1 /* 2 * Copyright (c) 2007, 2015, 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 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 // shared iterator instance 35 private final BreakPtrIterator iterator = new BreakPtrIterator(); 36 37 Curve() { 38 } 39 40 void set(float[] points, int type) { 41 switch(type) { 42 case 8: 43 set(points[0], points[1], 44 points[2], points[3], 45 points[4], points[5], 46 points[6], points[7]); 47 return; 48 case 6: 49 set(points[0], points[1], 50 points[2], points[3], 51 points[4], points[5]); 52 return; 53 default: 54 throw new InternalError("Curves can only be cubic or quadratic"); 55 } 56 } 57 58 void set(float x1, float y1, 59 float x2, float y2, 60 float x3, float y3, 61 float x4, float y4) 62 { 63 ax = 3f * (x2 - x3) + x4 - x1; 64 ay = 3f * (y2 - y3) + y4 - y1; 65 bx = 3f * (x1 - 2f * x2 + x3); 66 by = 3f * (y1 - 2f * y2 + y3); 67 cx = 3f * (x2 - x1); 68 cy = 3f * (y2 - y1); 69 dx = x1; 70 dy = y1; 71 dax = 3f * ax; day = 3f * ay; 72 dbx = 2f * bx; dby = 2f * by; 73 } 74 75 void set(float x1, float y1, 76 float x2, float y2, 77 float x3, float y3) 78 { 79 ax = 0f; ay = 0f; 80 bx = x1 - 2f * x2 + x3; 81 by = y1 - 2f * y2 + y3; 82 cx = 2f * (x2 - x1); 83 cy = 2f * (y2 - y1); 84 dx = x1; 85 dy = y1; 86 dax = 0f; day = 0f; 87 dbx = 2f * bx; dby = 2f * by; 88 } 89 90 float xat(float t) { 91 return t * (t * (t * ax + bx) + cx) + dx; 92 } 93 float yat(float t) { 94 return t * (t * (t * ay + by) + cy) + dy; 95 } 96 97 float dxat(float t) { 98 return t * (t * dax + dbx) + cx; 99 } 100 101 float dyat(float t) { 102 return t * (t * day + dby) + cy; 103 } 104 105 int dxRoots(float[] roots, int off) { 106 return Helpers.quadraticRoots(dax, dbx, cx, roots, off); 107 } 108 109 int dyRoots(float[] roots, int off) { 110 return Helpers.quadraticRoots(day, dby, cy, roots, off); 111 } 112 113 int infPoints(float[] pts, int off) { 114 // inflection point at t if -f'(t)x*f''(t)y + f'(t)y*f''(t)x == 0 115 // Fortunately, this turns out to be quadratic, so there are at 116 // most 2 inflection points. 117 final float a = dax * dby - dbx * day; 118 final float b = 2f * (cy * dax - day * cx); 119 final float c = cy * dbx - cx * dby; 120 121 return Helpers.quadraticRoots(a, b, c, pts, off); 122 } 123 124 // finds points where the first and second derivative are 125 // perpendicular. This happens when g(t) = f'(t)*f''(t) == 0 (where 126 // * is a dot product). Unfortunately, we have to solve a cubic. 127 private int perpendiculardfddf(float[] pts, int off) { 128 assert pts.length >= off + 4; 129 130 // these are the coefficients of some multiple of g(t) (not g(t), 131 // because the roots of a polynomial are not changed after multiplication 132 // by a constant, and this way we save a few multiplications). 133 final float a = 2f * (dax*dax + day*day); 134 final float b = 3f * (dax*dbx + day*dby); 135 final float c = 2f * (dax*cx + day*cy) + dbx*dbx + dby*dby; 136 final float d = dbx*cx + dby*cy; 137 return Helpers.cubicRootsInAB(a, b, c, d, pts, off, 0f, 1f); 138 } 139 140 // Tries to find the roots of the function ROC(t)-w in [0, 1). It uses 141 // a variant of the false position algorithm to find the roots. False 142 // position requires that 2 initial values x0,x1 be given, and that the 143 // function must have opposite signs at those values. To find such 144 // values, we need the local extrema of the ROC function, for which we 145 // need the roots of its derivative; however, it's harder to find the 146 // roots of the derivative in this case than it is to find the roots 147 // of the original function. So, we find all points where this curve's 148 // first and second derivative are perpendicular, and we pretend these 149 // are our local extrema. There are at most 3 of these, so we will check 150 // at most 4 sub-intervals of (0,1). ROC has asymptotes at inflection 151 // points, so roc-w can have at least 6 roots. This shouldn't be a 152 // problem for what we're trying to do (draw a nice looking curve). 153 int rootsOfROCMinusW(float[] roots, int off, final float w, final float err) { 154 // no OOB exception, because by now off<=6, and roots.length >= 10 155 assert off <= 6 && roots.length >= 10; 156 int ret = off; 157 int numPerpdfddf = perpendiculardfddf(roots, off); 158 float t0 = 0, ft0 = ROCsq(t0) - w*w; 159 roots[off + numPerpdfddf] = 1f; // always check interval end points 160 numPerpdfddf++; 161 for (int i = off; i < off + numPerpdfddf; i++) { 162 float t1 = roots[i], ft1 = ROCsq(t1) - w*w; 163 if (ft0 == 0f) { 164 roots[ret++] = t0; 165 } else if (ft1 * ft0 < 0f) { // have opposite signs 166 // (ROC(t)^2 == w^2) == (ROC(t) == w) is true because 167 // ROC(t) >= 0 for all t. 168 roots[ret++] = falsePositionROCsqMinusX(t0, t1, w*w, err); 169 } 170 t0 = t1; 171 ft0 = ft1; 172 } 173 174 return ret - off; 175 } 176 177 private static float eliminateInf(float x) { 178 return (x == Float.POSITIVE_INFINITY ? Float.MAX_VALUE : 179 (x == Float.NEGATIVE_INFINITY ? Float.MIN_VALUE : x)); 180 } 181 182 // A slight modification of the false position algorithm on wikipedia. 183 // This only works for the ROCsq-x functions. It might be nice to have 184 // the function as an argument, but that would be awkward in java6. 185 // TODO: It is something to consider for java8 (or whenever lambda 186 // expressions make it into the language), depending on how closures 187 // and turn out. Same goes for the newton's method 188 // algorithm in Helpers.java 189 private float falsePositionROCsqMinusX(float x0, float x1, 190 final float x, final float err) 191 { 192 final int iterLimit = 100; 193 int side = 0; 194 float t = x1, ft = eliminateInf(ROCsq(t) - x); 195 float s = x0, fs = eliminateInf(ROCsq(s) - x); 196 float r = s, fr; 197 for (int i = 0; i < iterLimit && Math.abs(t - s) > err * Math.abs(t + s); i++) { 198 r = (fs * t - ft * s) / (fs - ft); 199 fr = ROCsq(r) - x; 200 if (sameSign(fr, ft)) { 201 ft = fr; t = r; 202 if (side < 0) { 203 fs /= (1 << (-side)); 204 side--; 205 } else { 206 side = -1; 207 } 208 } else if (fr * fs > 0) { 209 fs = fr; s = r; 210 if (side > 0) { 211 ft /= (1 << side); 212 side++; 213 } else { 214 side = 1; 215 } 216 } else { 217 break; 218 } 219 } 220 return r; 221 } 222 223 private static boolean sameSign(float x, float y) { 224 // another way is to test if x*y > 0. This is bad for small x, y. 225 return (x < 0f && y < 0f) || (x > 0f && y > 0f); 226 } 227 228 // returns the radius of curvature squared at t of this curve 229 // see http://en.wikipedia.org/wiki/Radius_of_curvature_(applications) 230 private float ROCsq(final float t) { 231 // dx=xat(t) and dy=yat(t). These calls have been inlined for efficiency 232 final float dx = t * (t * dax + dbx) + cx; 233 final float dy = t * (t * day + dby) + cy; 234 final float ddx = 2f * dax * t + dbx; 235 final float ddy = 2f * day * t + dby; 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 return dx2dy2*((dx2dy2*dx2dy2) / (dx2dy2 * ddx2ddy2 - ddxdxddydy*ddxdxddydy)); 240 } 241 242 // curve to be broken should be in pts 243 // this will change the contents of pts but not Ts 244 // TODO: There's no reason for Ts to be an array. All we need is a sequence 245 // of t values at which to subdivide. An array statisfies this condition, 246 // but is unnecessarily restrictive. Ts should be an Iterator<Float> instead. 247 // Doing this will also make dashing easier, since we could easily make 248 // LengthIterator an Iterator<Float> and feed it to this function to simplify 249 // the loop in Dasher.somethingTo. 250 BreakPtrIterator breakPtsAtTs(final float[] pts, final int type, 251 final float[] Ts, final int numTs) 252 { 253 assert pts.length >= 2*type && numTs <= Ts.length; 254 255 // initialize shared iterator: 256 iterator.init(pts, type, Ts, numTs); 257 258 return iterator; 259 } 260 261 static final class BreakPtrIterator { 262 private int nextCurveIdx; 263 private int curCurveOff; 264 private float prevT; 265 private float[] pts; 266 private int type; 267 private float[] ts; 268 private int numTs; 269 270 void init(final float[] pts, final int type, 271 final float[] ts, final int numTs) { 272 this.pts = pts; 273 this.type = type; 274 this.ts = ts; 275 this.numTs = numTs; 276 277 nextCurveIdx = 0; 278 curCurveOff = 0; 279 prevT = 0f; 280 } 281 282 public boolean hasNext() { 283 return nextCurveIdx <= numTs; 284 } 285 286 public int next() { 287 int ret; 288 if (nextCurveIdx < numTs) { 289 float curT = ts[nextCurveIdx]; 290 float splitT = (curT - prevT) / (1f - prevT); 291 Helpers.subdivideAt(splitT, 292 pts, curCurveOff, 293 pts, 0, 294 pts, type, type); 295 prevT = curT; 296 ret = 0; 297 curCurveOff = type; 298 } else { 299 ret = curCurveOff; 300 } 301 nextCurveIdx++; 302 return ret; 303 } 304 } 305 } 306