1 /*
2 * Copyright (c) 2007, 2018, 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.marlin;
27
28 final class Curve {
29
30 float ax, ay, bx, by, cx, cy, dx, dy;
31 float dax, day, dbx, dby;
32
33 Curve() {
34 }
35
36 void set(final float[] points, final int type) {
37 // if instead of switch (perf + most probable cases first)
38 if (type == 8) {
39 set(points[0], points[1],
40 points[2], points[3],
41 points[4], points[5],
42 points[6], points[7]);
43 } else if (type == 4) {
44 set(points[0], points[1],
45 points[2], points[3]);
46 } else {
47 set(points[0], points[1],
48 points[2], points[3],
49 points[4], points[5]);
50 }
51 }
52
53 void set(final float x1, final float y1,
54 final float x2, final float y2,
55 final float x3, final float y3,
56 final float x4, final float y4)
57 {
58 final float dx32 = 3.0f * (x3 - x2);
59 final float dy32 = 3.0f * (y3 - y2);
60 final float dx21 = 3.0f * (x2 - x1);
61 final float dy21 = 3.0f * (y2 - y1);
62 ax = (x4 - x1) - dx32; // A = P3 - P0 - 3 (P2 - P1) = (P3 - P0) + 3 (P1 - P2)
63 ay = (y4 - y1) - dy32;
64 bx = (dx32 - dx21); // B = 3 (P2 - P1) - 3(P1 - P0) = 3 (P2 + P0) - 6 P1
65 by = (dy32 - dy21);
66 cx = dx21; // C = 3 (P1 - P0)
67 cy = dy21;
68 dx = x1; // D = P0
69 dy = y1;
70 dax = 3.0f * ax;
71 day = 3.0f * ay;
72 dbx = 2.0f * bx;
73 dby = 2.0f * by;
74 }
75
76 void set(final float x1, final float y1,
77 final float x2, final float y2,
78 final float x3, final float y3)
79 {
80 final float dx21 = (x2 - x1);
81 final float dy21 = (y2 - y1);
82 ax = 0.0f; // A = 0
83 ay = 0.0f;
84 bx = (x3 - x2) - dx21; // B = P3 - P0 - 2 P2
85 by = (y3 - y2) - dy21;
86 cx = 2.0f * dx21; // C = 2 (P2 - P1)
87 cy = 2.0f * dy21;
88 dx = x1; // D = P1
89 dy = y1;
90 dax = 0.0f;
91 day = 0.0f;
92 dbx = 2.0f * bx;
93 dby = 2.0f * by;
94 }
95
96 void set(final float x1, final float y1,
97 final float x2, final float y2)
98 {
99 final float dx21 = (x2 - x1);
100 final float dy21 = (y2 - y1);
101 ax = 0.0f; // A = 0
102 ay = 0.0f;
103 bx = 0.0f; // B = 0
104 by = 0.0f;
105 cx = dx21; // C = (P2 - P1)
106 cy = dy21;
107 dx = x1; // D = P1
108 dy = y1;
109 // useless derivatives for lines
110 if (false) {
111 dax = 0.0f;
112 day = 0.0f;
113 dbx = 0.0f;
114 dby = 0.0f;
115 }
116 }
117
118 int dxRoots(final float[] roots, final int off) {
119 return Helpers.quadraticRoots(dax, dbx, cx, roots, off);
120 }
121
122 int dyRoots(final float[] roots, final int off) {
123 return Helpers.quadraticRoots(day, dby, cy, roots, off);
124 }
125
126 int infPoints(final float[] pts, final int off) {
127 // inflection point at t if -f'(t)x*f''(t)y + f'(t)y*f''(t)x == 0
128 // Fortunately, this turns out to be quadratic, so there are at
129 // most 2 inflection points.
130 final float a = dax * dby - dbx * day;
131 final float b = 2.0f * (cy * dax - day * cx);
132 final float c = cy * dbx - cx * dby;
133
134 return Helpers.quadraticRoots(a, b, c, pts, off);
135 }
136
137 int xPoints(final float[] ts, final int off, final float x)
138 {
139 return Helpers.cubicRootsInAB(ax, bx, cx, dx - x, ts, off, 0.0f, 1.0f);
140 }
141
142 int yPoints(final float[] ts, final int off, final float y)
143 {
144 return Helpers.cubicRootsInAB(ay, by, cy, dy - y, ts, off, 0.0f, 1.0f);
145 }
146
147 // finds points where the first and second derivative are
148 // perpendicular. This happens when g(t) = f'(t)*f''(t) == 0 (where
149 // * is a dot product). Unfortunately, we have to solve a cubic.
150 private int perpendiculardfddf(final float[] pts, final int off) {
151 assert pts.length >= off + 4;
152
153 // these are the coefficients of some multiple of g(t) (not g(t),
154 // because the roots of a polynomial are not changed after multiplication
155 // by a constant, and this way we save a few multiplications).
156 final float a = 2.0f * (dax * dax + day * day);
157 final float b = 3.0f * (dax * dbx + day * dby);
158 final float c = 2.0f * (dax * cx + day * cy) + dbx * dbx + dby * dby;
159 final float d = dbx * cx + dby * cy;
160
161 return Helpers.cubicRootsInAB(a, b, c, d, pts, off, 0.0f, 1.0f);
162 }
163
164 // Tries to find the roots of the function ROC(t)-w in [0, 1). It uses
165 // a variant of the false position algorithm to find the roots. False
166 // position requires that 2 initial values x0,x1 be given, and that the
167 // function must have opposite signs at those values. To find such
168 // values, we need the local extrema of the ROC function, for which we
169 // need the roots of its derivative; however, it's harder to find the
170 // roots of the derivative in this case than it is to find the roots
171 // of the original function. So, we find all points where this curve's
172 // first and second derivative are perpendicular, and we pretend these
173 // are our local extrema. There are at most 3 of these, so we will check
174 // at most 4 sub-intervals of (0,1). ROC has asymptotes at inflection
175 // points, so roc-w can have at least 6 roots. This shouldn't be a
176 // problem for what we're trying to do (draw a nice looking curve).
177 int rootsOfROCMinusW(final float[] roots, final int off, final float w2, final float err) {
178 // no OOB exception, because by now off<=6, and roots.length >= 10
179 assert off <= 6 && roots.length >= 10;
180
181 int ret = off;
182 final int end = off + perpendiculardfddf(roots, off);
183 roots[end] = 1.0f; // always check interval end points
184
185 float t0 = 0.0f, ft0 = ROCsq(t0) - w2;
186
187 for (int i = off; i <= end; i++) {
188 float t1 = roots[i], ft1 = ROCsq(t1) - w2;
189 if (ft0 == 0.0f) {
190 roots[ret++] = t0;
191 } else if (ft1 * ft0 < 0.0f) { // have opposite signs
192 // (ROC(t)^2 == w^2) == (ROC(t) == w) is true because
193 // ROC(t) >= 0 for all t.
194 roots[ret++] = falsePositionROCsqMinusX(t0, t1, w2, err);
195 }
196 t0 = t1;
197 ft0 = ft1;
198 }
199
200 return ret - off;
201 }
202
203 private static float eliminateInf(final float x) {
204 return (x == Float.POSITIVE_INFINITY ? Float.MAX_VALUE :
205 (x == Float.NEGATIVE_INFINITY ? Float.MIN_VALUE : x));
206 }
207
208 // A slight modification of the false position algorithm on wikipedia.
209 // This only works for the ROCsq-x functions. It might be nice to have
210 // the function as an argument, but that would be awkward in java6.
211 // TODO: It is something to consider for java8 (or whenever lambda
212 // expressions make it into the language), depending on how closures
213 // and turn out. Same goes for the newton's method
214 // algorithm in Helpers.java
215 private float falsePositionROCsqMinusX(final float t0, final float t1,
216 final float w2, final float err)
217 {
218 final int iterLimit = 100;
219 int side = 0;
220 float t = t1, ft = eliminateInf(ROCsq(t) - w2);
221 float s = t0, fs = eliminateInf(ROCsq(s) - w2);
222 float r = s, fr;
223
224 for (int i = 0; i < iterLimit && Math.abs(t - s) > err * Math.abs(t + s); i++) {
225 r = (fs * t - ft * s) / (fs - ft);
226 fr = ROCsq(r) - w2;
227 if (sameSign(fr, ft)) {
228 ft = fr; t = r;
229 if (side < 0) {
230 fs /= (1 << (-side));
231 side--;
232 } else {
233 side = -1;
234 }
235 } else if (fr * fs > 0.0f) {
236 fs = fr; s = r;
237 if (side > 0) {
238 ft /= (1 << side);
239 side++;
240 } else {
241 side = 1;
242 }
243 } else {
244 break;
245 }
246 }
247 return r;
248 }
249
250 private static boolean sameSign(final float x, final float y) {
251 // another way is to test if x*y > 0. This is bad for small x, y.
252 return (x < 0.0f && y < 0.0f) || (x > 0.0f && y > 0.0f);
253 }
254
255 // returns the radius of curvature squared at t of this curve
256 // see http://en.wikipedia.org/wiki/Radius_of_curvature_(applications)
257 private float ROCsq(final float t) {
258 final float dx = t * (t * dax + dbx) + cx;
259 final float dy = t * (t * day + dby) + cy;
260 final float ddx = 2.0f * dax * t + dbx;
261 final float ddy = 2.0f * day * t + dby;
262 final float dx2dy2 = dx * dx + dy * dy;
263 final float ddx2ddy2 = ddx * ddx + ddy * ddy;
264 final float ddxdxddydy = ddx * dx + ddy * dy;
265 return dx2dy2 * ((dx2dy2 * dx2dy2) / (dx2dy2 * ddx2ddy2 - ddxdxddydy * ddxdxddydy));
266 }
267 }