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| 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 | + | |