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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 | |||