001/* CubicCurve2D.java -- represents a parameterized cubic curve in 2-D space
002   Copyright (C) 2002, 2003, 2004 Free Software Foundation
003
004This file is part of GNU Classpath.
005
006GNU Classpath is free software; you can redistribute it and/or modify
007it under the terms of the GNU General Public License as published by
008the Free Software Foundation; either version 2, or (at your option)
009any later version.
010
011GNU Classpath is distributed in the hope that it will be useful, but
012WITHOUT ANY WARRANTY; without even the implied warranty of
013MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
014General Public License for more details.
015
016You should have received a copy of the GNU General Public License
017along with GNU Classpath; see the file COPYING.  If not, write to the
018Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
01902110-1301 USA.
020
021Linking this library statically or dynamically with other modules is
022making a combined work based on this library.  Thus, the terms and
023conditions of the GNU General Public License cover the whole
024combination.
025
026As a special exception, the copyright holders of this library give you
027permission to link this library with independent modules to produce an
028executable, regardless of the license terms of these independent
029modules, and to copy and distribute the resulting executable under
030terms of your choice, provided that you also meet, for each linked
031independent module, the terms and conditions of the license of that
032module.  An independent module is a module which is not derived from
033or based on this library.  If you modify this library, you may extend
034this exception to your version of the library, but you are not
035obligated to do so.  If you do not wish to do so, delete this
036exception statement from your version. */
037
038package java.awt.geom;
039
040import java.awt.Rectangle;
041import java.awt.Shape;
042import java.util.NoSuchElementException;
043
044
045/**
046 * A two-dimensional curve that is parameterized with a cubic
047 * function.
048 *
049 * <p><img src="doc-files/CubicCurve2D-1.png" width="350" height="180"
050 * alt="A drawing of a CubicCurve2D" />
051 *
052 * @author Eric Blake (ebb9@email.byu.edu)
053 * @author Graydon Hoare (graydon@redhat.com)
054 * @author Sascha Brawer (brawer@dandelis.ch)
055 * @author Sven de Marothy (sven@physto.se)
056 *
057 * @since 1.2
058 */
059public abstract class CubicCurve2D implements Shape, Cloneable
060{
061  private static final double BIG_VALUE = java.lang.Double.MAX_VALUE / 10.0;
062  private static final double EPSILON = 1E-10;
063
064  /**
065   * Constructs a new CubicCurve2D. Typical users will want to
066   * construct instances of a subclass, such as {@link
067   * CubicCurve2D.Float} or {@link CubicCurve2D.Double}.
068   */
069  protected CubicCurve2D()
070  {
071  }
072
073  /**
074   * Returns the <i>x</i> coordinate of the curve&#x2019;s start
075   * point.
076   */
077  public abstract double getX1();
078
079  /**
080   * Returns the <i>y</i> coordinate of the curve&#x2019;s start
081   * point.
082   */
083  public abstract double getY1();
084
085  /**
086   * Returns the curve&#x2019;s start point.
087   */
088  public abstract Point2D getP1();
089
090  /**
091   * Returns the <i>x</i> coordinate of the curve&#x2019;s first
092   * control point.
093   */
094  public abstract double getCtrlX1();
095
096  /**
097   * Returns the <i>y</i> coordinate of the curve&#x2019;s first
098   * control point.
099   */
100  public abstract double getCtrlY1();
101
102  /**
103   * Returns the curve&#x2019;s first control point.
104   */
105  public abstract Point2D getCtrlP1();
106
107  /**
108   * Returns the <i>x</i> coordinate of the curve&#x2019;s second
109   * control point.
110   */
111  public abstract double getCtrlX2();
112
113  /**
114   * Returns the <i>y</i> coordinate of the curve&#x2019;s second
115   * control point.
116   */
117  public abstract double getCtrlY2();
118
119  /**
120   * Returns the curve&#x2019;s second control point.
121   */
122  public abstract Point2D getCtrlP2();
123
124  /**
125   * Returns the <i>x</i> coordinate of the curve&#x2019;s end
126   * point.
127   */
128  public abstract double getX2();
129
130  /**
131   * Returns the <i>y</i> coordinate of the curve&#x2019;s end
132   * point.
133   */
134  public abstract double getY2();
135
136  /**
137   * Returns the curve&#x2019;s end point.
138   */
139  public abstract Point2D getP2();
140
141  /**
142   * Changes the curve geometry, separately specifying each coordinate
143   * value.
144   *
145   * <p><img src="doc-files/CubicCurve2D-1.png" width="350" height="180"
146   * alt="A drawing of a CubicCurve2D" />
147   *
148   * @param x1 the <i>x</i> coordinate of the curve&#x2019;s new start
149   * point.
150   *
151   * @param y1 the <i>y</i> coordinate of the curve&#x2019;s new start
152   * point.
153   *
154   * @param cx1 the <i>x</i> coordinate of the curve&#x2019;s new
155   * first control point.
156   *
157   * @param cy1 the <i>y</i> coordinate of the curve&#x2019;s new
158   * first control point.
159   *
160   * @param cx2 the <i>x</i> coordinate of the curve&#x2019;s new
161   * second control point.
162   *
163   * @param cy2 the <i>y</i> coordinate of the curve&#x2019;s new
164   * second control point.
165   *
166   * @param x2 the <i>x</i> coordinate of the curve&#x2019;s new end
167   * point.
168   *
169   * @param y2 the <i>y</i> coordinate of the curve&#x2019;s new end
170   * point.
171   */
172  public abstract void setCurve(double x1, double y1, double cx1, double cy1,
173                                double cx2, double cy2, double x2, double y2);
174
175  /**
176   * Changes the curve geometry, specifying coordinate values in an
177   * array.
178   *
179   * @param coords an array containing the new coordinate values.  The
180   * <i>x</i> coordinate of the new start point is located at
181   * <code>coords[offset]</code>, its <i>y</i> coordinate at
182   * <code>coords[offset + 1]</code>.  The <i>x</i> coordinate of the
183   * new first control point is located at <code>coords[offset +
184   * 2]</code>, its <i>y</i> coordinate at <code>coords[offset +
185   * 3]</code>.  The <i>x</i> coordinate of the new second control
186   * point is located at <code>coords[offset + 4]</code>, its <i>y</i>
187   * coordinate at <code>coords[offset + 5]</code>.  The <i>x</i>
188   * coordinate of the new end point is located at <code>coords[offset
189   * + 6]</code>, its <i>y</i> coordinate at <code>coords[offset +
190   * 7]</code>.
191   *
192   * @param offset the offset of the first coordinate value in
193   * <code>coords</code>.
194   */
195  public void setCurve(double[] coords, int offset)
196  {
197    setCurve(coords[offset++], coords[offset++], coords[offset++],
198             coords[offset++], coords[offset++], coords[offset++],
199             coords[offset++], coords[offset++]);
200  }
201
202  /**
203   * Changes the curve geometry, specifying coordinate values in
204   * separate Point objects.
205   *
206   * <p><img src="doc-files/CubicCurve2D-1.png" width="350" height="180"
207   * alt="A drawing of a CubicCurve2D" />
208   *
209   * <p>The curve does not keep any reference to the passed point
210   * objects. Therefore, a later change to <code>p1</code>,
211   * <code>c1</code>, <code>c2</code> or <code>p2</code> will not
212   * affect the curve geometry.
213   *
214   * @param p1 the new start point.
215   * @param c1 the new first control point.
216   * @param c2 the new second control point.
217   * @param p2 the new end point.
218   */
219  public void setCurve(Point2D p1, Point2D c1, Point2D c2, Point2D p2)
220  {
221    setCurve(p1.getX(), p1.getY(), c1.getX(), c1.getY(), c2.getX(), c2.getY(),
222             p2.getX(), p2.getY());
223  }
224
225  /**
226   * Changes the curve geometry, specifying coordinate values in an
227   * array of Point objects.
228   *
229   * <p><img src="doc-files/CubicCurve2D-1.png" width="350" height="180"
230   * alt="A drawing of a CubicCurve2D" />
231   *
232   * <p>The curve does not keep references to the passed point
233   * objects. Therefore, a later change to the <code>pts</code> array
234   * or any of its elements will not affect the curve geometry.
235   *
236   * @param pts an array containing the points. The new start point
237   * is located at <code>pts[offset]</code>, the new first control
238   * point at <code>pts[offset + 1]</code>, the new second control
239   * point at <code>pts[offset + 2]</code>, and the new end point
240   * at <code>pts[offset + 3]</code>.
241   *
242   * @param offset the offset of the start point in <code>pts</code>.
243   */
244  public void setCurve(Point2D[] pts, int offset)
245  {
246    setCurve(pts[offset].getX(), pts[offset++].getY(), pts[offset].getX(),
247             pts[offset++].getY(), pts[offset].getX(), pts[offset++].getY(),
248             pts[offset].getX(), pts[offset++].getY());
249  }
250
251  /**
252   * Changes the curve geometry to that of another curve.
253   *
254   * @param c the curve whose coordinates will be copied.
255   */
256  public void setCurve(CubicCurve2D c)
257  {
258    setCurve(c.getX1(), c.getY1(), c.getCtrlX1(), c.getCtrlY1(),
259             c.getCtrlX2(), c.getCtrlY2(), c.getX2(), c.getY2());
260  }
261
262  /**
263   * Calculates the squared flatness of a cubic curve, directly
264   * specifying each coordinate value. The flatness is the maximal
265   * distance of a control point to the line between start and end
266   * point.
267   *
268   * <p><img src="doc-files/CubicCurve2D-4.png" width="350" height="180"
269   * alt="A drawing that illustrates the flatness" />
270   *
271   * <p>In the above drawing, the straight line connecting start point
272   * P1 and end point P2 is depicted in gray.  In comparison to C1,
273   * control point C2 is father away from the gray line. Therefore,
274   * the result will be the square of the distance between C2 and the
275   * gray line, i.e. the squared length of the red line.
276   *
277   * @param x1 the <i>x</i> coordinate of the start point P1.
278   * @param y1 the <i>y</i> coordinate of the start point P1.
279   * @param cx1 the <i>x</i> coordinate of the first control point C1.
280   * @param cy1 the <i>y</i> coordinate of the first control point C1.
281   * @param cx2 the <i>x</i> coordinate of the second control point C2.
282   * @param cy2 the <i>y</i> coordinate of the second control point C2.
283   * @param x2 the <i>x</i> coordinate of the end point P2.
284   * @param y2 the <i>y</i> coordinate of the end point P2.
285   */
286  public static double getFlatnessSq(double x1, double y1, double cx1,
287                                     double cy1, double cx2, double cy2,
288                                     double x2, double y2)
289  {
290    return Math.max(Line2D.ptSegDistSq(x1, y1, x2, y2, cx1, cy1),
291                    Line2D.ptSegDistSq(x1, y1, x2, y2, cx2, cy2));
292  }
293
294  /**
295   * Calculates the flatness of a cubic curve, directly specifying
296   * each coordinate value. The flatness is the maximal distance of a
297   * control point to the line between start and end point.
298   *
299   * <p><img src="doc-files/CubicCurve2D-4.png" width="350" height="180"
300   * alt="A drawing that illustrates the flatness" />
301   *
302   * <p>In the above drawing, the straight line connecting start point
303   * P1 and end point P2 is depicted in gray.  In comparison to C1,
304   * control point C2 is father away from the gray line. Therefore,
305   * the result will be the distance between C2 and the gray line,
306   * i.e. the length of the red line.
307   *
308   * @param x1 the <i>x</i> coordinate of the start point P1.
309   * @param y1 the <i>y</i> coordinate of the start point P1.
310   * @param cx1 the <i>x</i> coordinate of the first control point C1.
311   * @param cy1 the <i>y</i> coordinate of the first control point C1.
312   * @param cx2 the <i>x</i> coordinate of the second control point C2.
313   * @param cy2 the <i>y</i> coordinate of the second control point C2.
314   * @param x2 the <i>x</i> coordinate of the end point P2.
315   * @param y2 the <i>y</i> coordinate of the end point P2.
316   */
317  public static double getFlatness(double x1, double y1, double cx1,
318                                   double cy1, double cx2, double cy2,
319                                   double x2, double y2)
320  {
321    return Math.sqrt(getFlatnessSq(x1, y1, cx1, cy1, cx2, cy2, x2, y2));
322  }
323
324  /**
325   * Calculates the squared flatness of a cubic curve, specifying the
326   * coordinate values in an array. The flatness is the maximal
327   * distance of a control point to the line between start and end
328   * point.
329   *
330   * <p><img src="doc-files/CubicCurve2D-4.png" width="350" height="180"
331   * alt="A drawing that illustrates the flatness" />
332   *
333   * <p>In the above drawing, the straight line connecting start point
334   * P1 and end point P2 is depicted in gray.  In comparison to C1,
335   * control point C2 is father away from the gray line. Therefore,
336   * the result will be the square of the distance between C2 and the
337   * gray line, i.e. the squared length of the red line.
338   *
339   * @param coords an array containing the coordinate values.  The
340   * <i>x</i> coordinate of the start point P1 is located at
341   * <code>coords[offset]</code>, its <i>y</i> coordinate at
342   * <code>coords[offset + 1]</code>.  The <i>x</i> coordinate of the
343   * first control point C1 is located at <code>coords[offset +
344   * 2]</code>, its <i>y</i> coordinate at <code>coords[offset +
345   * 3]</code>. The <i>x</i> coordinate of the second control point C2
346   * is located at <code>coords[offset + 4]</code>, its <i>y</i>
347   * coordinate at <code>coords[offset + 5]</code>. The <i>x</i>
348   * coordinate of the end point P2 is located at <code>coords[offset
349   * + 6]</code>, its <i>y</i> coordinate at <code>coords[offset +
350   * 7]</code>.
351   *
352   * @param offset the offset of the first coordinate value in
353   * <code>coords</code>.
354   */
355  public static double getFlatnessSq(double[] coords, int offset)
356  {
357    return getFlatnessSq(coords[offset++], coords[offset++], coords[offset++],
358                         coords[offset++], coords[offset++], coords[offset++],
359                         coords[offset++], coords[offset++]);
360  }
361
362  /**
363   * Calculates the flatness of a cubic curve, specifying the
364   * coordinate values in an array. The flatness is the maximal
365   * distance of a control point to the line between start and end
366   * point.
367   *
368   * <p><img src="doc-files/CubicCurve2D-4.png" width="350" height="180"
369   * alt="A drawing that illustrates the flatness" />
370   *
371   * <p>In the above drawing, the straight line connecting start point
372   * P1 and end point P2 is depicted in gray.  In comparison to C1,
373   * control point C2 is father away from the gray line. Therefore,
374   * the result will be the distance between C2 and the gray line,
375   * i.e. the length of the red line.
376   *
377   * @param coords an array containing the coordinate values.  The
378   * <i>x</i> coordinate of the start point P1 is located at
379   * <code>coords[offset]</code>, its <i>y</i> coordinate at
380   * <code>coords[offset + 1]</code>.  The <i>x</i> coordinate of the
381   * first control point C1 is located at <code>coords[offset +
382   * 2]</code>, its <i>y</i> coordinate at <code>coords[offset +
383   * 3]</code>. The <i>x</i> coordinate of the second control point C2
384   * is located at <code>coords[offset + 4]</code>, its <i>y</i>
385   * coordinate at <code>coords[offset + 5]</code>. The <i>x</i>
386   * coordinate of the end point P2 is located at <code>coords[offset
387   * + 6]</code>, its <i>y</i> coordinate at <code>coords[offset +
388   * 7]</code>.
389   *
390   * @param offset the offset of the first coordinate value in
391   * <code>coords</code>.
392   */
393  public static double getFlatness(double[] coords, int offset)
394  {
395    return Math.sqrt(getFlatnessSq(coords[offset++], coords[offset++],
396                                   coords[offset++], coords[offset++],
397                                   coords[offset++], coords[offset++],
398                                   coords[offset++], coords[offset++]));
399  }
400
401  /**
402   * Calculates the squared flatness of this curve.  The flatness is
403   * the maximal distance of a control point to the line between start
404   * and end point.
405   *
406   * <p><img src="doc-files/CubicCurve2D-4.png" width="350" height="180"
407   * alt="A drawing that illustrates the flatness" />
408   *
409   * <p>In the above drawing, the straight line connecting start point
410   * P1 and end point P2 is depicted in gray.  In comparison to C1,
411   * control point C2 is father away from the gray line. Therefore,
412   * the result will be the square of the distance between C2 and the
413   * gray line, i.e. the squared length of the red line.
414   */
415  public double getFlatnessSq()
416  {
417    return getFlatnessSq(getX1(), getY1(), getCtrlX1(), getCtrlY1(),
418                         getCtrlX2(), getCtrlY2(), getX2(), getY2());
419  }
420
421  /**
422   * Calculates the flatness of this curve.  The flatness is the
423   * maximal distance of a control point to the line between start and
424   * end point.
425   *
426   * <p><img src="doc-files/CubicCurve2D-4.png" width="350" height="180"
427   * alt="A drawing that illustrates the flatness" />
428   *
429   * <p>In the above drawing, the straight line connecting start point
430   * P1 and end point P2 is depicted in gray.  In comparison to C1,
431   * control point C2 is father away from the gray line. Therefore,
432   * the result will be the distance between C2 and the gray line,
433   * i.e. the length of the red line.
434   */
435  public double getFlatness()
436  {
437    return Math.sqrt(getFlatnessSq(getX1(), getY1(), getCtrlX1(), getCtrlY1(),
438                                   getCtrlX2(), getCtrlY2(), getX2(), getY2()));
439  }
440
441  /**
442   * Subdivides this curve into two halves.
443   *
444   * <p><img src="doc-files/CubicCurve2D-3.png" width="700"
445   * height="180" alt="A drawing that illustrates the effects of
446   * subdividing a CubicCurve2D" />
447   *
448   * @param left a curve whose geometry will be set to the left half
449   * of this curve, or <code>null</code> if the caller is not
450   * interested in the left half.
451   *
452   * @param right a curve whose geometry will be set to the right half
453   * of this curve, or <code>null</code> if the caller is not
454   * interested in the right half.
455   */
456  public void subdivide(CubicCurve2D left, CubicCurve2D right)
457  {
458    // Use empty slots at end to share single array.
459    double[] d = new double[]
460                 {
461                   getX1(), getY1(), getCtrlX1(), getCtrlY1(), getCtrlX2(),
462                   getCtrlY2(), getX2(), getY2(), 0, 0, 0, 0, 0, 0
463                 };
464    subdivide(d, 0, d, 0, d, 6);
465    if (left != null)
466      left.setCurve(d, 0);
467    if (right != null)
468      right.setCurve(d, 6);
469  }
470
471  /**
472   * Subdivides a cubic curve into two halves.
473   *
474   * <p><img src="doc-files/CubicCurve2D-3.png" width="700"
475   * height="180" alt="A drawing that illustrates the effects of
476   * subdividing a CubicCurve2D" />
477   *
478   * @param src the curve to be subdivided.
479   *
480   * @param left a curve whose geometry will be set to the left half
481   * of <code>src</code>, or <code>null</code> if the caller is not
482   * interested in the left half.
483   *
484   * @param right a curve whose geometry will be set to the right half
485   * of <code>src</code>, or <code>null</code> if the caller is not
486   * interested in the right half.
487   */
488  public static void subdivide(CubicCurve2D src, CubicCurve2D left,
489                               CubicCurve2D right)
490  {
491    src.subdivide(left, right);
492  }
493
494  /**
495   * Subdivides a cubic curve into two halves, passing all coordinates
496   * in an array.
497   *
498   * <p><img src="doc-files/CubicCurve2D-3.png" width="700"
499   * height="180" alt="A drawing that illustrates the effects of
500   * subdividing a CubicCurve2D" />
501   *
502   * <p>The left end point and the right start point will always be
503   * identical. Memory-concious programmers thus may want to pass the
504   * same array for both <code>left</code> and <code>right</code>, and
505   * set <code>rightOff</code> to <code>leftOff + 6</code>.
506   *
507   * @param src an array containing the coordinates of the curve to be
508   * subdivided.  The <i>x</i> coordinate of the start point P1 is
509   * located at <code>src[srcOff]</code>, its <i>y</i> at
510   * <code>src[srcOff + 1]</code>.  The <i>x</i> coordinate of the
511   * first control point C1 is located at <code>src[srcOff +
512   * 2]</code>, its <i>y</i> at <code>src[srcOff + 3]</code>.  The
513   * <i>x</i> coordinate of the second control point C2 is located at
514   * <code>src[srcOff + 4]</code>, its <i>y</i> at <code>src[srcOff +
515   * 5]</code>. The <i>x</i> coordinate of the end point is located at
516   * <code>src[srcOff + 6]</code>, its <i>y</i> at <code>src[srcOff +
517   * 7]</code>.
518   *
519   * @param srcOff an offset into <code>src</code>, specifying
520   * the index of the start point&#x2019;s <i>x</i> coordinate.
521   *
522   * @param left an array that will receive the coordinates of the
523   * left half of <code>src</code>. It is acceptable to pass
524   * <code>src</code>. A caller who is not interested in the left half
525   * can pass <code>null</code>.
526   *
527   * @param leftOff an offset into <code>left</code>, specifying the
528   * index where the start point&#x2019;s <i>x</i> coordinate will be
529   * stored.
530   *
531   * @param right an array that will receive the coordinates of the
532   * right half of <code>src</code>. It is acceptable to pass
533   * <code>src</code> or <code>left</code>. A caller who is not
534   * interested in the right half can pass <code>null</code>.
535   *
536   * @param rightOff an offset into <code>right</code>, specifying the
537   * index where the start point&#x2019;s <i>x</i> coordinate will be
538   * stored.
539   */
540  public static void subdivide(double[] src, int srcOff, double[] left,
541                               int leftOff, double[] right, int rightOff)
542  {
543    // To understand this code, please have a look at the image
544    // "CubicCurve2D-3.png" in the sub-directory "doc-files".
545    double src_C1_x;
546    double src_C1_y;
547    double src_C2_x;
548    double src_C2_y;
549    double left_P1_x;
550    double left_P1_y;
551    double left_C1_x;
552    double left_C1_y;
553    double left_C2_x;
554    double left_C2_y;
555    double right_C1_x;
556    double right_C1_y;
557    double right_C2_x;
558    double right_C2_y;
559    double right_P2_x;
560    double right_P2_y;
561    double Mid_x; // Mid = left.P2 = right.P1
562    double Mid_y; // Mid = left.P2 = right.P1
563
564    left_P1_x = src[srcOff];
565    left_P1_y = src[srcOff + 1];
566    src_C1_x = src[srcOff + 2];
567    src_C1_y = src[srcOff + 3];
568    src_C2_x = src[srcOff + 4];
569    src_C2_y = src[srcOff + 5];
570    right_P2_x = src[srcOff + 6];
571    right_P2_y = src[srcOff + 7];
572
573    left_C1_x = (left_P1_x + src_C1_x) / 2;
574    left_C1_y = (left_P1_y + src_C1_y) / 2;
575    right_C2_x = (right_P2_x + src_C2_x) / 2;
576    right_C2_y = (right_P2_y + src_C2_y) / 2;
577    Mid_x = (src_C1_x + src_C2_x) / 2;
578    Mid_y = (src_C1_y + src_C2_y) / 2;
579    left_C2_x = (left_C1_x + Mid_x) / 2;
580    left_C2_y = (left_C1_y + Mid_y) / 2;
581    right_C1_x = (Mid_x + right_C2_x) / 2;
582    right_C1_y = (Mid_y + right_C2_y) / 2;
583    Mid_x = (left_C2_x + right_C1_x) / 2;
584    Mid_y = (left_C2_y + right_C1_y) / 2;
585
586    if (left != null)
587      {
588        left[leftOff] = left_P1_x;
589        left[leftOff + 1] = left_P1_y;
590        left[leftOff + 2] = left_C1_x;
591        left[leftOff + 3] = left_C1_y;
592        left[leftOff + 4] = left_C2_x;
593        left[leftOff + 5] = left_C2_y;
594        left[leftOff + 6] = Mid_x;
595        left[leftOff + 7] = Mid_y;
596      }
597
598    if (right != null)
599      {
600        right[rightOff] = Mid_x;
601        right[rightOff + 1] = Mid_y;
602        right[rightOff + 2] = right_C1_x;
603        right[rightOff + 3] = right_C1_y;
604        right[rightOff + 4] = right_C2_x;
605        right[rightOff + 5] = right_C2_y;
606        right[rightOff + 6] = right_P2_x;
607        right[rightOff + 7] = right_P2_y;
608      }
609  }
610
611  /**
612   * Finds the non-complex roots of a cubic equation, placing the
613   * results into the same array as the equation coefficients. The
614   * following equation is being solved:
615   *
616   * <blockquote><code>eqn[3]</code> &#xb7; <i>x</i><sup>3</sup>
617   * + <code>eqn[2]</code> &#xb7; <i>x</i><sup>2</sup>
618   * + <code>eqn[1]</code> &#xb7; <i>x</i>
619   * + <code>eqn[0]</code>
620   * = 0
621   * </blockquote>
622   *
623   * <p>For some background about solving cubic equations, see the
624   * article <a
625   * href="http://planetmath.org/encyclopedia/CubicFormula.html"
626   * >&#x201c;Cubic Formula&#x201d;</a> in <a
627   * href="http://planetmath.org/" >PlanetMath</a>.  For an extensive
628   * library of numerical algorithms written in the C programming
629   * language, see the <a href= "http://www.gnu.org/software/gsl/">GNU
630   * Scientific Library</a>, from which this implementation was
631   * adapted.
632   *
633   * @param eqn an array with the coefficients of the equation. When
634   * this procedure has returned, <code>eqn</code> will contain the
635   * non-complex solutions of the equation, in no particular order.
636   *
637   * @return the number of non-complex solutions. A result of 0
638   * indicates that the equation has no non-complex solutions. A
639   * result of -1 indicates that the equation is constant (i.e.,
640   * always or never zero).
641   *
642   * @see #solveCubic(double[], double[])
643   * @see QuadCurve2D#solveQuadratic(double[],double[])
644   *
645   * @author Brian Gough (bjg@network-theory.com)
646   * (original C implementation in the <a href=
647   * "http://www.gnu.org/software/gsl/">GNU Scientific Library</a>)
648   *
649   * @author Sascha Brawer (brawer@dandelis.ch)
650   * (adaptation to Java)
651   */
652  public static int solveCubic(double[] eqn)
653  {
654    return solveCubic(eqn, eqn);
655  }
656
657  /**
658   * Finds the non-complex roots of a cubic equation. The following
659   * equation is being solved:
660   *
661   * <blockquote><code>eqn[3]</code> &#xb7; <i>x</i><sup>3</sup>
662   * + <code>eqn[2]</code> &#xb7; <i>x</i><sup>2</sup>
663   * + <code>eqn[1]</code> &#xb7; <i>x</i>
664   * + <code>eqn[0]</code>
665   * = 0
666   * </blockquote>
667   *
668   * <p>For some background about solving cubic equations, see the
669   * article <a
670   * href="http://planetmath.org/encyclopedia/CubicFormula.html"
671   * >&#x201c;Cubic Formula&#x201d;</a> in <a
672   * href="http://planetmath.org/" >PlanetMath</a>.  For an extensive
673   * library of numerical algorithms written in the C programming
674   * language, see the <a href= "http://www.gnu.org/software/gsl/">GNU
675   * Scientific Library</a>, from which this implementation was
676   * adapted.
677   *
678   * @see QuadCurve2D#solveQuadratic(double[],double[])
679   *
680   * @param eqn an array with the coefficients of the equation.
681   *
682   * @param res an array into which the non-complex roots will be
683   * stored.  The results may be in an arbitrary order. It is safe to
684   * pass the same array object reference for both <code>eqn</code>
685   * and <code>res</code>.
686   *
687   * @return the number of non-complex solutions. A result of 0
688   * indicates that the equation has no non-complex solutions. A
689   * result of -1 indicates that the equation is constant (i.e.,
690   * always or never zero).
691   *
692   * @author Brian Gough (bjg@network-theory.com)
693   * (original C implementation in the <a href=
694   * "http://www.gnu.org/software/gsl/">GNU Scientific Library</a>)
695   *
696   * @author Sascha Brawer (brawer@dandelis.ch)
697   * (adaptation to Java)
698   */
699  public static int solveCubic(double[] eqn, double[] res)
700  {
701    // Adapted from poly/solve_cubic.c in the GNU Scientific Library
702    // (GSL), revision 1.7 of 2003-07-26. For the original source, see
703    // http://www.gnu.org/software/gsl/
704    //
705    // Brian Gough, the author of that code, has granted the
706    // permission to use it in GNU Classpath under the GNU Classpath
707    // license, and has assigned the copyright to the Free Software
708    // Foundation.
709    //
710    // The Java implementation is very similar to the GSL code, but
711    // not a strict one-to-one copy. For example, GSL would sort the
712    // result.
713    
714    double a;
715    double b;
716    double c;
717    double q;
718    double r;
719    double Q;
720    double R;
721    double c3;
722    double Q3;
723    double R2;
724    double CR2;
725    double CQ3;
726
727    // If the cubic coefficient is zero, we have a quadratic equation.
728    c3 = eqn[3];
729    if (c3 == 0)
730      return QuadCurve2D.solveQuadratic(eqn, res);
731
732    // Divide the equation by the cubic coefficient.
733    c = eqn[0] / c3;
734    b = eqn[1] / c3;
735    a = eqn[2] / c3;
736
737    // We now need to solve x^3 + ax^2 + bx + c = 0.
738    q = a * a - 3 * b;
739    r = 2 * a * a * a - 9 * a * b + 27 * c;
740
741    Q = q / 9;
742    R = r / 54;
743
744    Q3 = Q * Q * Q;
745    R2 = R * R;
746
747    CR2 = 729 * r * r;
748    CQ3 = 2916 * q * q * q;
749
750    if (R == 0 && Q == 0)
751      {
752        // The GNU Scientific Library would return three identical
753        // solutions in this case.
754        res[0] = -a / 3;
755        return 1;
756      }
757
758    if (CR2 == CQ3)
759      {
760        /* this test is actually R2 == Q3, written in a form suitable
761           for exact computation with integers */
762        /* Due to finite precision some double roots may be missed, and
763           considered to be a pair of complex roots z = x +/- epsilon i
764           close to the real axis. */
765        double sqrtQ = Math.sqrt(Q);
766
767        if (R > 0)
768          {
769            res[0] = -2 * sqrtQ - a / 3;
770            res[1] = sqrtQ - a / 3;
771          }
772        else
773          {
774            res[0] = -sqrtQ - a / 3;
775            res[1] = 2 * sqrtQ - a / 3;
776          }
777        return 2;
778      }
779
780    if (CR2 < CQ3) /* equivalent to R2 < Q3 */
781      {
782        double sqrtQ = Math.sqrt(Q);
783        double sqrtQ3 = sqrtQ * sqrtQ * sqrtQ;
784        double theta = Math.acos(R / sqrtQ3);
785        double norm = -2 * sqrtQ;
786        res[0] = norm * Math.cos(theta / 3) - a / 3;
787        res[1] = norm * Math.cos((theta + 2.0 * Math.PI) / 3) - a / 3;
788        res[2] = norm * Math.cos((theta - 2.0 * Math.PI) / 3) - a / 3;
789
790        // The GNU Scientific Library sorts the results. We don't.
791        return 3;
792      }
793
794    double sgnR = (R >= 0 ? 1 : -1);
795    double A = -sgnR * Math.pow(Math.abs(R) + Math.sqrt(R2 - Q3), 1.0 / 3.0);
796    double B = Q / A;
797    res[0] = A + B - a / 3;
798    return 1;
799  }
800
801  /**
802   * Determines whether a position lies inside the area bounded
803   * by the curve and the straight line connecting its end points.
804   *
805   * <p><img src="doc-files/CubicCurve2D-5.png" width="350" height="180"
806   * alt="A drawing of the area spanned by the curve" />
807   *
808   * <p>The above drawing illustrates in which area points are
809   * considered &#x201c;inside&#x201d; a CubicCurve2D.
810   */
811  public boolean contains(double x, double y)
812  {
813    if (! getBounds2D().contains(x, y))
814      return false;
815
816    return ((getAxisIntersections(x, y, true, BIG_VALUE) & 1) != 0);
817  }
818
819  /**
820   * Determines whether a point lies inside the area bounded
821   * by the curve and the straight line connecting its end points.
822   *
823   * <p><img src="doc-files/CubicCurve2D-5.png" width="350" height="180"
824   * alt="A drawing of the area spanned by the curve" />
825   *
826   * <p>The above drawing illustrates in which area points are
827   * considered &#x201c;inside&#x201d; a CubicCurve2D.
828   */
829  public boolean contains(Point2D p)
830  {
831    return contains(p.getX(), p.getY());
832  }
833
834  /**
835   * Determines whether any part of a rectangle is inside the area bounded
836   * by the curve and the straight line connecting its end points.
837   *
838   * <p><img src="doc-files/CubicCurve2D-5.png" width="350" height="180"
839   * alt="A drawing of the area spanned by the curve" />
840   *
841   * <p>The above drawing illustrates in which area points are
842   * considered &#x201c;inside&#x201d; in a CubicCurve2D.
843   * @see #contains(double, double)
844   */
845  public boolean intersects(double x, double y, double w, double h)
846  {
847    if (! getBounds2D().contains(x, y, w, h))
848      return false;
849
850    /* Does any edge intersect? */
851    if (getAxisIntersections(x, y, true, w) != 0 /* top */
852        || getAxisIntersections(x, y + h, true, w) != 0 /* bottom */
853        || getAxisIntersections(x + w, y, false, h) != 0 /* right */
854        || getAxisIntersections(x, y, false, h) != 0) /* left */
855      return true;
856
857    /* No intersections, is any point inside? */
858    if ((getAxisIntersections(x, y, true, BIG_VALUE) & 1) != 0)
859      return true;
860
861    return false;
862  }
863
864  /**
865   * Determines whether any part of a Rectangle2D is inside the area bounded 
866   * by the curve and the straight line connecting its end points.
867   * @see #intersects(double, double, double, double)
868   */
869  public boolean intersects(Rectangle2D r)
870  {
871    return intersects(r.getX(), r.getY(), r.getWidth(), r.getHeight());
872  }
873
874  /**
875   * Determine whether a rectangle is entirely inside the area that is bounded
876   * by the curve and the straight line connecting its end points.
877   *
878   * <p><img src="doc-files/CubicCurve2D-5.png" width="350" height="180"
879   * alt="A drawing of the area spanned by the curve" />
880   *
881   * <p>The above drawing illustrates in which area points are
882   * considered &#x201c;inside&#x201d; a CubicCurve2D.
883   * @see #contains(double, double)
884   */
885  public boolean contains(double x, double y, double w, double h)
886  {
887    if (! getBounds2D().intersects(x, y, w, h))
888      return false;
889
890    /* Does any edge intersect? */
891    if (getAxisIntersections(x, y, true, w) != 0 /* top */
892        || getAxisIntersections(x, y + h, true, w) != 0 /* bottom */
893        || getAxisIntersections(x + w, y, false, h) != 0 /* right */
894        || getAxisIntersections(x, y, false, h) != 0) /* left */
895      return false;
896
897    /* No intersections, is any point inside? */
898    if ((getAxisIntersections(x, y, true, BIG_VALUE) & 1) != 0)
899      return true;
900
901    return false;
902  }
903
904  /**
905   * Determine whether a Rectangle2D is entirely inside the area that is 
906   * bounded by the curve and the straight line connecting its end points.
907   *
908   * <p><img src="doc-files/CubicCurve2D-5.png" width="350" height="180"
909   * alt="A drawing of the area spanned by the curve" />
910   *
911   * <p>The above drawing illustrates in which area points are
912   * considered &#x201c;inside&#x201d; a CubicCurve2D.
913   * @see #contains(double, double)
914   */
915  public boolean contains(Rectangle2D r)
916  {
917    return contains(r.getX(), r.getY(), r.getWidth(), r.getHeight());
918  }
919
920  /**
921   * Determines the smallest rectangle that encloses the
922   * curve&#x2019;s start, end and control points.
923   */
924  public Rectangle getBounds()
925  {
926    return getBounds2D().getBounds();
927  }
928
929  public PathIterator getPathIterator(final AffineTransform at)
930  {
931    return new PathIterator()
932      {
933        /** Current coordinate. */
934        private int current = 0;
935
936        public int getWindingRule()
937        {
938          return WIND_NON_ZERO;
939        }
940
941        public boolean isDone()
942        {
943          return current >= 2;
944        }
945
946        public void next()
947        {
948          current++;
949        }
950
951        public int currentSegment(float[] coords)
952        {
953          int result;
954          switch (current)
955            {
956            case 0:
957              coords[0] = (float) getX1();
958              coords[1] = (float) getY1();
959              result = SEG_MOVETO;
960              break;
961            case 1:
962              coords[0] = (float) getCtrlX1();
963              coords[1] = (float) getCtrlY1();
964              coords[2] = (float) getCtrlX2();
965              coords[3] = (float) getCtrlY2();
966              coords[4] = (float) getX2();
967              coords[5] = (float) getY2();
968              result = SEG_CUBICTO;
969              break;
970            default:
971              throw new NoSuchElementException("cubic iterator out of bounds");
972            }
973          if (at != null)
974            at.transform(coords, 0, coords, 0, 3);
975          return result;
976        }
977
978        public int currentSegment(double[] coords)
979        {
980          int result;
981          switch (current)
982            {
983            case 0:
984              coords[0] = getX1();
985              coords[1] = getY1();
986              result = SEG_MOVETO;
987              break;
988            case 1:
989              coords[0] = getCtrlX1();
990              coords[1] = getCtrlY1();
991              coords[2] = getCtrlX2();
992              coords[3] = getCtrlY2();
993              coords[4] = getX2();
994              coords[5] = getY2();
995              result = SEG_CUBICTO;
996              break;
997            default:
998              throw new NoSuchElementException("cubic iterator out of bounds");
999            }
1000          if (at != null)
1001            at.transform(coords, 0, coords, 0, 3);
1002          return result;
1003        }
1004      };
1005  }
1006
1007  public PathIterator getPathIterator(AffineTransform at, double flatness)
1008  {
1009    return new FlatteningPathIterator(getPathIterator(at), flatness);
1010  }
1011
1012  /**
1013   * Create a new curve with the same contents as this one.
1014   *
1015   * @return the clone.
1016   */
1017  public Object clone()
1018  {
1019    try
1020      {
1021        return super.clone();
1022      }
1023    catch (CloneNotSupportedException e)
1024      {
1025        throw (Error) new InternalError().initCause(e); // Impossible
1026      }
1027  }
1028
1029  /**
1030   * Helper method used by contains() and intersects() methods, that
1031   * returns the number of curve/line intersections on a given axis
1032   * extending from a certain point.
1033   *
1034   * @param x x coordinate of the origin point
1035   * @param y y coordinate of the origin point
1036   * @param useYaxis axis used, if true the positive Y axis is used,
1037   * false uses the positive X axis.
1038   *
1039   * This is an implementation of the line-crossings algorithm,
1040   * Detailed in an article on Eric Haines' page:
1041   * http://www.acm.org/tog/editors/erich/ptinpoly/
1042   *
1043   * A special-case not adressed in this code is self-intersections
1044   * of the curve, e.g. if the axis intersects the self-itersection,
1045   * the degenerate roots of the polynomial will erroneously count as 
1046   * a single intersection of the curve, and not two.
1047   */
1048  private int getAxisIntersections(double x, double y, boolean useYaxis,
1049                                   double distance)
1050  {
1051    int nCrossings = 0;
1052    double a0;
1053    double a1;
1054    double a2;
1055    double a3;
1056    double b0;
1057    double b1;
1058    double b2;
1059    double b3;
1060    double[] r = new double[4];
1061    int nRoots;
1062
1063    a0 = a3 = 0.0;
1064
1065    if (useYaxis)
1066      {
1067        a0 = getY1() - y;
1068        a1 = getCtrlY1() - y;
1069        a2 = getCtrlY2() - y;
1070        a3 = getY2() - y;
1071        b0 = getX1() - x;
1072        b1 = getCtrlX1() - x;
1073        b2 = getCtrlX2() - x;
1074        b3 = getX2() - x;
1075      }
1076    else
1077      {
1078        a0 = getX1() - x;
1079        a1 = getCtrlX1() - x;
1080        a2 = getCtrlX2() - x;
1081        a3 = getX2() - x;
1082        b0 = getY1() - y;
1083        b1 = getCtrlY1() - y;
1084        b2 = getCtrlY2() - y;
1085        b3 = getY2() - y;
1086      }
1087
1088    /* If the axis intersects a start/endpoint, shift it up by some small 
1089       amount to guarantee the line is 'inside'
1090       If this is not done, bad behaviour may result for points on that axis.*/
1091    if (a0 == 0.0 || a3 == 0.0)
1092      {
1093        double small = getFlatness() * EPSILON;
1094        if (a0 == 0.0)
1095          a0 -= small;
1096        if (a3 == 0.0)
1097          a3 -= small;
1098      }
1099
1100    if (useYaxis)
1101      {
1102        if (Line2D.linesIntersect(b0, a0, b3, a3, EPSILON, 0.0, distance, 0.0))
1103          nCrossings++;
1104      }
1105    else
1106      {
1107        if (Line2D.linesIntersect(a0, b0, a3, b3, 0.0, EPSILON, 0.0, distance))
1108          nCrossings++;
1109      }
1110
1111    r[0] = a0;
1112    r[1] = 3 * (a1 - a0);
1113    r[2] = 3 * (a2 + a0 - 2 * a1);
1114    r[3] = a3 - 3 * a2 + 3 * a1 - a0;
1115
1116    if ((nRoots = solveCubic(r)) != 0)
1117      for (int i = 0; i < nRoots; i++)
1118        {
1119          double t = r[i];
1120          if (t >= 0.0 && t <= 1.0)
1121            {
1122              double crossing = -(t * t * t) * (b0 - 3 * b1 + 3 * b2 - b3)
1123                                + 3 * t * t * (b0 - 2 * b1 + b2)
1124                                + 3 * t * (b1 - b0) + b0;
1125              if (crossing > 0.0 && crossing <= distance)
1126                nCrossings++;
1127            }
1128        }
1129
1130    return (nCrossings);
1131  }
1132
1133  /**
1134   * A two-dimensional curve that is parameterized with a cubic
1135   * function and stores coordinate values in double-precision
1136   * floating-point format.
1137   *
1138   * @see CubicCurve2D.Float
1139   *
1140   * @author Eric Blake (ebb9@email.byu.edu)
1141   * @author Sascha Brawer (brawer@dandelis.ch)
1142   */
1143  public static class Double extends CubicCurve2D
1144  {
1145    /**
1146     * The <i>x</i> coordinate of the curve&#x2019;s start point.
1147     */
1148    public double x1;
1149
1150    /**
1151     * The <i>y</i> coordinate of the curve&#x2019;s start point.
1152     */
1153    public double y1;
1154
1155    /**
1156     * The <i>x</i> coordinate of the curve&#x2019;s first control point.
1157     */
1158    public double ctrlx1;
1159
1160    /**
1161     * The <i>y</i> coordinate of the curve&#x2019;s first control point.
1162     */
1163    public double ctrly1;
1164
1165    /**
1166     * The <i>x</i> coordinate of the curve&#x2019;s second control point.
1167     */
1168    public double ctrlx2;
1169
1170    /**
1171     * The <i>y</i> coordinate of the curve&#x2019;s second control point.
1172     */
1173    public double ctrly2;
1174
1175    /**
1176     * The <i>x</i> coordinate of the curve&#x2019;s end point.
1177     */
1178    public double x2;
1179
1180    /**
1181     * The <i>y</i> coordinate of the curve&#x2019;s end point.
1182     */
1183    public double y2;
1184
1185    /**
1186     * Constructs a new CubicCurve2D that stores its coordinate values
1187     * in double-precision floating-point format. All points are
1188     * initially at position (0, 0).
1189     */
1190    public Double()
1191    {
1192    }
1193
1194    /**
1195     * Constructs a new CubicCurve2D that stores its coordinate values
1196     * in double-precision floating-point format, specifying the
1197     * initial position of each point.
1198     *
1199     * <p><img src="doc-files/CubicCurve2D-1.png" width="350" height="180"
1200     * alt="A drawing of a CubicCurve2D" />
1201     *
1202     * @param x1 the <i>x</i> coordinate of the curve&#x2019;s start
1203     * point.
1204     *
1205     * @param y1 the <i>y</i> coordinate of the curve&#x2019;s start
1206     * point.
1207     *
1208     * @param cx1 the <i>x</i> coordinate of the curve&#x2019;s first
1209     * control point.
1210     *
1211     * @param cy1 the <i>y</i> coordinate of the curve&#x2019;s first
1212     * control point.
1213     *
1214     * @param cx2 the <i>x</i> coordinate of the curve&#x2019;s second
1215     * control point.
1216     *
1217     * @param cy2 the <i>y</i> coordinate of the curve&#x2019;s second
1218     * control point.
1219     *
1220     * @param x2 the <i>x</i> coordinate of the curve&#x2019;s end
1221     * point.
1222     *
1223     * @param y2 the <i>y</i> coordinate of the curve&#x2019;s end
1224     * point.
1225     */
1226    public Double(double x1, double y1, double cx1, double cy1, double cx2,
1227                  double cy2, double x2, double y2)
1228    {
1229      this.x1 = x1;
1230      this.y1 = y1;
1231      ctrlx1 = cx1;
1232      ctrly1 = cy1;
1233      ctrlx2 = cx2;
1234      ctrly2 = cy2;
1235      this.x2 = x2;
1236      this.y2 = y2;
1237    }
1238
1239    /**
1240     * Returns the <i>x</i> coordinate of the curve&#x2019;s start
1241     * point.
1242     */
1243    public double getX1()
1244    {
1245      return x1;
1246    }
1247
1248    /**
1249     * Returns the <i>y</i> coordinate of the curve&#x2019;s start
1250     * point.
1251     */
1252    public double getY1()
1253    {
1254      return y1;
1255    }
1256
1257    /**
1258     * Returns the curve&#x2019;s start point.
1259     */
1260    public Point2D getP1()
1261    {
1262      return new Point2D.Double(x1, y1);
1263    }
1264
1265    /**
1266     * Returns the <i>x</i> coordinate of the curve&#x2019;s first
1267     * control point.
1268     */
1269    public double getCtrlX1()
1270    {
1271      return ctrlx1;
1272    }
1273
1274    /**
1275     * Returns the <i>y</i> coordinate of the curve&#x2019;s first
1276     * control point.
1277     */
1278    public double getCtrlY1()
1279    {
1280      return ctrly1;
1281    }
1282
1283    /**
1284     * Returns the curve&#x2019;s first control point.
1285     */
1286    public Point2D getCtrlP1()
1287    {
1288      return new Point2D.Double(ctrlx1, ctrly1);
1289    }
1290
1291    /**
1292     * Returns the <i>x</i> coordinate of the curve&#x2019;s second
1293     * control point.
1294     */
1295    public double getCtrlX2()
1296    {
1297      return ctrlx2;
1298    }
1299
1300    /**
1301     * Returns the <i>y</i> coordinate of the curve&#x2019;s second
1302     * control point.
1303     */
1304    public double getCtrlY2()
1305    {
1306      return ctrly2;
1307    }
1308
1309    /**
1310     * Returns the curve&#x2019;s second control point.
1311     */
1312    public Point2D getCtrlP2()
1313    {
1314      return new Point2D.Double(ctrlx2, ctrly2);
1315    }
1316
1317    /**
1318     * Returns the <i>x</i> coordinate of the curve&#x2019;s end
1319     * point.
1320     */
1321    public double getX2()
1322    {
1323      return x2;
1324    }
1325
1326    /**
1327     * Returns the <i>y</i> coordinate of the curve&#x2019;s end
1328     * point.
1329     */
1330    public double getY2()
1331    {
1332      return y2;
1333    }
1334
1335    /**
1336     * Returns the curve&#x2019;s end point.
1337     */
1338    public Point2D getP2()
1339    {
1340      return new Point2D.Double(x2, y2);
1341    }
1342
1343    /**
1344     * Changes the curve geometry, separately specifying each coordinate
1345     * value.
1346     *
1347     * <p><img src="doc-files/CubicCurve2D-1.png" width="350" height="180"
1348     * alt="A drawing of a CubicCurve2D" />
1349     *
1350     * @param x1 the <i>x</i> coordinate of the curve&#x2019;s new start
1351     * point.
1352     *
1353     * @param y1 the <i>y</i> coordinate of the curve&#x2019;s new start
1354     * point.
1355     *
1356     * @param cx1 the <i>x</i> coordinate of the curve&#x2019;s new
1357     * first control point.
1358     *
1359     * @param cy1 the <i>y</i> coordinate of the curve&#x2019;s new
1360     * first control point.
1361     *
1362     * @param cx2 the <i>x</i> coordinate of the curve&#x2019;s new
1363     * second control point.
1364     *
1365     * @param cy2 the <i>y</i> coordinate of the curve&#x2019;s new
1366     * second control point.
1367     *
1368     * @param x2 the <i>x</i> coordinate of the curve&#x2019;s new end
1369     * point.
1370     *
1371     * @param y2 the <i>y</i> coordinate of the curve&#x2019;s new end
1372     * point.
1373     */
1374    public void setCurve(double x1, double y1, double cx1, double cy1,
1375                         double cx2, double cy2, double x2, double y2)
1376    {
1377      this.x1 = x1;
1378      this.y1 = y1;
1379      ctrlx1 = cx1;
1380      ctrly1 = cy1;
1381      ctrlx2 = cx2;
1382      ctrly2 = cy2;
1383      this.x2 = x2;
1384      this.y2 = y2;
1385    }
1386
1387    /**
1388     * Determines the smallest rectangle that encloses the
1389     * curve&#x2019;s start, end and control points. As the
1390     * illustration below shows, the invisible control points may cause
1391     * the bounds to be much larger than the area that is actually
1392     * covered by the curve.
1393     *
1394     * <p><img src="doc-files/CubicCurve2D-2.png" width="350" height="180"
1395     * alt="An illustration of the bounds of a CubicCurve2D" />
1396     */
1397    public Rectangle2D getBounds2D()
1398    {
1399      double nx1 = Math.min(Math.min(x1, ctrlx1), Math.min(ctrlx2, x2));
1400      double ny1 = Math.min(Math.min(y1, ctrly1), Math.min(ctrly2, y2));
1401      double nx2 = Math.max(Math.max(x1, ctrlx1), Math.max(ctrlx2, x2));
1402      double ny2 = Math.max(Math.max(y1, ctrly1), Math.max(ctrly2, y2));
1403      return new Rectangle2D.Double(nx1, ny1, nx2 - nx1, ny2 - ny1);
1404    }
1405  }
1406
1407  /**
1408   * A two-dimensional curve that is parameterized with a cubic
1409   * function and stores coordinate values in single-precision
1410   * floating-point format.
1411   *
1412   * @see CubicCurve2D.Float
1413   *
1414   * @author Eric Blake (ebb9@email.byu.edu)
1415   * @author Sascha Brawer (brawer@dandelis.ch)
1416   */
1417  public static class Float extends CubicCurve2D
1418  {
1419    /**
1420     * The <i>x</i> coordinate of the curve&#x2019;s start point.
1421     */
1422    public float x1;
1423
1424    /**
1425     * The <i>y</i> coordinate of the curve&#x2019;s start point.
1426     */
1427    public float y1;
1428
1429    /**
1430     * The <i>x</i> coordinate of the curve&#x2019;s first control point.
1431     */
1432    public float ctrlx1;
1433
1434    /**
1435     * The <i>y</i> coordinate of the curve&#x2019;s first control point.
1436     */
1437    public float ctrly1;
1438
1439    /**
1440     * The <i>x</i> coordinate of the curve&#x2019;s second control point.
1441     */
1442    public float ctrlx2;
1443
1444    /**
1445     * The <i>y</i> coordinate of the curve&#x2019;s second control point.
1446     */
1447    public float ctrly2;
1448
1449    /**
1450     * The <i>x</i> coordinate of the curve&#x2019;s end point.
1451     */
1452    public float x2;
1453
1454    /**
1455     * The <i>y</i> coordinate of the curve&#x2019;s end point.
1456     */
1457    public float y2;
1458
1459    /**
1460     * Constructs a new CubicCurve2D that stores its coordinate values
1461     * in single-precision floating-point format. All points are
1462     * initially at position (0, 0).
1463     */
1464    public Float()
1465    {
1466    }
1467
1468    /**
1469     * Constructs a new CubicCurve2D that stores its coordinate values
1470     * in single-precision floating-point format, specifying the
1471     * initial position of each point.
1472     *
1473     * <p><img src="doc-files/CubicCurve2D-1.png" width="350" height="180"
1474     * alt="A drawing of a CubicCurve2D" />
1475     *
1476     * @param x1 the <i>x</i> coordinate of the curve&#x2019;s start
1477     * point.
1478     *
1479     * @param y1 the <i>y</i> coordinate of the curve&#x2019;s start
1480     * point.
1481     *
1482     * @param cx1 the <i>x</i> coordinate of the curve&#x2019;s first
1483     * control point.
1484     *
1485     * @param cy1 the <i>y</i> coordinate of the curve&#x2019;s first
1486     * control point.
1487     *
1488     * @param cx2 the <i>x</i> coordinate of the curve&#x2019;s second
1489     * control point.
1490     *
1491     * @param cy2 the <i>y</i> coordinate of the curve&#x2019;s second
1492     * control point.
1493     *
1494     * @param x2 the <i>x</i> coordinate of the curve&#x2019;s end
1495     * point.
1496     *
1497     * @param y2 the <i>y</i> coordinate of the curve&#x2019;s end
1498     * point.
1499     */
1500    public Float(float x1, float y1, float cx1, float cy1, float cx2,
1501                 float cy2, float x2, float y2)
1502    {
1503      this.x1 = x1;
1504      this.y1 = y1;
1505      ctrlx1 = cx1;
1506      ctrly1 = cy1;
1507      ctrlx2 = cx2;
1508      ctrly2 = cy2;
1509      this.x2 = x2;
1510      this.y2 = y2;
1511    }
1512
1513    /**
1514     * Returns the <i>x</i> coordinate of the curve&#x2019;s start
1515     * point.
1516     */
1517    public double getX1()
1518    {
1519      return x1;
1520    }
1521
1522    /**
1523     * Returns the <i>y</i> coordinate of the curve&#x2019;s start
1524     * point.
1525     */
1526    public double getY1()
1527    {
1528      return y1;
1529    }
1530
1531    /**
1532     * Returns the curve&#x2019;s start point.
1533     */
1534    public Point2D getP1()
1535    {
1536      return new Point2D.Float(x1, y1);
1537    }
1538
1539    /**
1540     * Returns the <i>x</i> coordinate of the curve&#x2019;s first
1541     * control point.
1542     */
1543    public double getCtrlX1()
1544    {
1545      return ctrlx1;
1546    }
1547
1548    /**
1549     * Returns the <i>y</i> coordinate of the curve&#x2019;s first
1550     * control point.
1551     */
1552    public double getCtrlY1()
1553    {
1554      return ctrly1;
1555    }
1556
1557    /**
1558     * Returns the curve&#x2019;s first control point.
1559     */
1560    public Point2D getCtrlP1()
1561    {
1562      return new Point2D.Float(ctrlx1, ctrly1);
1563    }
1564
1565    /**
1566     * Returns the <i>s</i> coordinate of the curve&#x2019;s second
1567     * control point.
1568     */
1569    public double getCtrlX2()
1570    {
1571      return ctrlx2;
1572    }
1573
1574    /**
1575     * Returns the <i>y</i> coordinate of the curve&#x2019;s second
1576     * control point.
1577     */
1578    public double getCtrlY2()
1579    {
1580      return ctrly2;
1581    }
1582
1583    /**
1584     * Returns the curve&#x2019;s second control point.
1585     */
1586    public Point2D getCtrlP2()
1587    {
1588      return new Point2D.Float(ctrlx2, ctrly2);
1589    }
1590
1591    /**
1592     * Returns the <i>x</i> coordinate of the curve&#x2019;s end
1593     * point.
1594     */
1595    public double getX2()
1596    {
1597      return x2;
1598    }
1599
1600    /**
1601     * Returns the <i>y</i> coordinate of the curve&#x2019;s end
1602     * point.
1603     */
1604    public double getY2()
1605    {
1606      return y2;
1607    }
1608
1609    /**
1610     * Returns the curve&#x2019;s end point.
1611     */
1612    public Point2D getP2()
1613    {
1614      return new Point2D.Float(x2, y2);
1615    }
1616
1617    /**
1618     * Changes the curve geometry, separately specifying each coordinate
1619     * value as a double-precision floating-point number.
1620     *
1621     * <p><img src="doc-files/CubicCurve2D-1.png" width="350" height="180"
1622     * alt="A drawing of a CubicCurve2D" />
1623     *
1624     * @param x1 the <i>x</i> coordinate of the curve&#x2019;s new start
1625     * point.
1626     *
1627     * @param y1 the <i>y</i> coordinate of the curve&#x2019;s new start
1628     * point.
1629     *
1630     * @param cx1 the <i>x</i> coordinate of the curve&#x2019;s new
1631     * first control point.
1632     *
1633     * @param cy1 the <i>y</i> coordinate of the curve&#x2019;s new
1634     * first control point.
1635     *
1636     * @param cx2 the <i>x</i> coordinate of the curve&#x2019;s new
1637     * second control point.
1638     *
1639     * @param cy2 the <i>y</i> coordinate of the curve&#x2019;s new
1640     * second control point.
1641     *
1642     * @param x2 the <i>x</i> coordinate of the curve&#x2019;s new end
1643     * point.
1644     *
1645     * @param y2 the <i>y</i> coordinate of the curve&#x2019;s new end
1646     * point.
1647     */
1648    public void setCurve(double x1, double y1, double cx1, double cy1,
1649                         double cx2, double cy2, double x2, double y2)
1650    {
1651      this.x1 = (float) x1;
1652      this.y1 = (float) y1;
1653      ctrlx1 = (float) cx1;
1654      ctrly1 = (float) cy1;
1655      ctrlx2 = (float) cx2;
1656      ctrly2 = (float) cy2;
1657      this.x2 = (float) x2;
1658      this.y2 = (float) y2;
1659    }
1660
1661    /**
1662     * Changes the curve geometry, separately specifying each coordinate
1663     * value as a single-precision floating-point number.
1664     *
1665     * <p><img src="doc-files/CubicCurve2D-1.png" width="350" height="180"
1666     * alt="A drawing of a CubicCurve2D" />
1667     *
1668     * @param x1 the <i>x</i> coordinate of the curve&#x2019;s new start
1669     * point.
1670     *
1671     * @param y1 the <i>y</i> coordinate of the curve&#x2019;s new start
1672     * point.
1673     *
1674     * @param cx1 the <i>x</i> coordinate of the curve&#x2019;s new
1675     * first control point.
1676     *
1677     * @param cy1 the <i>y</i> coordinate of the curve&#x2019;s new
1678     * first control point.
1679     *
1680     * @param cx2 the <i>x</i> coordinate of the curve&#x2019;s new
1681     * second control point.
1682     *
1683     * @param cy2 the <i>y</i> coordinate of the curve&#x2019;s new
1684     * second control point.
1685     *
1686     * @param x2 the <i>x</i> coordinate of the curve&#x2019;s new end
1687     * point.
1688     *
1689     * @param y2 the <i>y</i> coordinate of the curve&#x2019;s new end
1690     * point.
1691     */
1692    public void setCurve(float x1, float y1, float cx1, float cy1, float cx2,
1693                         float cy2, float x2, float y2)
1694    {
1695      this.x1 = x1;
1696      this.y1 = y1;
1697      ctrlx1 = cx1;
1698      ctrly1 = cy1;
1699      ctrlx2 = cx2;
1700      ctrly2 = cy2;
1701      this.x2 = x2;
1702      this.y2 = y2;
1703    }
1704
1705    /**
1706     * Determines the smallest rectangle that encloses the
1707     * curve&#x2019;s start, end and control points. As the
1708     * illustration below shows, the invisible control points may cause
1709     * the bounds to be much larger than the area that is actually
1710     * covered by the curve.
1711     *
1712     * <p><img src="doc-files/CubicCurve2D-2.png" width="350" height="180"
1713     * alt="An illustration of the bounds of a CubicCurve2D" />
1714     */
1715    public Rectangle2D getBounds2D()
1716    {
1717      float nx1 = Math.min(Math.min(x1, ctrlx1), Math.min(ctrlx2, x2));
1718      float ny1 = Math.min(Math.min(y1, ctrly1), Math.min(ctrly2, y2));
1719      float nx2 = Math.max(Math.max(x1, ctrlx1), Math.max(ctrlx2, x2));
1720      float ny2 = Math.max(Math.max(y1, ctrly1), Math.max(ctrly2, y2));
1721      return new Rectangle2D.Float(nx1, ny1, nx2 - nx1, ny2 - ny1);
1722    }
1723  }
1724}