1 /*
2 * Copyright (c) 2007, 2010, Oracle and/or its affiliates. All rights reserved.
3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
4 *
5 * This code is free software; you can redistribute it and/or modify it
6 * under the terms of the GNU General Public License version 2 only, as
7 * published by the Free Software Foundation. Oracle designates this
8 * particular file as subject to the "Classpath" exception as provided
9 * by Oracle in the LICENSE file that accompanied this code.
10 *
11 * This code is distributed in the hope that it will be useful, but WITHOUT
12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
14 * version 2 for more details (a copy is included in the LICENSE file that
15 * accompanied this code).
16 *
17 * You should have received a copy of the GNU General Public License version
18 * 2 along with this work; if not, write to the Free Software Foundation,
19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
20 *
21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
22 * or visit www.oracle.com if you need additional information or have any
23 * questions.
24 */
25
26 package sun.java2d.pisces;
27
28 import java.util.Iterator;
29
30 class Curve {
31
32 float ax, ay, bx, by, cx, cy, dx, dy;
33 float dax, day, dbx, dby;
34
35 Curve() {
36 }
37
38 void set(float[] points, int type) {
39 switch(type) {
40 case 8:
41 set(points[0], points[1],
42 points[2], points[3],
43 points[4], points[5],
44 points[6], points[7]);
45 break;
46 case 6:
47 set(points[0], points[1],
48 points[2], points[3],
49 points[4], points[5]);
50 break;
51 default:
52 throw new InternalError("Curves can only be cubic or quadratic");
53 }
54 }
55
56 void set(float x1, float y1,
57 float x2, float y2,
58 float x3, float y3,
59 float x4, float y4)
60 {
61 ax = 3 * (x2 - x3) + x4 - x1;
62 ay = 3 * (y2 - y3) + y4 - y1;
63 bx = 3 * (x1 - 2 * x2 + x3);
64 by = 3 * (y1 - 2 * y2 + y3);
65 cx = 3 * (x2 - x1);
66 cy = 3 * (y2 - y1);
67 dx = x1;
68 dy = y1;
69 dax = 3 * ax; day = 3 * ay;
70 dbx = 2 * bx; dby = 2 * by;
71 }
72
73 void set(float x1, float y1,
74 float x2, float y2,
75 float x3, float y3)
76 {
77 ax = ay = 0f;
78
79 bx = x1 - 2 * x2 + x3;
80 by = y1 - 2 * y2 + y3;
81 cx = 2 * (x2 - x1);
82 cy = 2 * (y2 - y1);
83 dx = x1;
84 dy = y1;
85 dax = 0; day = 0;
86 dbx = 2 * bx; dby = 2 * by;
87 }
88
89 float xat(float t) {
90 return t * (t * (t * ax + bx) + cx) + dx;
91 }
92 float yat(float t) {
93 return t * (t * (t * ay + by) + cy) + dy;
94 }
95
96 float dxat(float t) {
97 return t * (t * dax + dbx) + cx;
98 }
99
100 float dyat(float t) {
101 return t * (t * day + dby) + cy;
102 }
103
104 private float ddxat(float t) {
105 return 2 * dax * t + dbx;
106 }
107
108 private float ddyat(float t) {
109 return 2 * day * t + dby;
110 }
111
112 int dxRoots(float[] roots, int off) {
113 return Helpers.quadraticRoots(dax, dbx, cx, roots, off);
114 }
115
116 int dyRoots(float[] roots, int off) {
117 return Helpers.quadraticRoots(day, dby, cy, roots, off);
118 }
119
120 int infPoints(float[] pts, int off) {
121 // inflection point at t if -f'(t)x*f''(t)y + f'(t)y*f''(t)x == 0
122 // Fortunately, this turns out to be quadratic, so there are at
123 // most 2 inflection points.
124 final float a = dax * dby - dbx * day;
125 final float b = 2 * (cy * dax - day * cx);
126 final float c = cy * dbx - cx * dby;
127
128 return Helpers.quadraticRoots(a, b, c, pts, off);
129 }
130
131 // finds points where the first and second derivative are
132 // perpendicular. This happens when g(t) = f'(t)*f''(t) == 0 (where
133 // * is a dot product). Unfortunately, we have to solve a cubic.
134 private int perpendiculardfddf(float[] pts, int off, final float err) {
135 assert pts.length >= off + 4;
136
137 // these are the coefficients of g(t):
138 final float a = 2*(dax*dax + day*day);
139 final float b = 3*(dax*dbx + day*dby);
140 final float c = 2*(dax*cx + day*cy) + dbx*dbx + dby*dby;
141 final float d = dbx*cx + dby*cy;
142 // TODO: We might want to divide the polynomial by a to make the
143 // coefficients smaller. This won't change the roots.
144 return Helpers.cubicRootsInAB(a, b, c, d, pts, off, err, 0f, 1f);
145 }
146
147 // Tries to find the roots of the function ROC(t)-w in [0, 1). It uses
148 // a variant of the false position algorithm to find the roots. False
149 // position requires that 2 initial values x0,x1 be given, and that the
150 // function must have opposite signs at those values. To find such
151 // values, we need the local extrema of the ROC function, for which we
152 // need the roots of its derivative; however, it's harder to find the
153 // roots of the derivative in this case than it is to find the roots
154 // of the original function. So, we find all points where this curve's
155 // first and second derivative are perpendicular, and we pretend these
156 // are our local extrema. There are at most 3 of these, so we will check
157 // at most 4 sub-intervals of (0,1). ROC has asymptotes at inflection
158 // points, so roc-w can have at least 6 roots. This shouldn't be a
159 // problem for what we're trying to do (draw a nice looking curve).
160 int rootsOfROCMinusW(float[] roots, int off, final float w, final float err) {
161 // no OOB exception, because by now off<=6, and roots.length >= 10
162 assert off <= 6 && roots.length >= 10;
163 int ret = off;
164 int numPerpdfddf = perpendiculardfddf(roots, off, err);
165 float t0 = 0, ft0 = ROCsq(t0) - w*w;
166 roots[off + numPerpdfddf] = 1f; // always check interval end points
167 numPerpdfddf++;
168 for (int i = off; i < off + numPerpdfddf; i++) {
169 float t1 = roots[i], ft1 = ROCsq(t1) - w*w;
170 if (ft0 == 0f) {
171 roots[ret++] = t0;
172 } else if (ft1 * ft0 < 0f) { // have opposite signs
173 // (ROC(t)^2 == w^2) == (ROC(t) == w) is true because
174 // ROC(t) >= 0 for all t.
175 roots[ret++] = falsePositionROCsqMinusX(t0, t1, w*w, err);
176 }
177 t0 = t1;
178 ft0 = ft1;
179 }
180
181 return ret - off;
182 }
183
184 private static float eliminateInf(float x) {
185 return (x == Float.POSITIVE_INFINITY ? Float.MAX_VALUE :
186 (x == Float.NEGATIVE_INFINITY ? Float.MIN_VALUE : x));
187 }
188
189 // A slight modification of the false position algorithm on wikipedia.
190 // This only works for the ROCsq-x functions. It might be nice to have
191 // the function as an argument, but that would be awkward in java6.
192 // It is something to consider for java7, depending on how closures
193 // and function objects turn out. Same goes for the newton's method
194 // algorithm in Helpers.java
195 private float falsePositionROCsqMinusX(float x0, float x1,
196 final float x, final float err)
197 {
198 final int iterLimit = 100;
199 int side = 0;
200 float t = x1, ft = eliminateInf(ROCsq(t) - x);
201 float s = x0, fs = eliminateInf(ROCsq(s) - x);
202 float r = s, fr;
203 for (int i = 0; i < iterLimit && Math.abs(t - s) > err * Math.abs(t + s); i++) {
204 r = (fs * t - ft * s) / (fs - ft);
205 fr = ROCsq(r) - x;
206 if (fr * ft > 0) {// have the same sign
207 ft = fr; t = r;
208 if (side < 0) {
209 fs /= (1 << (-side));
210 side--;
211 } else {
212 side = -1;
213 }
214 } else if (fr * fs > 0) {
215 fs = fr; s = r;
216 if (side > 0) {
217 ft /= (1 << side);
218 side++;
219 } else {
220 side = 1;
221 }
222 } else {
223 break;
224 }
225 }
226 return r;
227 }
228
229 // returns the radius of curvature squared at t of this curve
230 // see http://en.wikipedia.org/wiki/Radius_of_curvature_(applications)
231 private float ROCsq(final float t) {
232 final float dx = dxat(t);
233 final float dy = dyat(t);
234 final float ddx = ddxat(t);
235 final float ddy = ddyat(t);
236 final float dx2dy2 = dx*dx + dy*dy;
237 final float ddx2ddy2 = ddx*ddx + ddy*ddy;
238 final float ddxdxddydy = ddx*dx + ddy*dy;
239 float ret = ((dx2dy2*dx2dy2) / (dx2dy2 * ddx2ddy2 - ddxdxddydy*ddxdxddydy))*dx2dy2;
240 return ret;
241 }
242
243 // curve to be broken should be in pts[0]
244 // this will change the contents of both pts and Ts
245 // TODO: There's no reason for Ts to be an array. All we need is a sequence
246 // of t values at which to subdivide. An array statisfies this condition,
247 // but is unnecessarily restrictive. Ts should be an Iterator<Float> instead.
248 // Doing this will also make dashing easier, since we could easily make
249 // LengthIterator an Iterator<Float> and feed it to this function to simplify
250 // the loop in Dasher.somethingTo.
251 static Iterator<float[]> breakPtsAtTs(final float[][] pts, final int type,
252 final float[] Ts, final int numTs)
253 {
254 assert pts.length >= 2 && pts[0].length >= 8 && numTs <= Ts.length;
255 return new Iterator<float[]>() {
256 int nextIdx = 0;
257 int nextCurveIdx = 0;
258 float prevT = 0;
259
260 @Override public boolean hasNext() {
261 return nextCurveIdx < numTs + 1;
262 }
263
264 @Override public float[] next() {
265 float[] ret;
266 if (nextCurveIdx < numTs) {
267 float curT = Ts[nextCurveIdx];
268 float splitT = (curT - prevT) / (1 - prevT);
269 Helpers.subdivideAt(splitT,
270 pts[nextIdx], 0,
271 pts[nextIdx], 0,
272 pts[1-nextIdx], 0, type);
273 updateTs(Ts, Ts[nextCurveIdx], nextCurveIdx + 1, numTs - nextCurveIdx - 1);
274 ret = pts[nextIdx];
275 nextIdx = 1 - nextIdx;
276 } else {
277 ret = pts[nextIdx];
278 }
279 nextCurveIdx++;
280 return ret;
281 }
282
283 @Override public void remove() {}
284 };
285 }
286
287 // precondition: ts[off]...ts[off+len-1] must all be greater than t.
288 private static void updateTs(float[] ts, final float t, final int off, final int len) {
289 for (int i = off; i < off + len; i++) {
290 ts[i] = (ts[i] - t) / (1 - t);
291 }
292 }
293 }
294