caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 1 | /* |
| 2 | * Copyright 2012 Google Inc. |
| 3 | * |
| 4 | * Use of this source code is governed by a BSD-style license that can be |
| 5 | * found in the LICENSE file. |
| 6 | */ |
| 7 | #include "SkReduceOrder.h" |
| 8 | |
| 9 | int SkReduceOrder::reduce(const SkDLine& line) { |
| 10 | fLine[0] = line[0]; |
| 11 | int different = line[0] != line[1]; |
| 12 | fLine[1] = line[different]; |
| 13 | return 1 + different; |
| 14 | } |
| 15 | |
| 16 | static double interp_quad_coords(double a, double b, double c, double t) { |
| 17 | double ab = SkDInterp(a, b, t); |
| 18 | double bc = SkDInterp(b, c, t); |
| 19 | return SkDInterp(ab, bc, t); |
| 20 | } |
| 21 | |
| 22 | static int coincident_line(const SkDQuad& quad, SkDQuad& reduction) { |
| 23 | reduction[0] = reduction[1] = quad[0]; |
| 24 | return 1; |
| 25 | } |
| 26 | |
| 27 | static int reductionLineCount(const SkDQuad& reduction) { |
| 28 | return 1 + !reduction[0].approximatelyEqual(reduction[1]); |
| 29 | } |
| 30 | |
| 31 | static int vertical_line(const SkDQuad& quad, SkReduceOrder::Style reduceStyle, |
| 32 | SkDQuad& reduction) { |
| 33 | double tValue; |
| 34 | reduction[0] = quad[0]; |
| 35 | reduction[1] = quad[2]; |
| 36 | if (reduceStyle == SkReduceOrder::kFill_Style) { |
| 37 | return reductionLineCount(reduction); |
| 38 | } |
| 39 | int smaller = reduction[1].fY > reduction[0].fY; |
| 40 | int larger = smaller ^ 1; |
| 41 | if (SkDQuad::FindExtrema(quad[0].fY, quad[1].fY, quad[2].fY, &tValue)) { |
| 42 | double yExtrema = interp_quad_coords(quad[0].fY, quad[1].fY, quad[2].fY, tValue); |
| 43 | if (reduction[smaller].fY > yExtrema) { |
| 44 | reduction[smaller].fY = yExtrema; |
| 45 | } else if (reduction[larger].fY < yExtrema) { |
| 46 | reduction[larger].fY = yExtrema; |
| 47 | } |
| 48 | } |
| 49 | return reductionLineCount(reduction); |
| 50 | } |
| 51 | |
| 52 | static int horizontal_line(const SkDQuad& quad, SkReduceOrder::Style reduceStyle, |
| 53 | SkDQuad& reduction) { |
| 54 | double tValue; |
| 55 | reduction[0] = quad[0]; |
| 56 | reduction[1] = quad[2]; |
| 57 | if (reduceStyle == SkReduceOrder::kFill_Style) { |
| 58 | return reductionLineCount(reduction); |
| 59 | } |
| 60 | int smaller = reduction[1].fX > reduction[0].fX; |
| 61 | int larger = smaller ^ 1; |
| 62 | if (SkDQuad::FindExtrema(quad[0].fX, quad[1].fX, quad[2].fX, &tValue)) { |
| 63 | double xExtrema = interp_quad_coords(quad[0].fX, quad[1].fX, quad[2].fX, tValue); |
| 64 | if (reduction[smaller].fX > xExtrema) { |
| 65 | reduction[smaller].fX = xExtrema; |
| 66 | } else if (reduction[larger].fX < xExtrema) { |
| 67 | reduction[larger].fX = xExtrema; |
| 68 | } |
| 69 | } |
| 70 | return reductionLineCount(reduction); |
| 71 | } |
| 72 | |
| 73 | static int check_linear(const SkDQuad& quad, SkReduceOrder::Style reduceStyle, |
| 74 | int minX, int maxX, int minY, int maxY, SkDQuad& reduction) { |
| 75 | int startIndex = 0; |
| 76 | int endIndex = 2; |
| 77 | while (quad[startIndex].approximatelyEqual(quad[endIndex])) { |
| 78 | --endIndex; |
| 79 | if (endIndex == 0) { |
| 80 | SkDebugf("%s shouldn't get here if all four points are about equal", __FUNCTION__); |
| 81 | SkASSERT(0); |
| 82 | } |
| 83 | } |
| 84 | if (!quad.isLinear(startIndex, endIndex)) { |
| 85 | return 0; |
| 86 | } |
| 87 | // four are colinear: return line formed by outside |
| 88 | reduction[0] = quad[0]; |
| 89 | reduction[1] = quad[2]; |
| 90 | if (reduceStyle == SkReduceOrder::kFill_Style) { |
| 91 | return reductionLineCount(reduction); |
| 92 | } |
| 93 | int sameSide; |
| 94 | bool useX = quad[maxX].fX - quad[minX].fX >= quad[maxY].fY - quad[minY].fY; |
| 95 | if (useX) { |
| 96 | sameSide = SkDSign(quad[0].fX - quad[1].fX) + SkDSign(quad[2].fX - quad[1].fX); |
| 97 | } else { |
| 98 | sameSide = SkDSign(quad[0].fY - quad[1].fY) + SkDSign(quad[2].fY - quad[1].fY); |
| 99 | } |
| 100 | if ((sameSide & 3) != 2) { |
| 101 | return reductionLineCount(reduction); |
| 102 | } |
| 103 | double tValue; |
| 104 | int root; |
| 105 | if (useX) { |
| 106 | root = SkDQuad::FindExtrema(quad[0].fX, quad[1].fX, quad[2].fX, &tValue); |
| 107 | } else { |
| 108 | root = SkDQuad::FindExtrema(quad[0].fY, quad[1].fY, quad[2].fY, &tValue); |
| 109 | } |
| 110 | if (root) { |
| 111 | SkDPoint extrema; |
| 112 | extrema.fX = interp_quad_coords(quad[0].fX, quad[1].fX, quad[2].fX, tValue); |
| 113 | extrema.fY = interp_quad_coords(quad[0].fY, quad[1].fY, quad[2].fY, tValue); |
| 114 | // sameSide > 0 means mid is smaller than either [0] or [2], so replace smaller |
| 115 | int replace; |
| 116 | if (useX) { |
| 117 | if ((extrema.fX < quad[0].fX) ^ (extrema.fX < quad[2].fX)) { |
| 118 | return reductionLineCount(reduction); |
| 119 | } |
| 120 | replace = ((extrema.fX < quad[0].fX) | (extrema.fX < quad[2].fX)) |
| 121 | ^ (quad[0].fX < quad[2].fX); |
| 122 | } else { |
| 123 | if ((extrema.fY < quad[0].fY) ^ (extrema.fY < quad[2].fY)) { |
| 124 | return reductionLineCount(reduction); |
| 125 | } |
| 126 | replace = ((extrema.fY < quad[0].fY) | (extrema.fY < quad[2].fY)) |
| 127 | ^ (quad[0].fY < quad[2].fY); |
| 128 | } |
| 129 | reduction[replace] = extrema; |
| 130 | } |
| 131 | return reductionLineCount(reduction); |
| 132 | } |
| 133 | |
| 134 | // reduce to a quadratic or smaller |
| 135 | // look for identical points |
| 136 | // look for all four points in a line |
| 137 | // note that three points in a line doesn't simplify a cubic |
| 138 | // look for approximation with single quadratic |
| 139 | // save approximation with multiple quadratics for later |
| 140 | int SkReduceOrder::reduce(const SkDQuad& quad, Style reduceStyle) { |
| 141 | int index, minX, maxX, minY, maxY; |
| 142 | int minXSet, minYSet; |
| 143 | minX = maxX = minY = maxY = 0; |
| 144 | minXSet = minYSet = 0; |
| 145 | for (index = 1; index < 3; ++index) { |
| 146 | if (quad[minX].fX > quad[index].fX) { |
| 147 | minX = index; |
| 148 | } |
| 149 | if (quad[minY].fY > quad[index].fY) { |
| 150 | minY = index; |
| 151 | } |
| 152 | if (quad[maxX].fX < quad[index].fX) { |
| 153 | maxX = index; |
| 154 | } |
| 155 | if (quad[maxY].fY < quad[index].fY) { |
| 156 | maxY = index; |
| 157 | } |
| 158 | } |
| 159 | for (index = 0; index < 3; ++index) { |
| 160 | if (AlmostEqualUlps(quad[index].fX, quad[minX].fX)) { |
| 161 | minXSet |= 1 << index; |
| 162 | } |
| 163 | if (AlmostEqualUlps(quad[index].fY, quad[minY].fY)) { |
| 164 | minYSet |= 1 << index; |
| 165 | } |
| 166 | } |
| 167 | if (minXSet == 0x7) { // test for vertical line |
| 168 | if (minYSet == 0x7) { // return 1 if all four are coincident |
| 169 | return coincident_line(quad, fQuad); |
| 170 | } |
| 171 | return vertical_line(quad, reduceStyle, fQuad); |
| 172 | } |
| 173 | if (minYSet == 0xF) { // test for horizontal line |
| 174 | return horizontal_line(quad, reduceStyle, fQuad); |
| 175 | } |
| 176 | int result = check_linear(quad, reduceStyle, minX, maxX, minY, maxY, fQuad); |
| 177 | if (result) { |
| 178 | return result; |
| 179 | } |
| 180 | fQuad = quad; |
| 181 | return 3; |
| 182 | } |
| 183 | |
| 184 | //////////////////////////////////////////////////////////////////////////////////// |
| 185 | |
| 186 | static double interp_cubic_coords(const double* src, double t) { |
| 187 | double ab = SkDInterp(src[0], src[2], t); |
| 188 | double bc = SkDInterp(src[2], src[4], t); |
| 189 | double cd = SkDInterp(src[4], src[6], t); |
| 190 | double abc = SkDInterp(ab, bc, t); |
| 191 | double bcd = SkDInterp(bc, cd, t); |
| 192 | return SkDInterp(abc, bcd, t); |
| 193 | } |
| 194 | |
| 195 | static int coincident_line(const SkDCubic& cubic, SkDCubic& reduction) { |
| 196 | reduction[0] = reduction[1] = cubic[0]; |
| 197 | return 1; |
| 198 | } |
| 199 | |
| 200 | static int reductionLineCount(const SkDCubic& reduction) { |
| 201 | return 1 + !reduction[0].approximatelyEqual(reduction[1]); |
| 202 | } |
| 203 | |
| 204 | static int vertical_line(const SkDCubic& cubic, SkReduceOrder::Style reduceStyle, |
| 205 | SkDCubic& reduction) { |
| 206 | double tValues[2]; |
| 207 | reduction[0] = cubic[0]; |
| 208 | reduction[1] = cubic[3]; |
| 209 | if (reduceStyle == SkReduceOrder::kFill_Style) { |
| 210 | return reductionLineCount(reduction); |
| 211 | } |
| 212 | int smaller = reduction[1].fY > reduction[0].fY; |
| 213 | int larger = smaller ^ 1; |
| 214 | int roots = SkDCubic::FindExtrema(cubic[0].fY, cubic[1].fY, cubic[2].fY, cubic[3].fY, tValues); |
| 215 | for (int index = 0; index < roots; ++index) { |
| 216 | double yExtrema = interp_cubic_coords(&cubic[0].fY, tValues[index]); |
| 217 | if (reduction[smaller].fY > yExtrema) { |
| 218 | reduction[smaller].fY = yExtrema; |
| 219 | continue; |
| 220 | } |
| 221 | if (reduction[larger].fY < yExtrema) { |
| 222 | reduction[larger].fY = yExtrema; |
| 223 | } |
| 224 | } |
| 225 | return reductionLineCount(reduction); |
| 226 | } |
| 227 | |
| 228 | static int horizontal_line(const SkDCubic& cubic, SkReduceOrder::Style reduceStyle, |
| 229 | SkDCubic& reduction) { |
| 230 | double tValues[2]; |
| 231 | reduction[0] = cubic[0]; |
| 232 | reduction[1] = cubic[3]; |
| 233 | if (reduceStyle == SkReduceOrder::kFill_Style) { |
| 234 | return reductionLineCount(reduction); |
| 235 | } |
| 236 | int smaller = reduction[1].fX > reduction[0].fX; |
| 237 | int larger = smaller ^ 1; |
| 238 | int roots = SkDCubic::FindExtrema(cubic[0].fX, cubic[1].fX, cubic[2].fX, cubic[3].fX, tValues); |
| 239 | for (int index = 0; index < roots; ++index) { |
| 240 | double xExtrema = interp_cubic_coords(&cubic[0].fX, tValues[index]); |
| 241 | if (reduction[smaller].fX > xExtrema) { |
| 242 | reduction[smaller].fX = xExtrema; |
| 243 | continue; |
| 244 | } |
| 245 | if (reduction[larger].fX < xExtrema) { |
| 246 | reduction[larger].fX = xExtrema; |
| 247 | } |
| 248 | } |
| 249 | return reductionLineCount(reduction); |
| 250 | } |
| 251 | |
| 252 | // check to see if it is a quadratic or a line |
| 253 | static int check_quadratic(const SkDCubic& cubic, SkDCubic& reduction) { |
| 254 | double dx10 = cubic[1].fX - cubic[0].fX; |
| 255 | double dx23 = cubic[2].fX - cubic[3].fX; |
| 256 | double midX = cubic[0].fX + dx10 * 3 / 2; |
caryclark@google.com | 7ec5e39 | 2013-04-17 16:19:02 +0000 | [diff] [blame^] | 257 | double sideAx = midX - cubic[3].fX; |
| 258 | double sideBx = dx23 * 3 / 2; |
| 259 | if (approximately_zero(sideAx) ? !approximately_equal(sideAx, sideBx) |
| 260 | : !AlmostEqualUlps(sideAx, sideBx)) { |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 261 | return 0; |
| 262 | } |
| 263 | double dy10 = cubic[1].fY - cubic[0].fY; |
| 264 | double dy23 = cubic[2].fY - cubic[3].fY; |
| 265 | double midY = cubic[0].fY + dy10 * 3 / 2; |
caryclark@google.com | 7ec5e39 | 2013-04-17 16:19:02 +0000 | [diff] [blame^] | 266 | double sideAy = midY - cubic[3].fY; |
| 267 | double sideBy = dy23 * 3 / 2; |
| 268 | if (approximately_zero(sideAy) ? !approximately_equal(sideAy, sideBy) |
| 269 | : !AlmostEqualUlps(sideAy, sideBy)) { |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 270 | return 0; |
| 271 | } |
| 272 | reduction[0] = cubic[0]; |
| 273 | reduction[1].fX = midX; |
| 274 | reduction[1].fY = midY; |
| 275 | reduction[2] = cubic[3]; |
| 276 | return 3; |
| 277 | } |
| 278 | |
| 279 | static int check_linear(const SkDCubic& cubic, SkReduceOrder::Style reduceStyle, |
| 280 | int minX, int maxX, int minY, int maxY, SkDCubic& reduction) { |
| 281 | int startIndex = 0; |
| 282 | int endIndex = 3; |
| 283 | while (cubic[startIndex].approximatelyEqual(cubic[endIndex])) { |
| 284 | --endIndex; |
| 285 | if (endIndex == 0) { |
| 286 | SkDebugf("%s shouldn't get here if all four points are about equal\n", __FUNCTION__); |
| 287 | SkASSERT(0); |
| 288 | } |
| 289 | } |
| 290 | if (!cubic.isLinear(startIndex, endIndex)) { |
| 291 | return 0; |
| 292 | } |
| 293 | // four are colinear: return line formed by outside |
| 294 | reduction[0] = cubic[0]; |
| 295 | reduction[1] = cubic[3]; |
| 296 | if (reduceStyle == SkReduceOrder::kFill_Style) { |
| 297 | return reductionLineCount(reduction); |
| 298 | } |
| 299 | int sameSide1; |
| 300 | int sameSide2; |
| 301 | bool useX = cubic[maxX].fX - cubic[minX].fX >= cubic[maxY].fY - cubic[minY].fY; |
| 302 | if (useX) { |
| 303 | sameSide1 = SkDSign(cubic[0].fX - cubic[1].fX) + SkDSign(cubic[3].fX - cubic[1].fX); |
| 304 | sameSide2 = SkDSign(cubic[0].fX - cubic[2].fX) + SkDSign(cubic[3].fX - cubic[2].fX); |
| 305 | } else { |
| 306 | sameSide1 = SkDSign(cubic[0].fY - cubic[1].fY) + SkDSign(cubic[3].fY - cubic[1].fY); |
| 307 | sameSide2 = SkDSign(cubic[0].fY - cubic[2].fY) + SkDSign(cubic[3].fY - cubic[2].fY); |
| 308 | } |
| 309 | if (sameSide1 == sameSide2 && (sameSide1 & 3) != 2) { |
| 310 | return reductionLineCount(reduction); |
| 311 | } |
| 312 | double tValues[2]; |
| 313 | int roots; |
| 314 | if (useX) { |
| 315 | roots = SkDCubic::FindExtrema(cubic[0].fX, cubic[1].fX, cubic[2].fX, cubic[3].fX, tValues); |
| 316 | } else { |
| 317 | roots = SkDCubic::FindExtrema(cubic[0].fY, cubic[1].fY, cubic[2].fY, cubic[3].fY, tValues); |
| 318 | } |
| 319 | for (int index = 0; index < roots; ++index) { |
| 320 | SkDPoint extrema; |
| 321 | extrema.fX = interp_cubic_coords(&cubic[0].fX, tValues[index]); |
| 322 | extrema.fY = interp_cubic_coords(&cubic[0].fY, tValues[index]); |
| 323 | // sameSide > 0 means mid is smaller than either [0] or [3], so replace smaller |
| 324 | int replace; |
| 325 | if (useX) { |
| 326 | if ((extrema.fX < cubic[0].fX) ^ (extrema.fX < cubic[3].fX)) { |
| 327 | continue; |
| 328 | } |
| 329 | replace = ((extrema.fX < cubic[0].fX) | (extrema.fX < cubic[3].fX)) |
| 330 | ^ (cubic[0].fX < cubic[3].fX); |
| 331 | } else { |
| 332 | if ((extrema.fY < cubic[0].fY) ^ (extrema.fY < cubic[3].fY)) { |
| 333 | continue; |
| 334 | } |
| 335 | replace = ((extrema.fY < cubic[0].fY) | (extrema.fY < cubic[3].fY)) |
| 336 | ^ (cubic[0].fY < cubic[3].fY); |
| 337 | } |
| 338 | reduction[replace] = extrema; |
| 339 | } |
| 340 | return reductionLineCount(reduction); |
| 341 | } |
| 342 | |
| 343 | /* food for thought: |
| 344 | http://objectmix.com/graphics/132906-fast-precision-driven-cubic-quadratic-piecewise-degree-reduction-algos-2-a.html |
| 345 | |
| 346 | Given points c1, c2, c3 and c4 of a cubic Bezier, the points of the |
| 347 | corresponding quadratic Bezier are (given in convex combinations of |
| 348 | points): |
| 349 | |
| 350 | q1 = (11/13)c1 + (3/13)c2 -(3/13)c3 + (2/13)c4 |
| 351 | q2 = -c1 + (3/2)c2 + (3/2)c3 - c4 |
| 352 | q3 = (2/13)c1 - (3/13)c2 + (3/13)c3 + (11/13)c4 |
| 353 | |
| 354 | Of course, this curve does not interpolate the end-points, but it would |
| 355 | be interesting to see the behaviour of such a curve in an applet. |
| 356 | |
| 357 | -- |
| 358 | Kalle Rutanen |
| 359 | http://kaba.hilvi.org |
| 360 | |
| 361 | */ |
| 362 | |
| 363 | // reduce to a quadratic or smaller |
| 364 | // look for identical points |
| 365 | // look for all four points in a line |
| 366 | // note that three points in a line doesn't simplify a cubic |
| 367 | // look for approximation with single quadratic |
| 368 | // save approximation with multiple quadratics for later |
| 369 | int SkReduceOrder::reduce(const SkDCubic& cubic, Quadratics allowQuadratics, |
| 370 | Style reduceStyle) { |
| 371 | int index, minX, maxX, minY, maxY; |
| 372 | int minXSet, minYSet; |
| 373 | minX = maxX = minY = maxY = 0; |
| 374 | minXSet = minYSet = 0; |
| 375 | for (index = 1; index < 4; ++index) { |
| 376 | if (cubic[minX].fX > cubic[index].fX) { |
| 377 | minX = index; |
| 378 | } |
| 379 | if (cubic[minY].fY > cubic[index].fY) { |
| 380 | minY = index; |
| 381 | } |
| 382 | if (cubic[maxX].fX < cubic[index].fX) { |
| 383 | maxX = index; |
| 384 | } |
| 385 | if (cubic[maxY].fY < cubic[index].fY) { |
| 386 | maxY = index; |
| 387 | } |
| 388 | } |
| 389 | for (index = 0; index < 4; ++index) { |
| 390 | double cx = cubic[index].fX; |
| 391 | double cy = cubic[index].fY; |
| 392 | double denom = SkTMax(fabs(cx), SkTMax(fabs(cy), |
| 393 | SkTMax(fabs(cubic[minX].fX), fabs(cubic[minY].fY)))); |
| 394 | if (denom == 0) { |
| 395 | minXSet |= 1 << index; |
| 396 | minYSet |= 1 << index; |
| 397 | continue; |
| 398 | } |
| 399 | double inv = 1 / denom; |
| 400 | if (approximately_equal_half(cx * inv, cubic[minX].fX * inv)) { |
| 401 | minXSet |= 1 << index; |
| 402 | } |
| 403 | if (approximately_equal_half(cy * inv, cubic[minY].fY * inv)) { |
| 404 | minYSet |= 1 << index; |
| 405 | } |
| 406 | } |
| 407 | if (minXSet == 0xF) { // test for vertical line |
| 408 | if (minYSet == 0xF) { // return 1 if all four are coincident |
| 409 | return coincident_line(cubic, fCubic); |
| 410 | } |
| 411 | return vertical_line(cubic, reduceStyle, fCubic); |
| 412 | } |
| 413 | if (minYSet == 0xF) { // test for horizontal line |
| 414 | return horizontal_line(cubic, reduceStyle, fCubic); |
| 415 | } |
| 416 | int result = check_linear(cubic, reduceStyle, minX, maxX, minY, maxY, fCubic); |
| 417 | if (result) { |
| 418 | return result; |
| 419 | } |
| 420 | if (allowQuadratics == SkReduceOrder::kAllow_Quadratics |
| 421 | && (result = check_quadratic(cubic, fCubic))) { |
| 422 | return result; |
| 423 | } |
| 424 | fCubic = cubic; |
| 425 | return 4; |
| 426 | } |
| 427 | |
| 428 | SkPath::Verb SkReduceOrder::Quad(const SkPoint a[3], SkTDArray<SkPoint>* reducePts) { |
| 429 | SkDQuad quad; |
| 430 | quad.set(a); |
| 431 | SkReduceOrder reducer; |
| 432 | int order = reducer.reduce(quad, kFill_Style); |
| 433 | if (order == 2) { // quad became line |
| 434 | for (int index = 0; index < order; ++index) { |
| 435 | SkPoint* pt = reducePts->append(); |
| 436 | pt->fX = SkDoubleToScalar(reducer.fLine[index].fX); |
| 437 | pt->fY = SkDoubleToScalar(reducer.fLine[index].fY); |
| 438 | } |
| 439 | } |
| 440 | return (SkPath::Verb) (order - 1); |
| 441 | } |
| 442 | |
| 443 | SkPath::Verb SkReduceOrder::Cubic(const SkPoint a[4], SkTDArray<SkPoint>* reducePts) { |
| 444 | SkDCubic cubic; |
| 445 | cubic.set(a); |
| 446 | SkReduceOrder reducer; |
| 447 | int order = reducer.reduce(cubic, kAllow_Quadratics, kFill_Style); |
| 448 | if (order == 2 || order == 3) { // cubic became line or quad |
| 449 | for (int index = 0; index < order; ++index) { |
| 450 | SkPoint* pt = reducePts->append(); |
| 451 | pt->fX = SkDoubleToScalar(reducer.fQuad[index].fX); |
| 452 | pt->fY = SkDoubleToScalar(reducer.fQuad[index].fY); |
| 453 | } |
| 454 | } |
| 455 | return (SkPath::Verb) (order - 1); |
| 456 | } |