxx = 1.0 / mxx;
                        myy = 1.0 / myy;

                        while (--numPts >= 0) {
                            dstPts[dstOff++] = mxx * srcPts[srcOff++];
                            dstPts[dstOff++] = myy * srcPts[srcOff++];
                            dstPts[dstOff++] = srcPts[srcOff++];
                        }
                        return;
                    case APPLY_TRANSLATE:
                        tx = getTx();
                        ty = getTy();
                        while (--numPts >= 0) {
                            dstPts[dstOff++] = srcPts[srcOff++] - tx;
                            dstPts[dstOff++] = srcPts[srcOff++] - ty;
                            dstPts[dstOff++] = srcPts[srcOff++];
                        }
                        return;
                    case APPLY_IDENTITY:
                        if (srcPts != dstPts || srcOff != dstOff) {
                            System.arraycopy(srcPts, srcOff, dstPts, dstOff,
                                             numPts * 3);
                        }
                        return;
                }
                // cannot reach
            case APPLY_TRANSLATE:
                tx = getTx();
                ty = getTy();
                tz = getTz();
                while (--numPts >= 0) {
                    dstPts[dstOff++] = srcPts[srcOff++] - tx;
                    dstPts[dstOff++] = srcPts[srcOff++] - ty;
                    dstPts[dstOff++] = srcPts[srcOff++] - tz;
                }
                return;
            case APPLY_SCALE:
                mxx = getMxx();
                myy = getMyy();
                mzz = getMzz();
                if (mxx == 0.0 || myy == 0.0 | mzz == 0.0) {
                    throw new NonInvertibleTransformException("Determinant is 0");
                }

                mxx = 1.0 / mxx;
                myy = 1.0 / myy;
                mzz = 1.0 / mzz;

                while (--numPts >= 0) {
                    dstPts[dstOff++] = mxx * srcPts[srcOff++];
                    dstPts[dstOff++] = myy * srcPts[srcOff++];
                    dstPts[dstOff++] = mzz * srcPts[srcOff++];
                }
                return;
            case APPLY_SCALE | APPLY_TRANSLATE:
                mxx = getMxx(); tx = getTx();
                myy = getMyy(); ty = getTy();
                mzz = getMzz(); tz = getTz();
                if (mxx == 0.0 || myy == 0.0 || mzz == 0.0) {
                    throw new NonInvertibleTransformException("Determinant is 0");
                }

                tx = -tx / mxx;
                ty = -ty / myy;
                tz = -tz / mzz;
                mxx = 1.0 / mxx;
                myy = 1.0 / myy;
                mzz = 1.0 / mzz;

                while (--numPts >= 0) {
                    dstPts[dstOff++] = mxx * srcPts[srcOff++] + tx;
                    dstPts[dstOff++] = myy * srcPts[srcOff++] + ty;
                    dstPts[dstOff++] = mzz * srcPts[srcOff++] + tz;
                }
                return;
            case APPLY_3D_COMPLEX:
                super.inverseTransform3DPointsImpl(srcPts, srcOff,
                        dstPts, dstOff, numPts);
                return;
        }
    }

    @Override
    public Point2D inverseDeltaTransform(double x, double y)
            throws NonInvertibleTransformException {
        ensureCanTransform2DPoint();

        switch (state2d) {
            default:
                return super.inverseDeltaTransform(x, y);
            case APPLY_SHEAR | APPLY_TRANSLATE:
            case APPLY_SHEAR:
                final double mxy_s = getMxy();
                final double myx_s = getMyx();
                if (mxy_s == 0.0 || myx_s == 0.0) {
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                return new Point2D((1.0 / myx_s) * y, (1.0 / mxy_s) * x);
            case APPLY_SCALE | APPLY_TRANSLATE:
            case APPLY_SCALE:
                final double mxx_s = getMxx();
                final double myy_s = getMyy();
                if (mxx_s == 0.0 || myy_s == 0.0) {
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                return new Point2D((1.0 / mxx_s) * x, (1.0 / myy_s) * y);
            case APPLY_TRANSLATE:
            case APPLY_IDENTITY:
                return new Point2D(x, y);
        }
    }

    @Override
    public Point3D inverseDeltaTransform(double x, double y, double z)
            throws NonInvertibleTransformException {
        switch(state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                switch (state2d) {
                    default:
                        return super.inverseDeltaTransform(x, y, z);
                    case APPLY_SHEAR | APPLY_TRANSLATE:
                    case APPLY_SHEAR:
                        final double mxy_s = getMxy();
                        final double myx_s = getMyx();
                        if (mxy_s == 0.0 || myx_s == 0.0) {
                            throw new NonInvertibleTransformException(
                                    "Determinant is 0");
                        }
                        return new Point3D(
                                (1.0 / myx_s) * y,
                                (1.0 / mxy_s) * x, z);
                    case APPLY_SCALE | APPLY_TRANSLATE:
                    case APPLY_SCALE:
                        final double mxx_s = getMxx();
                        final double myy_s = getMyy();
                        if (mxx_s == 0.0 || myy_s == 0.0) {
                            throw new NonInvertibleTransformException(
                                    "Determinant is 0");
                        }
                        return new Point3D(
                                (1.0 / mxx_s) * x,
                                (1.0 / myy_s) * y, z);
                    case APPLY_TRANSLATE:
                    case APPLY_IDENTITY:
                        return new Point3D(x, y, z);
                }

            case APPLY_TRANSLATE:
                return new Point3D(x, y, z);
            case APPLY_SCALE | APPLY_TRANSLATE:
            case APPLY_SCALE:
                final double mxx_s = getMxx();
                final double myy_s = getMyy();
                final double mzz_s = getMzz();
                if (mxx_s == 0.0 || myy_s == 0.0 || mzz_s == 0.0) {
                    throw new NonInvertibleTransformException("Determinant is 0");
                }
                return new Point3D(
                        (1.0 / mxx_s) * x,
                        (1.0 / myy_s) * y,
                        (1.0 / mzz_s) * z);
            case APPLY_3D_COMPLEX:
                return super.inverseDeltaTransform(x, y, z);
        }
    }

    /* *************************************************************************
     *                                                                         *
     *                               Other API                                 *
     *                                                                         *
     **************************************************************************/

    /**
     * Returns a string representation of this {@code Affine} object.
     * @return a string representation of this {@code Affine} object.
     */
    @Override
    public String toString() {
       final StringBuilder sb = new StringBuilder("Affine [\n");

        sb.append("\t").append(getMxx());
        sb.append(", ").append(getMxy());
        sb.append(", ").append(getMxz());
        sb.append(", ").append(getTx());
        sb.append('\n');
        sb.append("\t").append(getMyx());
        sb.append(", ").append(getMyy());
        sb.append(", ").append(getMyz());
        sb.append(", ").append(getTy());
        sb.append('\n');
        sb.append("\t").append(getMzx());
        sb.append(", ").append(getMzy());
        sb.append(", ").append(getMzz());
        sb.append(", ").append(getTz());

        return sb.append("\n]").toString();
    }

    /* *************************************************************************
     *                                                                         *
     *                    Internal implementation stuff                        *
     *                                                                         *
     **************************************************************************/

    /**
     * Manually recalculates the state of the transform when the matrix
     * changes too much to predict the effects on the state.
     * The following tables specify what the various settings of the
     * state fields say about the values of the corresponding matrix
     * element fields.
     *
     * <h4>state3d:</h4>
     * <pre>
     *                     SCALE          TRANSLATE        OTHER ELEMENTS
     *                 mxx, myy, mzz      tx, ty, tz       all remaining
     *
     * TRANSLATE (TR)       1.0           not all 0.0           0.0
     * SCALE (SC)       not all 1.0           0.0               0.0
     * TR | SC          not all 1.0       not all 0.0           0.0
     * 3D_COMPLEX           any               any            not all 0.0
     * NON_3D: mxz, myz, mzx, mzy, tz are 0.0, mzz is 1.0, for the rest
     *         see state2d
     * </pre>
     *
     * <h4>state2d:</h4>
     * Contains meaningful value only if state3d == APPLY_NON_3D.
     * Note that the rules governing the SCALE fields are slightly
     * different depending on whether the SHEAR flag is also set.
     * <pre>
     *                     SCALE            SHEAR          TRANSLATE
     *                    mxx/myy          mxy/myx           tx/ty
     *
     * IDENTITY             1.0              0.0              0.0
     * TRANSLATE (TR)       1.0              0.0          not both 0.0
     * SCALE (SC)       not both 1.0         0.0              0.0
     * TR | SC          not both 1.0         0.0          not both 0.0
     * SHEAR (SH)           0.0          not both 0.0         0.0
     * TR | SH              0.0          not both 0.0     not both 0.0
     * SC | SH          not both 0.0     not both 0.0         0.0
     * TR | SC | SH     not both 0.0     not both 0.0     not both 0.0
     * </pre>
     */
    private void updateState() {
        updateState2D();

        state3d = APPLY_NON_3D;

        if (getMxz() != 0.0 ||
            getMyz() != 0.0 ||
            getMzx() != 0.0 ||
            getMzy() != 0.0)
        {
            state3d = APPLY_3D_COMPLEX;
        } else {
            if ((state2d & APPLY_SHEAR) == 0) {
                if (getTz() != 0.0) {
                    state3d |= APPLY_TRANSLATE;
                }
                if (getMzz() != 1.0) {
                    state3d |= APPLY_SCALE;
                }
                if (state3d != APPLY_NON_3D) {
                    state3d |= (state2d & (APPLY_SCALE | APPLY_TRANSLATE));
                }
            } else {
                if (getMzz() != 1.0 || getTz() != 0.0) {
                    state3d = APPLY_3D_COMPLEX;
                }
            }
        }
    }

    /**
     * 2D part of {@code updateState()}. It is sufficient to call this method
     * when we know this this a 2D transform and the operation was 2D-only
     * so it could not switch the transform to 3D.
     */
    private void updateState2D() {
        if (getMxy() == 0.0 && getMyx() == 0.0) {
            if (getMxx() == 1.0 && getMyy() == 1.0) {
                if (getTx() == 0.0 && getTy() == 0.0) {
                    state2d = APPLY_IDENTITY;
                } else {
                    state2d = APPLY_TRANSLATE;
                }
            } else {
                if (getTx() == 0.0 && getTy() == 0.0) {
                    state2d = APPLY_SCALE;
                } else {
                    state2d = (APPLY_SCALE | APPLY_TRANSLATE);
                }
            }
        } else {
            if (getMxx() == 0.0 && getMyy() == 0.0) {
                if (getTx() == 0.0 && getTy() == 0.0) {
                    state2d = APPLY_SHEAR;
                } else {
                    state2d = (APPLY_SHEAR | APPLY_TRANSLATE);
                }
            } else {
                if (getTx() == 0.0 && getTy() == 0.0) {
                    state2d = (APPLY_SHEAR | APPLY_SCALE);
                } else {
                    state2d = (APPLY_SHEAR | APPLY_SCALE | APPLY_TRANSLATE);
                }
            }
        }
    }

    /**
     * Convenience method used internally to throw exceptions when
     * a case was forgotten in a switch statement.
     */
    private static void stateError() {
        throw new InternalError("missing case in a switch");
    }

    /**
     * @treatAsPrivate implementation detail
     * @deprecated This is an internal API that is not intended for use and will be removed in the next version
     */
    @Deprecated
    @Override
    public void impl_apply(final Affine3D trans) {
        trans.concatenate(getMxx(), getMxy(), getMxz(), getTx(),
                          getMyx(), getMyy(), getMyz(), getTy(),
                          getMzx(), getMzy(), getMzz(), getTz());
    }

    /**
     * @treatAsPrivate implementation detail
     * @deprecated This is an internal API that is not intended for use and will be removed in the next version
     */
    @Deprecated
    @Override
    public BaseTransform impl_derive(final BaseTransform trans) {
        switch(state3d) {
            default:
                stateError();
                // cannot reach
            case APPLY_NON_3D:
                switch(state2d) {
                    case APPLY_IDENTITY:
                        return trans;
                    case APPLY_TRANSLATE:
                        return trans.deriveWithTranslation(getTx(), getTy());
                    case APPLY_SCALE:
                        return trans.deriveWithScale(getMxx(), getMyy(), 1.0);
                    case APPLY_SCALE | APPLY_TRANSLATE:
                        // fall through
                    default:
                        return trans.deriveWithConcatenation(
                                getMxx(), getMyx(),
                                getMxy(), getMyy(),
                                getTx(), getTy());
                }
            case APPLY_TRANSLATE:
                return trans.deriveWithTranslation(getTx(), getTy(), getTz());
            case APPLY_SCALE:
                return trans.deriveWithScale(getMxx(), getMyy(), getMzz());
            case APPLY_SCALE | APPLY_TRANSLATE:
                // fall through
            case APPLY_3D_COMPLEX:
                return trans.deriveWithConcatenation(
                        getMxx(), getMxy(), getMxz(), getTx(),
                        getMyx(), getMyy(), getMyz(), getTy(),
                        getMzx(), getMzy(), getMzz(), getTz());
        }
    }

    /**
     * Keeps track of the atomic changes of more elements.
     * Don't forget to end or cancel a running atomic operation
     * when an exception is to be thrown during one.
     */
    private class AffineAtomicChange {
        private boolean running = false;

        private void start() {
            if (running) {
                throw new InternalError("Affine internal error: "
                        + "trying to run inner atomic operation");
            }
            if (mxx != null) mxx.preProcessAtomicChange();
            if (mxy != null) mxy.preProcessAtomicChange();
            if (mxz != null) mxz.preProcessAtomicChange();
            if (tx != null) tx.preProcessAtomicChange();
            if (myx != null) myx.preProcessAtomicChange();
            if (myy != null) myy.preProcessAtomicChange();
            if (myz != null) myz.preProcessAtomicChange();
            if (ty != null) ty.preProcessAtomicChange();
            if (mzx != null) mzx.preProcessAtomicChange();
            if (mzy != null) mzy.preProcessAtomicChange();
            if (mzz != null) mzz.preProcessAtomicChange();
            if (tz != null) tz.preProcessAtomicChange();
            running = true;
        }

        private void end() {
            running = false;
            transformChanged();
            if (mxx != null) mxx.postProcessAtomicChange();
            if (mxy != null) mxy.postProcessAtomicChange();
            if (mxz != null) mxz.postProcessAtomicChange();
            if (tx != null) tx.postProcessAtomicChange();
            if (myx != null) myx.postProcessAtomicChange();
            if (myy != null) myy.postProcessAtomicChange();
            if (myz != null) myz.postProcessAtomicChange();
            if (ty != null) ty.postProcessAtomicChange();
            if (mzx != null) mzx.postProcessAtomicChange();
            if (mzy != null) mzy.postProcessAtomicChange();
            if (mzz != null) mzz.postProcessAtomicChange();
            if (tz != null) tz.postProcessAtomicChange();
        }

        private void cancel() {
            running = false;
        }

        private boolean runs() {
            return running;
        }
    }

    /**
     * Used only by tests to check the 2d matrix state
     */
    int getState2d() {
        return state2d;
    }

    /**
     * Used only by tests to check the 3d matrix state
     */
    int getState3d() {
        return state3d;
    }

    /**
     * Used only by tests to check the atomic operation state
     */
    boolean atomicChangeRuns() {
        return atomicChange.runs();
    }
}
