1 /*
2 * Copyright (c) 2007, 2011, 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 final 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 int dxRoots(float[] roots, int off) {
105 return Helpers.quadraticRoots(dax, dbx, cx, roots, off);
106 }
107
108 int dyRoots(float[] roots, int off) {
109 return Helpers.quadraticRoots(day, dby, cy, roots, off);
110 }
111
112 int infPoints(float[] pts, int off) {
113 // inflection point at t if -f'(t)x*f''(t)y + f'(t)y*f''(t)x == 0
114 // Fortunately, this turns out to be quadratic, so there are at
115 // most 2 inflection points.
116 final float a = dax * dby - dbx * day;
117 final float b = 2 * (cy * dax - day * cx);
118 final float c = cy * dbx - cx * dby;
119
120 return Helpers.quadraticRoots(a, b, c, pts, off);
121 }
122
123 // finds points where the first and second derivative are
124 // perpendicular. This happens when g(t) = f'(t)*f''(t) == 0 (where
125 // * is a dot product). Unfortunately, we have to solve a cubic.
126 private int perpendiculardfddf(float[] pts, int off) {
127 assert pts.length >= off + 4;
128
129 // these are the coefficients of some multiple of g(t) (not g(t),
130 // because the roots of a polynomial are not changed after multiplication
131 // by a constant, and this way we save a few multiplications).
132 final float a = 2*(dax*dax + day*day);
133 final float b = 3*(dax*dbx + day*dby);
134 final float c = 2*(dax*cx + day*cy) + dbx*dbx + dby*dby;
135 final float d = dbx*cx + dby*cy;
136 return Helpers.cubicRootsInAB(a, b, c, d, pts, off, 0f, 1f);
137 }
138
139 // Tries to find the roots of the function ROC(t)-w in [0, 1). It uses
140 // a variant of the false position algorithm to find the roots. False
141 // position requires that 2 initial values x0,x1 be given, and that the
142 // function must have opposite signs at those values. To find such
143 // values, we need the local extrema of the ROC function, for which we
144 // need the roots of its derivative; however, it's harder to find the
145 // roots of the derivative in this case than it is to find the roots
146 // of the original function. So, we find all points where this curve's
147 // first and second derivative are perpendicular, and we pretend these
148 // are our local extrema. There are at most 3 of these, so we will check
149 // at most 4 sub-intervals of (0,1). ROC has asymptotes at inflection
150 // points, so roc-w can have at least 6 roots. This shouldn't be a
151 // problem for what we're trying to do (draw a nice looking curve).
152 int rootsOfROCMinusW(float[] roots, int off, final float w, final float err) {
153 // no OOB exception, because by now off<=6, and roots.length >= 10
154 assert off <= 6 && roots.length >= 10;
155 int ret = off;
156 int numPerpdfddf = perpendiculardfddf(roots, off);
157 float t0 = 0, ft0 = ROCsq(t0) - w*w;
158 roots[off + numPerpdfddf] = 1f; // always check interval end points
159 numPerpdfddf++;
160 for (int i = off; i < off + numPerpdfddf; i++) {
161 float t1 = roots[i], ft1 = ROCsq(t1) - w*w;
162 if (ft0 == 0f) {
163 roots[ret++] = t0;
164 } else if (ft1 * ft0 < 0f) { // have opposite signs
165 // (ROC(t)^2 == w^2) == (ROC(t) == w) is true because
166 // ROC(t) >= 0 for all t.
167 roots[ret++] = falsePositionROCsqMinusX(t0, t1, w*w, err);
168 }
169 t0 = t1;
170 ft0 = ft1;
171 }
172
173 return ret - off;
174 }
175
176 private static float eliminateInf(float x) {
177 return (x == Float.POSITIVE_INFINITY ? Float.MAX_VALUE :
178 (x == Float.NEGATIVE_INFINITY ? Float.MIN_VALUE : x));
179 }
180
181 // A slight modification of the false position algorithm on wikipedia.
182 // This only works for the ROCsq-x functions. It might be nice to have
183 // the function as an argument, but that would be awkward in java6.
184 // TODO: It is something to consider for java8 (or whenever lambda
185 // expressions make it into the language), depending on how closures
186 // and turn out. Same goes for the newton's method
187 // algorithm in Helpers.java
188 private float falsePositionROCsqMinusX(float x0, float x1,
189 final float x, final float err)
190 {
191 final int iterLimit = 100;
192 int side = 0;
193 float t = x1, ft = eliminateInf(ROCsq(t) - x);
194 float s = x0, fs = eliminateInf(ROCsq(s) - x);
195 float r = s, fr;
196 for (int i = 0; i < iterLimit && Math.abs(t - s) > err * Math.abs(t + s); i++) {
197 r = (fs * t - ft * s) / (fs - ft);
198 fr = ROCsq(r) - x;
199 if (sameSign(fr, ft)) {
200 ft = fr; t = r;
201 if (side < 0) {
202 fs /= (1 << (-side));
203 side--;
204 } else {
205 side = -1;
206 }
207 } else if (fr * fs > 0) {
208 fs = fr; s = r;
209 if (side > 0) {
210 ft /= (1 << side);
211 side++;
212 } else {
213 side = 1;
214 }
215 } else {
216 break;
217 }
218 }
219 return r;
220 }
221
222 private static boolean sameSign(double x, double y) {
223 // another way is to test if x*y > 0. This is bad for small x, y.
224 return (x < 0 && y < 0) || (x > 0 && y > 0);
225 }
226
227 // returns the radius of curvature squared at t of this curve
228 // see http://en.wikipedia.org/wiki/Radius_of_curvature_(applications)
229 private float ROCsq(final float t) {
230 // dx=xat(t) and dy=yat(t). These calls have been inlined for efficiency
231 final float dx = t * (t * dax + dbx) + cx;
232 final float dy = t * (t * day + dby) + cy;
233 final float ddx = 2 * dax * t + dbx;
234 final float ddy = 2 * day * t + dby;
235 final float dx2dy2 = dx*dx + dy*dy;
236 final float ddx2ddy2 = ddx*ddx + ddy*ddy;
237 final float ddxdxddydy = ddx*dx + ddy*dy;
238 return dx2dy2*((dx2dy2*dx2dy2) / (dx2dy2 * ddx2ddy2 - ddxdxddydy*ddxdxddydy));
239 }
240
241 // curve to be broken should be in pts
242 // this will change the contents of pts but not Ts
243 // TODO: There's no reason for Ts to be an array. All we need is a sequence
244 // of t values at which to subdivide. An array statisfies this condition,
245 // but is unnecessarily restrictive. Ts should be an Iterator<Float> instead.
246 // Doing this will also make dashing easier, since we could easily make
247 // LengthIterator an Iterator<Float> and feed it to this function to simplify
248 // the loop in Dasher.somethingTo.
249 static Iterator<Integer> breakPtsAtTs(final float[] pts, final int type,
250 final float[] Ts, final int numTs)
251 {
252 assert pts.length >= 2*type && numTs <= Ts.length;
253 return new Iterator<Integer>() {
254 // these prevent object creation and destruction during autoboxing.
255 // Because of this, the compiler should be able to completely
256 // eliminate the boxing costs.
257 final Integer i0 = 0;
258 final Integer itype = type;
259 int nextCurveIdx = 0;
260 Integer curCurveOff = i0;
261 float prevT = 0;
262
263 @Override public boolean hasNext() {
264 return nextCurveIdx < numTs + 1;
265 }
266
267 @Override public Integer next() {
268 Integer ret;
269 if (nextCurveIdx < numTs) {
270 float curT = Ts[nextCurveIdx];
271 float splitT = (curT - prevT) / (1 - prevT);
272 Helpers.subdivideAt(splitT,
273 pts, curCurveOff,
274 pts, 0,
275 pts, type, type);
276 prevT = curT;
277 ret = i0;
278 curCurveOff = itype;
279 } else {
280 ret = curCurveOff;
281 }
282 nextCurveIdx++;
283 return ret;
284 }
285
286 @Override public void remove() {}
287 };
288 }
289 }
290