/*
 * @(#)CubicCurve2D.java	1.25 00/02/02
 *
 * Copyright 1997-2000 Sun Microsystems, Inc. All Rights Reserved.
 * 
 * This software is the proprietary information of Sun Microsystems, Inc.  
 * Use is subject to license terms.
 * 
 */

package java.awt.geom;

import java.awt.Shape;
import java.awt.Rectangle;
import java.util.Arrays;

/**
 * The <code>CubicCurve2D</code> class defines a cubic parametric curve 
 * segment in (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.
 *
 * @version 	1.25, 02/02/00
 * @author	Jim Graham
 */
public abstract class CubicCurve2D implements Shape, Cloneable {
    /**
     * A cubic parametric curve segment specified with float coordinates.
     */
    public static class Float extends CubicCurve2D {
	/**
	 * The X coordinate of the start point
	 * of the cubic curve segment.
	 */
	public float x1;

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

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

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

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

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

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

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

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

	/**
	 * Constructs and initializes a <code>CubicCurve2D</code> from
	 * the specified coordinates.
	 * @param x1, y1 the first specified coordinates for the start
	 *		point of the resulting <code>CubicCurve2D</code>
	 * @param ctrlx1, ctrly1 the second specified coordinates for the
	 *		first control point of the resulting 
	 *		<code>CubicCurve2D</code>
	 * @param ctrlx2, ctrly2 the third specified coordinates for the 
	 *		second control point of the resulting
	 *		<code>CubicCurve2D</code>
	 * @param x2, y2 the fourth specified coordinates for the end
	 *		point of the resulting <code>CubicCurve2D</code>
	 */
	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);
	}

	/**
	 * Returns the X coordinate of the start point
	 * in double precision.
	 * @return the X coordinate of the start point of the 
	 * 		<code>CubicCurve2D</code>.
	 */
	public double getX1() {
	    return (double) x1;
	}

	/**
	 * Returns the Y coordinate of the start point 
	 * in double precision.
	 * @return the Y coordinate of the start point of the 
	 * 		<code>CubicCurve2D</code>.
	 */
	public double getY1() {
	    return (double) y1;
	}

	/**
	 * Returns the start point.
	 * @return a {@link Point2D} that is the start point of the
	 *		<code>CubicCurve2D</code>.
	 */
	public Point2D getP1() {
	    return new Point2D.Float(x1, y1);
	}

	/**
	 * 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</code>.
	 */
	public double getCtrlX1() {
	    return (double) ctrlx1;
	}

	/**
	 * 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</code>.
	 */
	public double getCtrlY1() {
	    return (double) ctrly1;
	}

	/**
	 * Returns the first control point.
	 * @return a <code>Point2D</code> that is the first control point
	 *		of the <code>CubicCurve2D</code>.
	 */
	public Point2D getCtrlP1() {
	    return new Point2D.Float(ctrlx1, ctrly1);
	}

	/**
	 * 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</code>.
	 */
	public double getCtrlX2() {
	    return (double) ctrlx2;
	}

	/**
	 * 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</code>.
	 */
	public double getCtrlY2() {
	    return (double) ctrly2;
	}

	/**
	 * Returns the second control point.
	 * @return a <code>Point2D</code> that is the second control point
	 *		of the <code>CubicCurve2D</code>.
	 */
	public Point2D getCtrlP2() {
	    return new Point2D.Float(ctrlx2, ctrly2);
	}

	/**
	 * Returns the X coordinate of the end point
	 * in double precision.
	 * @return the X coordinate of the end point of the
	 *		<code>CubicCurve2D</code>.
	 */
	public double getX2() {
	    return (double) x2;
	}

	/**
	 * Returns the Y coordinate of the end point 
	 * in double precision.
	 * @return the Y coordinate of the end point of the
	 *		<code>CubicCurve2D</code>.
	 */
	public double getY2() {
	    return (double) y2;
	}

	/**
	 * Returns the end point.
	 * @return a <code>Point2D</code> that is the end point
	 *		of the <code>CubicCurve2D</code>.
	 */
	public Point2D getP2() {
	    return new Point2D.Float(x2, y2);
	}

	/**
	 * Sets the location of the endpoints and controlpoints
	 * of this <code>CubicCurve2D</code> to the specified double 
	 * coordinates.
	 * @param x1, y1 the first specified coordinates used to set the start
	 *		point of this <code>CubicCurve2D</code>
	 * @param ctrlx1, ctrly1 the second specified coordinates used to set the
	 *		first control point of this <code>CubicCurve2D</code>
	 * @param ctrlx2, ctrly2 the third specified coordinates used to set the 
	 *		second control point of this <code>CubicCurve2D</code>		
	 * @param x2, y2 the fourth specified coordinates used to set the end
	 *		point of this <code>CubicCurve2D</code>
	 */
	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 endpoints and controlpoints
	 * of this curve to the specified float coordinates.
	 * @param x1, y1 the first specified coordinates used to set the start
	 *		point of this <code>CubicCurve2D</code>
	 * @param ctrlx1, ctrly1 the second specified coordinates used to set the
	 *		first control point of this <code>CubicCurve2D</code>
	 * @param ctrlx2, ctrly2 the third specified coordinates used to set the 
	 *		second control point of this <code>CubicCurve2D</code>		
	 * @param x2, y2 the fourth specified coordinates used to set the end
	 *		point of this <code>CubicCurve2D</code>
	 */
	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;
	}

	/**
	 * Returns the bounding box of the shape.
	 * @return a {@link Rectangle2D} that is the bounding box of the
	 * 		shape.
	 */
	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);
	}
    }

    /**
     * A cubic parametric curve segment specified with double coordinates.
     */
    public static class Double extends CubicCurve2D {
	/**
	 * The X coordinate of the start point
	 * of the cubic curve segment.
	 */
	public double x1;

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

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

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

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

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

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

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

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

	/**
	 * Constructs and initializes a <code>CubicCurve2D</code> from
	 * the specified coordinates.
	 * @param x1, y1 the first specified coordinates for the start
	 *		point of the resulting <code>CubicCurve2D</code>
	 * @param ctrlx1, ctrly1 the second specified coordinates for the
	 *		first control point of the resulting 
	 *		<code>CubicCurve2D</code>
	 * @param ctrlx2, ctrly2 the third specified coordinates for the 
	 *		second control point of the resulting
	 *		<code>CubicCurve2D</code>
	 * @param x2, y2 the fourth specified coordinates for the end
	 *		point of the resulting <code>CubicCurve2D</code>
	 */
	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);
	}

	/**
	 * Returns the X coordinate of the start point
	 * in double precision.
	 * @return the X coordinate of the first control point of the
	 *		<code>CubicCurve2D</code>.
	 */
	public double getX1() {
	    return x1;
	}

	/**
	 * Returns the Y coordinate of the start point 
	 * in double precision.
	 * @return the Y coordinate of the start point of the
	 *		<code>CubicCurve2D</code>.
	 */
	public double getY1() {
	    return y1;
	}

	/**
	 * Returns the start point.
	 * @return a <code>Point2D</code> that is the start point of the
	 *		<code>CubicCurve2D</code>.
	 */
	public Point2D getP1() {
	    return new Point2D.Double(x1, y1);
	}

	/**
	 * 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</code>.
	 */
	public double getCtrlX1() {
	    return ctrlx1;
	}

	/**
	 * 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</code>.
	 */
	public double getCtrlY1() {
	    return ctrly1;
	}

	/**
	 * Returns the first control point.
	 * @return a <code>Point2D</code> that is the first control point of the
	 *		<code>CubicCurve2D</code>.
	 */
	public Point2D getCtrlP1() {
	    return new Point2D.Double(ctrlx1, ctrly1);
	}

	/**
	 * 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</code>.
	 */
	public double getCtrlX2() {
	    return ctrlx2;
	}

	/**
	 * 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</code>.
	 */
	public double getCtrlY2() {
	    return ctrly2;
	}

	/**
	 * Returns the second control point.
	 * @return a <code>Point2D</code> that is the second control point of 
	 *		the <code>CubicCurve2D</code>.	
	 */
	public Point2D getCtrlP2() {
	    return new Point2D.Double(ctrlx2, ctrly2);
	}

	/**
	 * Returns the X coordinate of the end point
	 * in double precision.
	 * @return the X coordinate of the end point of the
	 *		<code>CubicCurve2D</code>.
	 */
	public double getX2() {
	    return x2;
	}

	/**
	 * Returns the Y coordinate of the end point 
	 * in double precision.
	 * @return the Y coordinate of the end point of the
	 *		<code>CubicCurve2D</code>.
	 */
	public double getY2() {
	    return y2;
	}

	/**
	 * Returns the end point.
	 * @return a <code>Point2D</code> that is the end point of 
	 *		the <code>CubicCurve2D</code>.	
	 */
	public Point2D getP2() {
	    return new Point2D.Double(x2, y2);
	}

	/**
	 * Sets the location of the endpoints and controlpoints
	 * of this curve to the specified double coordinates.
	 * @param x1, y1 the first specified coordinates used to set the start
	 *		point of this <code>CubicCurve2D</code>
	 * @param ctrlx1, ctrly1 the second specified coordinates used to set the
	 *		first control point of this <code>CubicCurve2D</code>
	 * @param ctrlx2, ctrly2 the third specified coordinates used to set the 
	 *		second control point of this <code>CubicCurve2D</code>		
	 * @param x2, y2 the fourth specified coordinates used to set the end
	 *		point of this <code>CubicCurve2D</code>	 
	 */
	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;
	}

	/**
	 * Returns the bounding box of the shape.
	 * @return a <code>Rectangle2D</code> that is the bounding box
	 *		of the shape.
	 */
	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);
	}
    }

    /**
     * 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
     */
    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</code>.
     */
    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</code>.
     */
    public abstract double getY1();

    /**
     * Returns the start point.
     * @return a <code>Point2D</code> that is the start point of 
     *		the <code>CubicCurve2D</code>.
     */
    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</code>.
     */
    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</code>.
     */
    public abstract double getCtrlY1();

    /**
     * Returns the first control point.
     * @return a <code>Point2D</code> that is the first control point of 
     *		the <code>CubicCurve2D</code>.
     */
    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</code>.
     */
    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</code>.
     */
    public abstract double getCtrlY2();

    /**
     * Returns the second control point.
     * @return a <code>Point2D</code> that is the second control point of 
     *		the <code>CubicCurve2D</code>.
     */
    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</code>.
     */
    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</code>.
     */
    public abstract double getY2();

    /**
     * Returns the end point.
     * @return a <code>Point2D</code> that is the end point of 
     *		the <code>CubicCurve2D</code>.
     */
    public abstract Point2D getP2();

    /**
     * Sets the location of the endpoints and controlpoints of this curve
     * to the specified double coordinates.
     * @param x1, y1 the first specified coordinates used to set the start
     *		point of this <code>CubicCurve2D</code>
     * @param ctrlx1, ctrly1 the second specified coordinates used to set the
     *		first control point of this <code>CubicCurve2D</code>
     * @param ctrlx2, ctrly2 the third specified coordinates used to set the 
     *		second control point of this <code>CubicCurve2D</code>		
     * @param x2, y2 the fourth specified coordinates used to set the end
     *		point of this <code>CubicCurve2D</code>	 
     */
    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 endpoints and controlpoints 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> at which to begin 
     *		setting the endpoints and controlpoints of this curve
     *		to the coordinates contained in <code>coords</code>	
     */
    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 endpoints and controlpoints 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 p2 the second specified <code>Point2D</code> used to set the
     *		first control point of this curve
     * @param p3 the third specified <code>Point2D</code> used to set the
     *		second control point of this curve
     * @param p4 the fourth specified <code>Point2D</code> used to set the
     *		end point of this curve
     */
    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 endpoints and controlpoints 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> at which to begin setting
     *		the endpoints and controlpoints of this curve to the 
     *		points contained in <code>pts</code>
     */
    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 endpoints and controlpoints of this curve
     * to the same as those in the specified <code>CubicCurve2D</code>.
     * @param c the specified <code>CubicCurve2D</code>
     */
    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 controlpoints. The flatness is the maximum distance 
     * of a controlpoint from the line connecting the endpoints.
     * @param x1, y1 the first specified coordinates that specify the start
     *		point of a <code>CubicCurve2D</code>
     * @param ctrlx1, ctrly1 the second specified coordinates that specify the 
     * 		first control point of a <code>CubicCurve2D</code>
     * @param ctrlx2, ctrly2 the third specified coordinates that specify the 
     * 		second control point of a <code>CubicCurve2D</code>	
     * @param x2, y2 the fourth specified coordinates that specify the 
     * 		end point of a <code>CubicCurve2D</code>
     * @return the square of the flatness of the <code>CubicCurve2D</code>
     *		represented by the specified coordinates.
     */
    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 controlpoints. The flatness is the maximum distance 
     * of a controlpoint from the line connecting the endpoints.
     * @param x1, y1 the first specified coordinates that specify the start
     *		point of a <code>CubicCurve2D</code>
     * @param ctrlx1, ctrly1 the second specified coordinates that specify the 
     * 		first control point of a <code>CubicCurve2D</code>
     * @param ctrlx2, ctrly2 the third specified coordinates that specify the 
     * 		second control point of a <code>CubicCurve2D</code>	
     * @param x2, y2 the fourth specified coordinates that specify the 
     * 		end point of a <code>CubicCurve2D</code>
     * @return the flatness of the <code>CubicCurve2D</code>
     *		represented by the specified coordinates.
     */
    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 controlpoints stored in the indicated array at the 
     * indicated index. The flatness is the maximum distance 
     * of a controlpoint from the line connecting the endpoints.
     * @param coords an array containing coordinates
     * @param offset the index of <code>coords</code> at which to begin 
     *		setting the endpoints and controlpoints of this curve
     *		to the coordinates contained in <code>coords</code>
     * @return the square of the flatness of the <code>CubicCurve2D</code>
     *		specified by the coordinates in <code>coords</code> at
     *		the specified offset.
     */
    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 controlpoints stored in the indicated array at the 
     * indicated index.  The flatness is the maximum distance 
     * of a controlpoint from the line connecting the endpoints.
     * @param coords an array containing coordinates
     * @param offset the index of <code>coords</code> at which to begin 
     *		setting the endpoints and controlpoints of this curve
     *		to the coordinates contained in <code>coords</code>
     * @return the flatness of the <code>CubicCurve2D</code>
     *		specified by the coordinates in <code>coords</code> at
     *		the specified offset.
     */
    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 controlpoint from the line connecting the 
     * endpoints.
     * @return the square of the flatness of this curve.
     */
    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 controlpoint from the line connecting the 
     * endpoints.
     * @return the flatness of this curve.
     */
    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
     */
    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
     */
    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> + 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
     */
    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.
     */
    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.
     * @return the number of roots, or -1 if the equation is a constant
     */
    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;
	    }
	}
    }

    /**
     * Tests if a specified coordinate is inside the boundary of the shape.
     * @param x, y the specified coordinate to be tested
     * @return <code>true</code> if the coordinate is inside the boundary of
     *		the shape; <code>false</code> otherwise.
     */
    public boolean contains(double x, double y) {
	// We count the "Y" crossings to determine if the point is
	// inside the curve bounded by its closing line.
	int crossings = 0;
	double x1 = getX1();
	double y1 = getY1();
	double x2 = getX2();
	double y2 = getY2();
	// First check for a crossing of the line connecting the endpoints
	double dy = y2 - y1;
	if ((dy > 0.0 && y >= y1 && y <= y2) ||
	    (dy < 0.0 && y <= y1 && y >= y2))
	{
	    if (x < x1 + (y - y1) * (x2 - x1) / dy) {
		crossings++;
	    }
	}
	// Solve the Y parametric equation for intersections with y
	double ctrlx1 = getCtrlX1();
	double ctrly1 = getCtrlY1();
	double ctrlx2 = getCtrlX2();
	double ctrly2 = getCtrlY2();
	boolean include0 = ((y2 - y1) * (ctrly1 - y1) >= 0);
	boolean include1 = ((y1 - y2) * (ctrly2 - y2) >= 0);
	double eqn[] = new double[4];
	double res[] = new double[4];
	fillEqn(eqn, y, y1, ctrly1, ctrly2, y2);
	int roots = solveCubic(eqn, res);
	roots = evalCubic(res, roots,
			  include0, include1, eqn,
			  x1, ctrlx1, ctrlx2, x2);
	while (--roots >= 0) {
	    if (x < res[roots]) {
		crossings++;
	    }
	}
	return ((crossings & 1) == 1);
    }

    /**
     * Tests if a specified <code>Point2D</code> is inside the boundary of 
     * the shape.
     * @param p the specified <code>Point2D</code> to be tested
     * @return <code>true</code> if the <code>p</code> is inside the boundary
     *		of the shape; <code>false</code> otherwise.
     */
    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:
     *   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
     * @param x, y the coordinates of the upper left corner of the specified
     *		rectangular shape
     * @param w the width of the specified rectangular shape
     * @param h the height of the specified rectangular shape
     * @return <code>true</code> if the shape intersects the interior of the
     *		the specified set of rectangular coordinates; 
     *		<code>false</code> otherwise.
     */
    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);
	}
    }

    /**
     * Tests if the shape intersects the interior of a specified
     * set of rectangular coordinates.
     * @param x, y the coordinates of the upper left corner
     * 		of the specified rectangular area
     * @param w the width of the specified rectangular area
     * @param h the height of the specified rectangular area
     * @return <code>true</code> if the shape intersects the
     *		interior of the specified rectangular area;
     *		<code>false</code> otherwise.
     */
    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));
    }

    /**
     * Tests if the shape intersects the interior of a specified
     * <code>Rectangle2D</code>.
     * @param r the specified <code>Rectangle2D</code> to be tested
     * @return <code>true</code> if the shape intersects the interior of
     *		the specified <code>Rectangle2D</code>
     *		<code>false</code> otherwise.
     */
    public boolean intersects(Rectangle2D r) {
	return intersects(r.getX(), r.getY(), r.getWidth(), r.getHeight());
    }

    /**
     * Tests if the interior of the shape entirely contains the specified
     * set of rectangular coordinates.
     * @param x, y the coordinates of the upper left corner of the specified
     *		rectangular shape
     * @param w the width of the specified rectangular shape
     * @param h the height of the specified rectangular shape
     * @return <code>true</code> if the shape entirely contains
     *		the specified set of rectangular coordinates; 
     *		<code>false</code> otherwise.
     */
    public boolean contains(double x, double y, double w, double h) {
	// 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());
    }

    /**
     * Tests if the interior of the shape entirely contains the specified
     * <code>Rectangle2D</code>.
     * @param r the specified <code>Rectangle2D</code> to be tested
     * @return <code>true</code> if the shape entirely contains
     *		the specified <code>Rectangle2D</code> 
     *		<code>false</code> otherwise.
     */
    public boolean contains(Rectangle2D r) {
	return contains(r.getX(), r.getY(), r.getWidth(), r.getHeight());
    }

    /**
     * Returns the bounding box of the shape.
     * @return a {@link Rectangle} that is the bounding box of the shape.
     */
    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.
     */
    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 endpoints
     * @return    the <code>PathIterator</code> object that returns the
     * geometry of the outline of this <code>CubicCurve2D</code>, one segment at a time.
     */
    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();
	}
    }
}