/*
 * Copyright (c) 1997, 2006, Oracle and/or its affiliates. All rights reserved.
 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
 *
 * This code is free software; you can redistribute it and/or modify it
 * under the terms of the GNU General Public License version 2 only, as
 * published by the Free Software Foundation.  Oracle designates this
 * particular file as subject to the "Classpath" exception as provided
 * by Oracle in the LICENSE file that accompanied this code.
 *
 * This code is distributed in the hope that it will be useful, but WITHOUT
 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
 * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
 * version 2 for more details (a copy is included in the LICENSE file that
 * accompanied this code).
 *
 * You should have received a copy of the GNU General Public License version
 * 2 along with this work; if not, write to the Free Software Foundation,
 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
 *
 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
 * or visit www.oracle.com if you need additional information or have any
 * questions.
 */

package java.awt.geom;

import java.awt.Shape;
import java.awt.Rectangle;
import java.util.Arrays;
import java.io.Serializable;
import sun.awt.geom.Curve;

/**
 * The <code>CubicCurve2D</code> class defines a cubic parametric curve
 * segment in {@code (x,y)} coordinate space.
 * <p>
 * This class is only the abstract superclass for all objects which
 * store a 2D cubic curve segment.
 * The actual storage representation of the coordinates is left to
 * the subclass.
 *
 * @author      Jim Graham
 * @since 1.2
 */
public abstract class CubicCurve2D implements Shape, Cloneable {

    /**
     * A cubic parametric curve segment specified with
     * {@code float} coordinates.
     * @since 1.2
     */
    public static class Float extends CubicCurve2D implements Serializable {
        /**
         * The X coordinate of the start point
         * of the cubic curve segment.
         * @since 1.2
         * @serial
         */
        public float x1;

        /**
         * The Y coordinate of the start point
         * of the cubic curve segment.
         * @since 1.2
         * @serial
         */
        public float y1;

        /**
         * The X coordinate of the first control point
         * of the cubic curve segment.
         * @since 1.2
         * @serial
         */
        public float ctrlx1;

        /**
         * The Y coordinate of the first control point
         * of the cubic curve segment.
         * @since 1.2
         * @serial
         */
        public float ctrly1;

        /**
         * The X coordinate of the second control point
         * of the cubic curve segment.
         * @since 1.2
         * @serial
         */
        public float ctrlx2;

        /**
         * The Y coordinate of the second control point
         * of the cubic curve segment.
         * @since 1.2
         * @serial
         */
        public float ctrly2;

        /**
         * The X coordinate of the end point
         * of the cubic curve segment.
         * @since 1.2
         * @serial
         */
        public float x2;

        /**
         * The Y coordinate of the end point
         * of the cubic curve segment.
         * @since 1.2
         * @serial
         */
        public float y2;

        /**
         * Constructs and initializes a CubicCurve with coordinates
         * (0, 0, 0, 0, 0, 0, 0, 0).
         * @since 1.2
         */
        public Float() {
        }

        /**
         * Constructs and initializes a {@code CubicCurve2D} from
         * the specified {@code float} coordinates.
         *
         * @param x1 the X coordinate for the start point
         *           of the resulting {@code CubicCurve2D}
         * @param y1 the Y coordinate for the start point
         *           of the resulting {@code CubicCurve2D}
         * @param ctrlx1 the X coordinate for the first control point
         *               of the resulting {@code CubicCurve2D}
         * @param ctrly1 the Y coordinate for the first control point
         *               of the resulting {@code CubicCurve2D}
         * @param ctrlx2 the X coordinate for the second control point
         *               of the resulting {@code CubicCurve2D}
         * @param ctrly2 the Y coordinate for the second control point
         *               of the resulting {@code CubicCurve2D}
         * @param x2 the X coordinate for the end point
         *           of the resulting {@code CubicCurve2D}
         * @param y2 the Y coordinate for the end point
         *           of the resulting {@code CubicCurve2D}
         * @since 1.2
         */
        public Float(float x1, float y1,
                     float ctrlx1, float ctrly1,
                     float ctrlx2, float ctrly2,
                     float x2, float y2)
        {
            setCurve(x1, y1, ctrlx1, ctrly1, ctrlx2, ctrly2, x2, y2);
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public double getX1() {
            return (double) x1;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public double getY1() {
            return (double) y1;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public Point2D getP1() {
            return new Point2D.Float(x1, y1);
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public double getCtrlX1() {
            return (double) ctrlx1;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public double getCtrlY1() {
            return (double) ctrly1;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public Point2D getCtrlP1() {
            return new Point2D.Float(ctrlx1, ctrly1);
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public double getCtrlX2() {
            return (double) ctrlx2;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public double getCtrlY2() {
            return (double) ctrly2;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public Point2D getCtrlP2() {
            return new Point2D.Float(ctrlx2, ctrly2);
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public double getX2() {
            return (double) x2;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public double getY2() {
            return (double) y2;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public Point2D getP2() {
            return new Point2D.Float(x2, y2);
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public void setCurve(double x1, double y1,
                             double ctrlx1, double ctrly1,
                             double ctrlx2, double ctrly2,
                             double x2, double y2)
        {
            this.x1     = (float) x1;
            this.y1     = (float) y1;
            this.ctrlx1 = (float) ctrlx1;
            this.ctrly1 = (float) ctrly1;
            this.ctrlx2 = (float) ctrlx2;
            this.ctrly2 = (float) ctrly2;
            this.x2     = (float) x2;
            this.y2     = (float) y2;
        }

        /**
         * Sets the location of the end points and control points
         * of this curve to the specified {@code float} coordinates.
         *
         * @param x1 the X coordinate used to set the start point
         *           of this {@code CubicCurve2D}
         * @param y1 the Y coordinate used to set the start point
         *           of this {@code CubicCurve2D}
         * @param ctrlx1 the X coordinate used to set the first control point
         *               of this {@code CubicCurve2D}
         * @param ctrly1 the Y coordinate used to set the first control point
         *               of this {@code CubicCurve2D}
         * @param ctrlx2 the X coordinate used to set the second control point
         *               of this {@code CubicCurve2D}
         * @param ctrly2 the Y coordinate used to set the second control point
         *               of this {@code CubicCurve2D}
         * @param x2 the X coordinate used to set the end point
         *           of this {@code CubicCurve2D}
         * @param y2 the Y coordinate used to set the end point
         *           of this {@code CubicCurve2D}
         * @since 1.2
         */
        public void setCurve(float x1, float y1,
                             float ctrlx1, float ctrly1,
                             float ctrlx2, float ctrly2,
                             float x2, float y2)
        {
            this.x1     = x1;
            this.y1     = y1;
            this.ctrlx1 = ctrlx1;
            this.ctrly1 = ctrly1;
            this.ctrlx2 = ctrlx2;
            this.ctrly2 = ctrly2;
            this.x2     = x2;
            this.y2     = y2;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public Rectangle2D getBounds2D() {
            float left   = Math.min(Math.min(x1, x2),
                                    Math.min(ctrlx1, ctrlx2));
            float top    = Math.min(Math.min(y1, y2),
                                    Math.min(ctrly1, ctrly2));
            float right  = Math.max(Math.max(x1, x2),
                                    Math.max(ctrlx1, ctrlx2));
            float bottom = Math.max(Math.max(y1, y2),
                                    Math.max(ctrly1, ctrly2));
            return new Rectangle2D.Float(left, top,
                                         right - left, bottom - top);
        }

        /*
         * JDK 1.6 serialVersionUID
         */
        private static final long serialVersionUID = -1272015596714244385L;
    }

    /**
     * A cubic parametric curve segment specified with
     * {@code double} coordinates.
     * @since 1.2
     */
    public static class Double extends CubicCurve2D implements Serializable {
        /**
         * The X coordinate of the start point
         * of the cubic curve segment.
         * @since 1.2
         * @serial
         */
        public double x1;

        /**
         * The Y coordinate of the start point
         * of the cubic curve segment.
         * @since 1.2
         * @serial
         */
        public double y1;

        /**
         * The X coordinate of the first control point
         * of the cubic curve segment.
         * @since 1.2
         * @serial
         */
        public double ctrlx1;

        /**
         * The Y coordinate of the first control point
         * of the cubic curve segment.
         * @since 1.2
         * @serial
         */
        public double ctrly1;

        /**
         * The X coordinate of the second control point
         * of the cubic curve segment.
         * @since 1.2
         * @serial
         */
        public double ctrlx2;

        /**
         * The Y coordinate of the second control point
         * of the cubic curve segment.
         * @since 1.2
         * @serial
         */
        public double ctrly2;

        /**
         * The X coordinate of the end point
         * of the cubic curve segment.
         * @since 1.2
         * @serial
         */
        public double x2;

        /**
         * The Y coordinate of the end point
         * of the cubic curve segment.
         * @since 1.2
         * @serial
         */
        public double y2;

        /**
         * Constructs and initializes a CubicCurve with coordinates
         * (0, 0, 0, 0, 0, 0, 0, 0).
         * @since 1.2
         */
        public Double() {
        }

        /**
         * Constructs and initializes a {@code CubicCurve2D} from
         * the specified {@code double} coordinates.
         *
         * @param x1 the X coordinate for the start point
         *           of the resulting {@code CubicCurve2D}
         * @param y1 the Y coordinate for the start point
         *           of the resulting {@code CubicCurve2D}
         * @param ctrlx1 the X coordinate for the first control point
         *               of the resulting {@code CubicCurve2D}
         * @param ctrly1 the Y coordinate for the first control point
         *               of the resulting {@code CubicCurve2D}
         * @param ctrlx2 the X coordinate for the second control point
         *               of the resulting {@code CubicCurve2D}
         * @param ctrly2 the Y coordinate for the second control point
         *               of the resulting {@code CubicCurve2D}
         * @param x2 the X coordinate for the end point
         *           of the resulting {@code CubicCurve2D}
         * @param y2 the Y coordinate for the end point
         *           of the resulting {@code CubicCurve2D}
         * @since 1.2
         */
        public Double(double x1, double y1,
                      double ctrlx1, double ctrly1,
                      double ctrlx2, double ctrly2,
                      double x2, double y2)
        {
            setCurve(x1, y1, ctrlx1, ctrly1, ctrlx2, ctrly2, x2, y2);
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public double getX1() {
            return x1;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public double getY1() {
            return y1;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public Point2D getP1() {
            return new Point2D.Double(x1, y1);
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public double getCtrlX1() {
            return ctrlx1;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public double getCtrlY1() {
            return ctrly1;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public Point2D getCtrlP1() {
            return new Point2D.Double(ctrlx1, ctrly1);
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public double getCtrlX2() {
            return ctrlx2;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public double getCtrlY2() {
            return ctrly2;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public Point2D getCtrlP2() {
            return new Point2D.Double(ctrlx2, ctrly2);
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public double getX2() {
            return x2;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public double getY2() {
            return y2;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public Point2D getP2() {
            return new Point2D.Double(x2, y2);
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public void setCurve(double x1, double y1,
                             double ctrlx1, double ctrly1,
                             double ctrlx2, double ctrly2,
                             double x2, double y2)
        {
            this.x1     = x1;
            this.y1     = y1;
            this.ctrlx1 = ctrlx1;
            this.ctrly1 = ctrly1;
            this.ctrlx2 = ctrlx2;
            this.ctrly2 = ctrly2;
            this.x2     = x2;
            this.y2     = y2;
        }

        /**
         * {@inheritDoc}
         * @since 1.2
         */
        public Rectangle2D getBounds2D() {
            double left   = Math.min(Math.min(x1, x2),
                                     Math.min(ctrlx1, ctrlx2));
            double top    = Math.min(Math.min(y1, y2),
                                     Math.min(ctrly1, ctrly2));
            double right  = Math.max(Math.max(x1, x2),
                                     Math.max(ctrlx1, ctrlx2));
            double bottom = Math.max(Math.max(y1, y2),
                                     Math.max(ctrly1, ctrly2));
            return new Rectangle2D.Double(left, top,
                                          right - left, bottom - top);
        }

        /*
         * JDK 1.6 serialVersionUID
         */
        private static final long serialVersionUID = -4202960122839707295L;
    }

    /**
     * This is an abstract class that cannot be instantiated directly.
     * Type-specific implementation subclasses are available for
     * instantiation and provide a number of formats for storing
     * the information necessary to satisfy the various accessor
     * methods below.
     *
     * @see java.awt.geom.CubicCurve2D.Float
     * @see java.awt.geom.CubicCurve2D.Double
     * @since 1.2
     */
    protected CubicCurve2D() {
    }

    /**
     * Returns the X coordinate of the start point in double precision.
     * @return the X coordinate of the start point of the
     *         {@code CubicCurve2D}.
     * @since 1.2
     */
    public abstract double getX1();

    /**
     * Returns the Y coordinate of the start point in double precision.
     * @return the Y coordinate of the start point of the
     *         {@code CubicCurve2D}.
     * @since 1.2
     */
    public abstract double getY1();

    /**
     * Returns the start point.
     * @return a {@code Point2D} that is the start point of
     *         the {@code CubicCurve2D}.
     * @since 1.2
     */
    public abstract Point2D getP1();

    /**
     * Returns the X coordinate of the first control point in double precision.
     * @return the X coordinate of the first control point of the
     *         {@code CubicCurve2D}.
     * @since 1.2
     */
    public abstract double getCtrlX1();

    /**
     * Returns the Y coordinate of the first control point in double precision.
     * @return the Y coordinate of the first control point of the
     *         {@code CubicCurve2D}.
     * @since 1.2
     */
    public abstract double getCtrlY1();

    /**
     * Returns the first control point.
     * @return a {@code Point2D} that is the first control point of
     *         the {@code CubicCurve2D}.
     * @since 1.2
     */
    public abstract Point2D getCtrlP1();

    /**
     * Returns the X coordinate of the second control point
     * in double precision.
     * @return the X coordinate of the second control point of the
     *         {@code CubicCurve2D}.
     * @since 1.2
     */
    public abstract double getCtrlX2();

    /**
     * Returns the Y coordinate of the second control point
     * in double precision.
     * @return the Y coordinate of the second control point of the
     *         {@code CubicCurve2D}.
     * @since 1.2
     */
    public abstract double getCtrlY2();

    /**
     * Returns the second control point.
     * @return a {@code Point2D} that is the second control point of
     *         the {@code CubicCurve2D}.
     * @since 1.2
     */
    public abstract Point2D getCtrlP2();

    /**
     * Returns the X coordinate of the end point in double precision.
     * @return the X coordinate of the end point of the
     *         {@code CubicCurve2D}.
     * @since 1.2
     */
    public abstract double getX2();

    /**
     * Returns the Y coordinate of the end point in double precision.
     * @return the Y coordinate of the end point of the
     *         {@code CubicCurve2D}.
     * @since 1.2
     */
    public abstract double getY2();

    /**
     * Returns the end point.
     * @return a {@code Point2D} that is the end point of
     *         the {@code CubicCurve2D}.
     * @since 1.2
     */
    public abstract Point2D getP2();

    /**
     * Sets the location of the end points and control points of this curve
     * to the specified double coordinates.
     *
     * @param x1 the X coordinate used to set the start point
     *           of this {@code CubicCurve2D}
     * @param y1 the Y coordinate used to set the start point
     *           of this {@code CubicCurve2D}
     * @param ctrlx1 the X coordinate used to set the first control point
     *               of this {@code CubicCurve2D}
     * @param ctrly1 the Y coordinate used to set the first control point
     *               of this {@code CubicCurve2D}
     * @param ctrlx2 the X coordinate used to set the second control point
     *               of this {@code CubicCurve2D}
     * @param ctrly2 the Y coordinate used to set the second control point
     *               of this {@code CubicCurve2D}
     * @param x2 the X coordinate used to set the end point
     *           of this {@code CubicCurve2D}
     * @param y2 the Y coordinate used to set the end point
     *           of this {@code CubicCurve2D}
     * @since 1.2
     */
    public abstract void setCurve(double x1, double y1,
                                  double ctrlx1, double ctrly1,
                                  double ctrlx2, double ctrly2,
                                  double x2, double y2);

    /**
     * Sets the location of the end points and control points of this curve
     * to the double coordinates at the specified offset in the specified
     * array.
     * @param coords a double array containing coordinates
     * @param offset the index of <code>coords</code> from which to begin
     *          setting the end points and control points of this curve
     *          to the coordinates contained in <code>coords</code>
     * @since 1.2
     */
    public void setCurve(double[] coords, int offset) {
        setCurve(coords[offset + 0], coords[offset + 1],
                 coords[offset + 2], coords[offset + 3],
                 coords[offset + 4], coords[offset + 5],
                 coords[offset + 6], coords[offset + 7]);
    }

    /**
     * Sets the location of the end points and control points of this curve
     * to the specified <code>Point2D</code> coordinates.
     * @param p1 the first specified <code>Point2D</code> used to set the
     *          start point of this curve
     * @param cp1 the second specified <code>Point2D</code> used to set the
     *          first control point of this curve
     * @param cp2 the third specified <code>Point2D</code> used to set the
     *          second control point of this curve
     * @param p2 the fourth specified <code>Point2D</code> used to set the
     *          end point of this curve
     * @since 1.2
     */
    public void setCurve(Point2D p1, Point2D cp1, Point2D cp2, Point2D p2) {
        setCurve(p1.getX(), p1.getY(), cp1.getX(), cp1.getY(),
                 cp2.getX(), cp2.getY(), p2.getX(), p2.getY());
    }

    /**
     * Sets the location of the end points and control points of this curve
     * to the coordinates of the <code>Point2D</code> objects at the specified
     * offset in the specified array.
     * @param pts an array of <code>Point2D</code> objects
     * @param offset  the index of <code>pts</code> from which to begin setting
     *          the end points and control points of this curve to the
     *          points contained in <code>pts</code>
     * @since 1.2
     */
    public void setCurve(Point2D[] pts, int offset) {
        setCurve(pts[offset + 0].getX(), pts[offset + 0].getY(),
                 pts[offset + 1].getX(), pts[offset + 1].getY(),
                 pts[offset + 2].getX(), pts[offset + 2].getY(),
                 pts[offset + 3].getX(), pts[offset + 3].getY());
    }

    /**
     * Sets the location of the end points and control points of this curve
     * to the same as those in the specified <code>CubicCurve2D</code>.
     * @param c the specified <code>CubicCurve2D</code>
     * @since 1.2
     */
    public void setCurve(CubicCurve2D c) {
        setCurve(c.getX1(), c.getY1(), c.getCtrlX1(), c.getCtrlY1(),
                 c.getCtrlX2(), c.getCtrlY2(), c.getX2(), c.getY2());
    }

    /**
     * Returns the square of the flatness of the cubic curve specified
     * by the indicated control points. The flatness is the maximum distance
     * of a control point from the line connecting the end points.
     *
     * @param x1 the X coordinate that specifies the start point
     *           of a {@code CubicCurve2D}
     * @param y1 the Y coordinate that specifies the start point
     *           of a {@code CubicCurve2D}
     * @param ctrlx1 the X coordinate that specifies the first control point
     *               of a {@code CubicCurve2D}
     * @param ctrly1 the Y coordinate that specifies the first control point
     *               of a {@code CubicCurve2D}
     * @param ctrlx2 the X coordinate that specifies the second control point
     *               of a {@code CubicCurve2D}
     * @param ctrly2 the Y coordinate that specifies the second control point
     *               of a {@code CubicCurve2D}
     * @param x2 the X coordinate that specifies the end point
     *           of a {@code CubicCurve2D}
     * @param y2 the Y coordinate that specifies the end point
     *           of a {@code CubicCurve2D}
     * @return the square of the flatness of the {@code CubicCurve2D}
     *          represented by the specified coordinates.
     * @since 1.2
     */
    public static double getFlatnessSq(double x1, double y1,
                                       double ctrlx1, double ctrly1,
                                       double ctrlx2, double ctrly2,
                                       double x2, double y2) {
        return Math.max(Line2D.ptSegDistSq(x1, y1, x2, y2, ctrlx1, ctrly1),
                        Line2D.ptSegDistSq(x1, y1, x2, y2, ctrlx2, ctrly2));

    }

    /**
     * Returns the flatness of the cubic curve specified
     * by the indicated control points. The flatness is the maximum distance
     * of a control point from the line connecting the end points.
     *
     * @param x1 the X coordinate that specifies the start point
     *           of a {@code CubicCurve2D}
     * @param y1 the Y coordinate that specifies the start point
     *           of a {@code CubicCurve2D}
     * @param ctrlx1 the X coordinate that specifies the first control point
     *               of a {@code CubicCurve2D}
     * @param ctrly1 the Y coordinate that specifies the first control point
     *               of a {@code CubicCurve2D}
     * @param ctrlx2 the X coordinate that specifies the second control point
     *               of a {@code CubicCurve2D}
     * @param ctrly2 the Y coordinate that specifies the second control point
     *               of a {@code CubicCurve2D}
     * @param x2 the X coordinate that specifies the end point
     *           of a {@code CubicCurve2D}
     * @param y2 the Y coordinate that specifies the end point
     *           of a {@code CubicCurve2D}
     * @return the flatness of the {@code CubicCurve2D}
     *          represented by the specified coordinates.
     * @since 1.2
     */
    public static double getFlatness(double x1, double y1,
                                     double ctrlx1, double ctrly1,
                                     double ctrlx2, double ctrly2,
                                     double x2, double y2) {
        return Math.sqrt(getFlatnessSq(x1, y1, ctrlx1, ctrly1,
                                       ctrlx2, ctrly2, x2, y2));
    }

    /**
     * Returns the square of the flatness of the cubic curve specified
     * by the control points stored in the indicated array at the
     * indicated index. The flatness is the maximum distance
     * of a control point from the line connecting the end points.
     * @param coords an array containing coordinates
     * @param offset the index of <code>coords</code> from which to begin
     *          getting the end points and control points of the curve
     * @return the square of the flatness of the <code>CubicCurve2D</code>
     *          specified by the coordinates in <code>coords</code> at
     *          the specified offset.
     * @since 1.2
     */
    public static double getFlatnessSq(double coords[], int offset) {
        return getFlatnessSq(coords[offset + 0], coords[offset + 1],
                             coords[offset + 2], coords[offset + 3],
                             coords[offset + 4], coords[offset + 5],
                             coords[offset + 6], coords[offset + 7]);
    }

    /**
     * Returns the flatness of the cubic curve specified
     * by the control points stored in the indicated array at the
     * indicated index.  The flatness is the maximum distance
     * of a control point from the line connecting the end points.
     * @param coords an array containing coordinates
     * @param offset the index of <code>coords</code> from which to begin
     *          getting the end points and control points of the curve
     * @return the flatness of the <code>CubicCurve2D</code>
     *          specified by the coordinates in <code>coords</code> at
     *          the specified offset.
     * @since 1.2
     */
    public static double getFlatness(double coords[], int offset) {
        return getFlatness(coords[offset + 0], coords[offset + 1],
                           coords[offset + 2], coords[offset + 3],
                           coords[offset + 4], coords[offset + 5],
                           coords[offset + 6], coords[offset + 7]);
    }

    /**
     * Returns the square of the flatness of this curve.  The flatness is the
     * maximum distance of a control point from the line connecting the
     * end points.
     * @return the square of the flatness of this curve.
     * @since 1.2
     */
    public double getFlatnessSq() {
        return getFlatnessSq(getX1(), getY1(), getCtrlX1(), getCtrlY1(),
                             getCtrlX2(), getCtrlY2(), getX2(), getY2());
    }

    /**
     * Returns the flatness of this curve.  The flatness is the
     * maximum distance of a control point from the line connecting the
     * end points.
     * @return the flatness of this curve.
     * @since 1.2
     */
    public double getFlatness() {
        return getFlatness(getX1(), getY1(), getCtrlX1(), getCtrlY1(),
                           getCtrlX2(), getCtrlY2(), getX2(), getY2());
    }

    /**
     * Subdivides this cubic curve and stores the resulting two
     * subdivided curves into the left and right curve parameters.
     * Either or both of the left and right objects may be the same
     * as this object or null.
     * @param left the cubic curve object for storing for the left or
     * first half of the subdivided curve
     * @param right the cubic curve object for storing for the right or
     * second half of the subdivided curve
     * @since 1.2
     */
    public void subdivide(CubicCurve2D left, CubicCurve2D right) {
        subdivide(this, left, right);
    }

    /**
     * Subdivides the cubic curve specified by the <code>src</code> parameter
     * and stores the resulting two subdivided curves into the
     * <code>left</code> and <code>right</code> curve parameters.
     * Either or both of the <code>left</code> and <code>right</code> objects
     * may be the same as the <code>src</code> object or <code>null</code>.
     * @param src the cubic curve to be subdivided
     * @param left the cubic curve object for storing the left or
     * first half of the subdivided curve
     * @param right the cubic curve object for storing the right or
     * second half of the subdivided curve
     * @since 1.2
     */
    public static void subdivide(CubicCurve2D src,
                                 CubicCurve2D left,
                                 CubicCurve2D right) {
        double x1 = src.getX1();
        double y1 = src.getY1();
        double ctrlx1 = src.getCtrlX1();
        double ctrly1 = src.getCtrlY1();
        double ctrlx2 = src.getCtrlX2();
        double ctrly2 = src.getCtrlY2();
        double x2 = src.getX2();
        double y2 = src.getY2();
        double centerx = (ctrlx1 + ctrlx2) / 2.0;
        double centery = (ctrly1 + ctrly2) / 2.0;
        ctrlx1 = (x1 + ctrlx1) / 2.0;
        ctrly1 = (y1 + ctrly1) / 2.0;
        ctrlx2 = (x2 + ctrlx2) / 2.0;
        ctrly2 = (y2 + ctrly2) / 2.0;
        double ctrlx12 = (ctrlx1 + centerx) / 2.0;
        double ctrly12 = (ctrly1 + centery) / 2.0;
        double ctrlx21 = (ctrlx2 + centerx) / 2.0;
        double ctrly21 = (ctrly2 + centery) / 2.0;
        centerx = (ctrlx12 + ctrlx21) / 2.0;
        centery = (ctrly12 + ctrly21) / 2.0;
        if (left != null) {
            left.setCurve(x1, y1, ctrlx1, ctrly1,
                          ctrlx12, ctrly12, centerx, centery);
        }
        if (right != null) {
            right.setCurve(centerx, centery, ctrlx21, ctrly21,
                           ctrlx2, ctrly2, x2, y2);
        }
    }

    /**
     * Subdivides the cubic curve specified by the coordinates
     * stored in the <code>src</code> array at indices <code>srcoff</code>
     * through (<code>srcoff</code>&nbsp;+&nbsp;7) and stores the
     * resulting two subdivided curves into the two result arrays at the
     * corresponding indices.
     * Either or both of the <code>left</code> and <code>right</code>
     * arrays may be <code>null</code> or a reference to the same array
     * as the <code>src</code> array.
     * Note that the last point in the first subdivided curve is the
     * same as the first point in the second subdivided curve. Thus,
     * it is possible to pass the same array for <code>left</code>
     * and <code>right</code> and to use offsets, such as <code>rightoff</code>
     * equals (<code>leftoff</code> + 6), in order
     * to avoid allocating extra storage for this common point.
     * @param src the array holding the coordinates for the source curve
     * @param srcoff the offset into the array of the beginning of the
     * the 6 source coordinates
     * @param left the array for storing the coordinates for the first
     * half of the subdivided curve
     * @param leftoff the offset into the array of the beginning of the
     * the 6 left coordinates
     * @param right the array for storing the coordinates for the second
     * half of the subdivided curve
     * @param rightoff the offset into the array of the beginning of the
     * the 6 right coordinates
     * @since 1.2
     */
    public static void subdivide(double src[], int srcoff,
                                 double left[], int leftoff,
                                 double right[], int rightoff) {
        double x1 = src[srcoff + 0];
        double y1 = src[srcoff + 1];
        double ctrlx1 = src[srcoff + 2];
        double ctrly1 = src[srcoff + 3];
        double ctrlx2 = src[srcoff + 4];
        double ctrly2 = src[srcoff + 5];
        double x2 = src[srcoff + 6];
        double y2 = src[srcoff + 7];
        if (left != null) {
            left[leftoff + 0] = x1;
            left[leftoff + 1] = y1;
        }
        if (right != null) {
            right[rightoff + 6] = x2;
            right[rightoff + 7] = y2;
        }
        x1 = (x1 + ctrlx1) / 2.0;
        y1 = (y1 + ctrly1) / 2.0;
        x2 = (x2 + ctrlx2) / 2.0;
        y2 = (y2 + ctrly2) / 2.0;
        double centerx = (ctrlx1 + ctrlx2) / 2.0;
        double centery = (ctrly1 + ctrly2) / 2.0;
        ctrlx1 = (x1 + centerx) / 2.0;
        ctrly1 = (y1 + centery) / 2.0;
        ctrlx2 = (x2 + centerx) / 2.0;
        ctrly2 = (y2 + centery) / 2.0;
        centerx = (ctrlx1 + ctrlx2) / 2.0;
        centery = (ctrly1 + ctrly2) / 2.0;
        if (left != null) {
            left[leftoff + 2] = x1;
            left[leftoff + 3] = y1;
            left[leftoff + 4] = ctrlx1;
            left[leftoff + 5] = ctrly1;
            left[leftoff + 6] = centerx;
            left[leftoff + 7] = centery;
        }
        if (right != null) {
            right[rightoff + 0] = centerx;
            right[rightoff + 1] = centery;
            right[rightoff + 2] = ctrlx2;
            right[rightoff + 3] = ctrly2;
            right[rightoff + 4] = x2;
            right[rightoff + 5] = y2;
        }
    }

    /**
     * Solves the cubic whose coefficients are in the <code>eqn</code>
     * array and places the non-complex roots back into the same array,
     * returning the number of roots.  The solved cubic is represented
     * by the equation:
     * <pre>
     *     eqn = {c, b, a, d}
     *     dx^3 + ax^2 + bx + c = 0
     * </pre>
     * A return value of -1 is used to distinguish a constant equation
     * that might be always 0 or never 0 from an equation that has no
     * zeroes.
     * @param eqn an array containing coefficients for a cubic
     * @return the number of roots, or -1 if the equation is a constant.
     * @since 1.2
     */
    public static int solveCubic(double eqn[]) {
        return solveCubic(eqn, eqn);
    }

    /**
     * Solve the cubic whose coefficients are in the <code>eqn</code>
     * array and place the non-complex roots into the <code>res</code>
     * array, returning the number of roots.
     * The cubic solved is represented by the equation:
     *     eqn = {c, b, a, d}
     *     dx^3 + ax^2 + bx + c = 0
     * A return value of -1 is used to distinguish a constant equation,
     * which may be always 0 or never 0, from an equation which has no
     * zeroes.
     * @param eqn the specified array of coefficients to use to solve
     *        the cubic equation
     * @param res the array that contains the non-complex roots
     *        resulting from the solution of the cubic equation
     * @return the number of roots, or -1 if the equation is a constant
     * @since 1.3
     */
    public static int solveCubic(double eqn[], double res[]) {
        // From Numerical Recipes, 5.6, Quadratic and Cubic Equations
        double d = eqn[3];
        if (d == 0.0) {
            // The cubic has degenerated to quadratic (or line or ...).
            return QuadCurve2D.solveQuadratic(eqn, res);
        }
        double a = eqn[2] / d;
        double b = eqn[1] / d;
        double c = eqn[0] / d;
        int roots = 0;
        double Q = (a * a - 3.0 * b) / 9.0;
        double R = (2.0 * a * a * a - 9.0 * a * b + 27.0 * c) / 54.0;
        double R2 = R * R;
        double Q3 = Q * Q * Q;
        a = a / 3.0;
        if (R2 < Q3) {
            double theta = Math.acos(R / Math.sqrt(Q3));
            Q = -2.0 * Math.sqrt(Q);
            if (res == eqn) {
                // Copy the eqn so that we don't clobber it with the
                // roots.  This is needed so that fixRoots can do its
                // work with the original equation.
                eqn = new double[4];
                System.arraycopy(res, 0, eqn, 0, 4);
            }
            res[roots++] = Q * Math.cos(theta / 3.0) - a;
            res[roots++] = Q * Math.cos((theta + Math.PI * 2.0)/ 3.0) - a;
            res[roots++] = Q * Math.cos((theta - Math.PI * 2.0)/ 3.0) - a;
            fixRoots(res, eqn);
        } else {
            boolean neg = (R < 0.0);
            double S = Math.sqrt(R2 - Q3);
            if (neg) {
                R = -R;
            }
            double A = Math.pow(R + S, 1.0 / 3.0);
            if (!neg) {
                A = -A;
            }
            double B = (A == 0.0) ? 0.0 : (Q / A);
            res[roots++] = (A + B) - a;
        }
        return roots;
    }

    /*
     * This pruning step is necessary since solveCubic uses the
     * cosine function to calculate the roots when there are 3
     * of them.  Since the cosine method can have an error of
     * +/- 1E-14 we need to make sure that we don't make any
     * bad decisions due to an error.
     *
     * If the root is not near one of the endpoints, then we will
     * only have a slight inaccuracy in calculating the x intercept
     * which will only cause a slightly wrong answer for some
     * points very close to the curve.  While the results in that
     * case are not as accurate as they could be, they are not
     * disastrously inaccurate either.
     *
     * On the other hand, if the error happens near one end of
     * the curve, then our processing to reject values outside
     * of the t=[0,1] range will fail and the results of that
     * failure will be disastrous since for an entire horizontal
     * range of test points, we will either overcount or undercount
     * the crossings and get a wrong answer for all of them, even
     * when they are clearly and obviously inside or outside the
     * curve.
     *
     * To work around this problem, we try a couple of Newton-Raphson
     * iterations to see if the true root is closer to the endpoint
     * or further away.  If it is further away, then we can stop
     * since we know we are on the right side of the endpoint.  If
     * we change direction, then either we are now being dragged away
     * from the endpoint in which case the first condition will cause
     * us to stop, or we have passed the endpoint and are headed back.
     * In the second case, we simply evaluate the slope at the
     * endpoint itself and place ourselves on the appropriate side
     * of it or on it depending on that result.
     */
    private static void fixRoots(double res[], double eqn[]) {
        final double EPSILON = 1E-5;
        for (int i = 0; i < 3; i++) {
            double t = res[i];
            if (Math.abs(t) < EPSILON) {
                res[i] = findZero(t, 0, eqn);
            } else if (Math.abs(t - 1) < EPSILON) {
                res[i] = findZero(t, 1, eqn);
            }
        }
    }

    private static double solveEqn(double eqn[], int order, double t) {
        double v = eqn[order];
        while (--order >= 0) {
            v = v * t + eqn[order];
        }
        return v;
    }

    private static double findZero(double t, double target, double eqn[]) {
        double slopeqn[] = {eqn[1], 2*eqn[2], 3*eqn[3]};
        double slope;
        double origdelta = 0;
        double origt = t;
        while (true) {
            slope = solveEqn(slopeqn, 2, t);
            if (slope == 0) {
                // At a local minima - must return
                return t;
            }
            double y = solveEqn(eqn, 3, t);
            if (y == 0) {
                // Found it! - return it
                return t;
            }
            // assert(slope != 0 && y != 0);
            double delta = - (y / slope);
            // assert(delta != 0);
            if (origdelta == 0) {
                origdelta = delta;
            }
            if (t < target) {
                if (delta < 0) return t;
            } else if (t > target) {
                if (delta > 0) return t;
            } else { /* t == target */
                return (delta > 0
                        ? (target + java.lang.Double.MIN_VALUE)
                        : (target - java.lang.Double.MIN_VALUE));
            }
            double newt = t + delta;
            if (t == newt) {
                // The deltas are so small that we aren't moving...
                return t;
            }
            if (delta * origdelta < 0) {
                // We have reversed our path.
                int tag = (origt < t
                           ? getTag(target, origt, t)
                           : getTag(target, t, origt));
                if (tag != INSIDE) {
                    // Local minima found away from target - return the middle
                    return (origt + t) / 2;
                }
                // Local minima somewhere near target - move to target
                // and let the slope determine the resulting t.
                t = target;
            } else {
                t = newt;
            }
        }
    }

    /**
     * {@inheritDoc}
     * @since 1.2
     */
    public boolean contains(double x, double y) {
        if (!(x * 0.0 + y * 0.0 == 0.0)) {
            /* Either x or y was infinite or NaN.
             * A NaN always produces a negative response to any test
             * and Infinity values cannot be "inside" any path so
             * they should return false as well.
             */
            return false;
        }
        // We count the "Y" crossings to determine if the point is
        // inside the curve bounded by its closing line.
        double x1 = getX1();
        double y1 = getY1();
        double x2 = getX2();
        double y2 = getY2();
        int crossings =
            (Curve.pointCrossingsForLine(x, y, x1, y1, x2, y2) +
             Curve.pointCrossingsForCubic(x, y,
                                          x1, y1,
                                          getCtrlX1(), getCtrlY1(),
                                          getCtrlX2(), getCtrlY2(),
                                          x2, y2, 0));
        return ((crossings & 1) == 1);
    }

    /**
     * {@inheritDoc}
     * @since 1.2
     */
    public boolean contains(Point2D p) {
        return contains(p.getX(), p.getY());
    }

    /*
     * Fill an array with the coefficients of the parametric equation
     * in t, ready for solving against val with solveCubic.
     * We currently have:
     * <pre>
     *   val = P(t) = C1(1-t)^3 + 3CP1 t(1-t)^2 + 3CP2 t^2(1-t) + C2 t^3
     *              = C1 - 3C1t + 3C1t^2 - C1t^3 +
     *                3CP1t - 6CP1t^2 + 3CP1t^3 +
     *                3CP2t^2 - 3CP2t^3 +
     *                C2t^3
     *            0 = (C1 - val) +
     *                (3CP1 - 3C1) t +
     *                (3C1 - 6CP1 + 3CP2) t^2 +
     *                (C2 - 3CP2 + 3CP1 - C1) t^3
     *            0 = C + Bt + At^2 + Dt^3
     *     C = C1 - val
     *     B = 3*CP1 - 3*C1
     *     A = 3*CP2 - 6*CP1 + 3*C1
     *     D = C2 - 3*CP2 + 3*CP1 - C1
     * </pre>
     */
    private static void fillEqn(double eqn[], double val,
                                double c1, double cp1, double cp2, double c2) {
        eqn[0] = c1 - val;
        eqn[1] = (cp1 - c1) * 3.0;
        eqn[2] = (cp2 - cp1 - cp1 + c1) * 3.0;
        eqn[3] = c2 + (cp1 - cp2) * 3.0 - c1;
        return;
    }

    /*
     * Evaluate the t values in the first num slots of the vals[] array
     * and place the evaluated values back into the same array.  Only
     * evaluate t values that are within the range <0, 1>, including
     * the 0 and 1 ends of the range iff the include0 or include1
     * booleans are true.  If an "inflection" equation is handed in,
     * then any points which represent a point of inflection for that
     * cubic equation are also ignored.
     */
    private static int evalCubic(double vals[], int num,
                                 boolean include0,
                                 boolean include1,
                                 double inflect[],
                                 double c1, double cp1,
                                 double cp2, double c2) {
        int j = 0;
        for (int i = 0; i < num; i++) {
            double t = vals[i];
            if ((include0 ? t >= 0 : t > 0) &&
                (include1 ? t <= 1 : t < 1) &&
                (inflect == null ||
                 inflect[1] + (2*inflect[2] + 3*inflect[3]*t)*t != 0))
            {
                double u = 1 - t;
                vals[j++] = c1*u*u*u + 3*cp1*t*u*u + 3*cp2*t*t*u + c2*t*t*t;
            }
        }
        return j;
    }

    private static final int BELOW = -2;
    private static final int LOWEDGE = -1;
    private static final int INSIDE = 0;
    private static final int HIGHEDGE = 1;
    private static final int ABOVE = 2;

    /*
     * Determine where coord lies with respect to the range from
     * low to high.  It is assumed that low <= high.  The return
     * value is one of the 5 values BELOW, LOWEDGE, INSIDE, HIGHEDGE,
     * or ABOVE.
     */
    private static int getTag(double coord, double low, double high) {
        if (coord <= low) {
            return (coord < low ? BELOW : LOWEDGE);
        }
        if (coord >= high) {
            return (coord > high ? ABOVE : HIGHEDGE);
        }
        return INSIDE;
    }

    /*
     * Determine if the pttag represents a coordinate that is already
     * in its test range, or is on the border with either of the two
     * opttags representing another coordinate that is "towards the
     * inside" of that test range.  In other words, are either of the
     * two "opt" points "drawing the pt inward"?
     */
    private static boolean inwards(int pttag, int opt1tag, int opt2tag) {
        switch (pttag) {
        case BELOW:
        case ABOVE:
        default:
            return false;
        case LOWEDGE:
            return (opt1tag >= INSIDE || opt2tag >= INSIDE);
        case INSIDE:
            return true;
        case HIGHEDGE:
            return (opt1tag <= INSIDE || opt2tag <= INSIDE);
        }
    }

    /**
     * {@inheritDoc}
     * @since 1.2
     */
    public boolean intersects(double x, double y, double w, double h) {
        // Trivially reject non-existant rectangles
        if (w <= 0 || h <= 0) {
            return false;
        }

        // Trivially accept if either endpoint is inside the rectangle
        // (not on its border since it may end there and not go inside)
        // Record where they lie with respect to the rectangle.
        //     -1 => left, 0 => inside, 1 => right
        double x1 = getX1();
        double y1 = getY1();
        int x1tag = getTag(x1, x, x+w);
        int y1tag = getTag(y1, y, y+h);
        if (x1tag == INSIDE && y1tag == INSIDE) {
            return true;
        }
        double x2 = getX2();
        double y2 = getY2();
        int x2tag = getTag(x2, x, x+w);
        int y2tag = getTag(y2, y, y+h);
        if (x2tag == INSIDE && y2tag == INSIDE) {
            return true;
        }

        double ctrlx1 = getCtrlX1();
        double ctrly1 = getCtrlY1();
        double ctrlx2 = getCtrlX2();
        double ctrly2 = getCtrlY2();
        int ctrlx1tag = getTag(ctrlx1, x, x+w);
        int ctrly1tag = getTag(ctrly1, y, y+h);
        int ctrlx2tag = getTag(ctrlx2, x, x+w);
        int ctrly2tag = getTag(ctrly2, y, y+h);

        // Trivially reject if all points are entirely to one side of
        // the rectangle.
        if (x1tag < INSIDE && x2tag < INSIDE &&
            ctrlx1tag < INSIDE && ctrlx2tag < INSIDE)
        {
            return false;       // All points left
        }
        if (y1tag < INSIDE && y2tag < INSIDE &&
            ctrly1tag < INSIDE && ctrly2tag < INSIDE)
        {
            return false;       // All points above
        }
        if (x1tag > INSIDE && x2tag > INSIDE &&
            ctrlx1tag > INSIDE && ctrlx2tag > INSIDE)
        {
            return false;       // All points right
        }
        if (y1tag > INSIDE && y2tag > INSIDE &&
            ctrly1tag > INSIDE && ctrly2tag > INSIDE)
        {
            return false;       // All points below
        }

        // Test for endpoints on the edge where either the segment
        // or the curve is headed "inwards" from them
        // Note: These tests are a superset of the fast endpoint tests
        //       above and thus repeat those tests, but take more time
        //       and cover more cases
        if (inwards(x1tag, x2tag, ctrlx1tag) &&
            inwards(y1tag, y2tag, ctrly1tag))
        {
            // First endpoint on border with either edge moving inside
            return true;
        }
        if (inwards(x2tag, x1tag, ctrlx2tag) &&
            inwards(y2tag, y1tag, ctrly2tag))
        {
            // Second endpoint on border with either edge moving inside
            return true;
        }

        // Trivially accept if endpoints span directly across the rectangle
        boolean xoverlap = (x1tag * x2tag <= 0);
        boolean yoverlap = (y1tag * y2tag <= 0);
        if (x1tag == INSIDE && x2tag == INSIDE && yoverlap) {
            return true;
        }
        if (y1tag == INSIDE && y2tag == INSIDE && xoverlap) {
            return true;
        }

        // We now know that both endpoints are outside the rectangle
        // but the 4 points are not all on one side of the rectangle.
        // Therefore the curve cannot be contained inside the rectangle,
        // but the rectangle might be contained inside the curve, or
        // the curve might intersect the boundary of the rectangle.

        double[] eqn = new double[4];
        double[] res = new double[4];
        if (!yoverlap) {
            // Both y coordinates for the closing segment are above or
            // below the rectangle which means that we can only intersect
            // if the curve crosses the top (or bottom) of the rectangle
            // in more than one place and if those crossing locations
            // span the horizontal range of the rectangle.
            fillEqn(eqn, (y1tag < INSIDE ? y : y+h), y1, ctrly1, ctrly2, y2);
            int num = solveCubic(eqn, res);
            num = evalCubic(res, num, true, true, null,
                            x1, ctrlx1, ctrlx2, x2);
            // odd counts imply the crossing was out of [0,1] bounds
            // otherwise there is no way for that part of the curve to
            // "return" to meet its endpoint
            return (num == 2 &&
                    getTag(res[0], x, x+w) * getTag(res[1], x, x+w) <= 0);
        }

        // Y ranges overlap.  Now we examine the X ranges
        if (!xoverlap) {
            // Both x coordinates for the closing segment are left of
            // or right of the rectangle which means that we can only
            // intersect if the curve crosses the left (or right) edge
            // of the rectangle in more than one place and if those
            // crossing locations span the vertical range of the rectangle.
            fillEqn(eqn, (x1tag < INSIDE ? x : x+w), x1, ctrlx1, ctrlx2, x2);
            int num = solveCubic(eqn, res);
            num = evalCubic(res, num, true, true, null,
                            y1, ctrly1, ctrly2, y2);
            // odd counts imply the crossing was out of [0,1] bounds
            // otherwise there is no way for that part of the curve to
            // "return" to meet its endpoint
            return (num == 2 &&
                    getTag(res[0], y, y+h) * getTag(res[1], y, y+h) <= 0);
        }

        // The X and Y ranges of the endpoints overlap the X and Y
        // ranges of the rectangle, now find out how the endpoint
        // line segment intersects the Y range of the rectangle
        double dx = x2 - x1;
        double dy = y2 - y1;
        double k = y2 * x1 - x2 * y1;
        int c1tag, c2tag;
        if (y1tag == INSIDE) {
            c1tag = x1tag;
        } else {
            c1tag = getTag((k + dx * (y1tag < INSIDE ? y : y+h)) / dy, x, x+w);
        }
        if (y2tag == INSIDE) {
            c2tag = x2tag;
        } else {
            c2tag = getTag((k + dx * (y2tag < INSIDE ? y : y+h)) / dy, x, x+w);
        }
        // If the part of the line segment that intersects the Y range
        // of the rectangle crosses it horizontally - trivially accept
        if (c1tag * c2tag <= 0) {
            return true;
        }

        // Now we know that both the X and Y ranges intersect and that
        // the endpoint line segment does not directly cross the rectangle.
        //
        // We can almost treat this case like one of the cases above
        // where both endpoints are to one side, except that we may
        // get one or three intersections of the curve with the vertical
        // side of the rectangle.  This is because the endpoint segment
        // accounts for the other intersection in an even pairing.  Thus,
        // with the endpoint crossing we end up with 2 or 4 total crossings.
        //
        // (Remember there is overlap in both the X and Y ranges which
        //  means that the segment itself must cross at least one vertical
        //  edge of the rectangle - in particular, the "near vertical side"
        //  - leaving an odd number of intersections for the curve.)
        //
        // Now we calculate the y tags of all the intersections on the
        // "near vertical side" of the rectangle.  We will have one with
        // the endpoint segment, and one or three with the curve.  If
        // any pair of those vertical intersections overlap the Y range
        // of the rectangle, we have an intersection.  Otherwise, we don't.

        // c1tag = vertical intersection class of the endpoint segment
        //
        // Choose the y tag of the endpoint that was not on the same
        // side of the rectangle as the subsegment calculated above.
        // Note that we can "steal" the existing Y tag of that endpoint
        // since it will be provably the same as the vertical intersection.
        c1tag = ((c1tag * x1tag <= 0) ? y1tag : y2tag);

        // Now we have to calculate an array of solutions of the curve
        // with the "near vertical side" of the rectangle.  Then we
        // need to sort the tags and do a pairwise range test to see
        // if either of the pairs of crossings spans the Y range of
        // the rectangle.
        //
        // Note that the c2tag can still tell us which vertical edge
        // to test against.
        fillEqn(eqn, (c2tag < INSIDE ? x : x+w), x1, ctrlx1, ctrlx2, x2);
        int num = solveCubic(eqn, res);
        num = evalCubic(res, num, true, true, null, y1, ctrly1, ctrly2, y2);

        // Now put all of the tags into a bucket and sort them.  There
        // is an intersection iff one of the pairs of tags "spans" the
        // Y range of the rectangle.
        int tags[] = new int[num+1];
        for (int i = 0; i < num; i++) {
            tags[i] = getTag(res[i], y, y+h);
        }
        tags[num] = c1tag;
        Arrays.sort(tags);
        return ((num >= 1 && tags[0] * tags[1] <= 0) ||
                (num >= 3 && tags[2] * tags[3] <= 0));
    }

    /**
     * {@inheritDoc}
     * @since 1.2
     */
    public boolean intersects(Rectangle2D r) {
        return intersects(r.getX(), r.getY(), r.getWidth(), r.getHeight());
    }

    /**
     * {@inheritDoc}
     * @since 1.2
     */
    public boolean contains(double x, double y, double w, double h) {
        if (w <= 0 || h <= 0) {
            return false;
        }
        // Assertion: Cubic curves closed by connecting their
        // endpoints form either one or two convex halves with
        // the closing line segment as an edge of both sides.
        if (!(contains(x, y) &&
              contains(x + w, y) &&
              contains(x + w, y + h) &&
              contains(x, y + h))) {
            return false;
        }
        // Either the rectangle is entirely inside one of the convex
        // halves or it crosses from one to the other, in which case
        // it must intersect the closing line segment.
        Rectangle2D rect = new Rectangle2D.Double(x, y, w, h);
        return !rect.intersectsLine(getX1(), getY1(), getX2(), getY2());
    }

    /**
     * {@inheritDoc}
     * @since 1.2
     */
    public boolean contains(Rectangle2D r) {
        return contains(r.getX(), r.getY(), r.getWidth(), r.getHeight());
    }

    /**
     * {@inheritDoc}
     * @since 1.2
     */
    public Rectangle getBounds() {
        return getBounds2D().getBounds();
    }

    /**
     * Returns an iteration object that defines the boundary of the
     * shape.
     * The iterator for this class is not multi-threaded safe,
     * which means that this <code>CubicCurve2D</code> class does not
     * guarantee that modifications to the geometry of this
     * <code>CubicCurve2D</code> object do not affect any iterations of
     * that geometry that are already in process.
     * @param at an optional <code>AffineTransform</code> to be applied to the
     * coordinates as they are returned in the iteration, or <code>null</code>
     * if untransformed coordinates are desired
     * @return    the <code>PathIterator</code> object that returns the
     *          geometry of the outline of this <code>CubicCurve2D</code>, one
     *          segment at a time.
     * @since 1.2
     */
    public PathIterator getPathIterator(AffineTransform at) {
        return new CubicIterator(this, at);
    }

    /**
     * Return an iteration object that defines the boundary of the
     * flattened shape.
     * The iterator for this class is not multi-threaded safe,
     * which means that this <code>CubicCurve2D</code> class does not
     * guarantee that modifications to the geometry of this
     * <code>CubicCurve2D</code> object do not affect any iterations of
     * that geometry that are already in process.
     * @param at an optional <code>AffineTransform</code> to be applied to the
     * coordinates as they are returned in the iteration, or <code>null</code>
     * if untransformed coordinates are desired
     * @param flatness the maximum amount that the control points
     * for a given curve can vary from colinear before a subdivided
     * curve is replaced by a straight line connecting the end points
     * @return    the <code>PathIterator</code> object that returns the
     * geometry of the outline of this <code>CubicCurve2D</code>,
     * one segment at a time.
     * @since 1.2
     */
    public PathIterator getPathIterator(AffineTransform at, double flatness) {
        return new FlatteningPathIterator(getPathIterator(at), flatness);
    }

    /**
     * Creates a new object of the same class as this object.
     *
     * @return     a clone of this instance.
     * @exception  OutOfMemoryError            if there is not enough memory.
     * @see        java.lang.Cloneable
     * @since      1.2
     */
    public Object clone() {
        try {
            return super.clone();
        } catch (CloneNotSupportedException e) {
            // this shouldn't happen, since we are Cloneable
            throw new InternalError();
        }
    }
}