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 "SkIntersections.h" 8#include "SkPathOpsCubic.h" 9#include "SkPathOpsLine.h" 10 11/* 12Find the interection of a line and cubic by solving for valid t values. 13 14Analogous to line-quadratic intersection, solve line-cubic intersection by 15representing the cubic as: 16 x = a(1-t)^3 + 2b(1-t)^2t + c(1-t)t^2 + dt^3 17 y = e(1-t)^3 + 2f(1-t)^2t + g(1-t)t^2 + ht^3 18and the line as: 19 y = i*x + j (if the line is more horizontal) 20or: 21 x = i*y + j (if the line is more vertical) 22 23Then using Mathematica, solve for the values of t where the cubic intersects the 24line: 25 26 (in) Resultant[ 27 a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - x, 28 e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - i*x - j, x] 29 (out) -e + j + 30 3 e t - 3 f t - 31 3 e t^2 + 6 f t^2 - 3 g t^2 + 32 e t^3 - 3 f t^3 + 3 g t^3 - h t^3 + 33 i ( a - 34 3 a t + 3 b t + 35 3 a t^2 - 6 b t^2 + 3 c t^2 - 36 a t^3 + 3 b t^3 - 3 c t^3 + d t^3 ) 37 38if i goes to infinity, we can rewrite the line in terms of x. Mathematica: 39 40 (in) Resultant[ 41 a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - i*y - j, 42 e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - y, y] 43 (out) a - j - 44 3 a t + 3 b t + 45 3 a t^2 - 6 b t^2 + 3 c t^2 - 46 a t^3 + 3 b t^3 - 3 c t^3 + d t^3 - 47 i ( e - 48 3 e t + 3 f t + 49 3 e t^2 - 6 f t^2 + 3 g t^2 - 50 e t^3 + 3 f t^3 - 3 g t^3 + h t^3 ) 51 52Solving this with Mathematica produces an expression with hundreds of terms; 53instead, use Numeric Solutions recipe to solve the cubic. 54 55The near-horizontal case, in terms of: Ax^3 + Bx^2 + Cx + D == 0 56 A = (-(-e + 3*f - 3*g + h) + i*(-a + 3*b - 3*c + d) ) 57 B = 3*(-( e - 2*f + g ) + i*( a - 2*b + c ) ) 58 C = 3*(-(-e + f ) + i*(-a + b ) ) 59 D = (-( e ) + i*( a ) + j ) 60 61The near-vertical case, in terms of: Ax^3 + Bx^2 + Cx + D == 0 62 A = ( (-a + 3*b - 3*c + d) - i*(-e + 3*f - 3*g + h) ) 63 B = 3*( ( a - 2*b + c ) - i*( e - 2*f + g ) ) 64 C = 3*( (-a + b ) - i*(-e + f ) ) 65 D = ( ( a ) - i*( e ) - j ) 66 67For horizontal lines: 68(in) Resultant[ 69 a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - j, 70 e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - y, y] 71(out) e - j - 72 3 e t + 3 f t + 73 3 e t^2 - 6 f t^2 + 3 g t^2 - 74 e t^3 + 3 f t^3 - 3 g t^3 + h t^3 75 */ 76 77class LineCubicIntersections { 78public: 79 enum PinTPoint { 80 kPointUninitialized, 81 kPointInitialized 82 }; 83 84 LineCubicIntersections(const SkDCubic& c, const SkDLine& l, SkIntersections* i) 85 : fCubic(c) 86 , fLine(l) 87 , fIntersections(i) 88 , fAllowNear(true) { 89 i->setMax(3); 90 } 91 92 void allowNear(bool allow) { 93 fAllowNear = allow; 94 } 95 96 // see parallel routine in line quadratic intersections 97 int intersectRay(double roots[3]) { 98 double adj = fLine[1].fX - fLine[0].fX; 99 double opp = fLine[1].fY - fLine[0].fY; 100 SkDCubic r; 101 for (int n = 0; n < 4; ++n) { 102 r[n].fX = (fCubic[n].fY - fLine[0].fY) * adj - (fCubic[n].fX - fLine[0].fX) * opp; 103 } 104 double A, B, C, D; 105 SkDCubic::Coefficients(&r[0].fX, &A, &B, &C, &D); 106 return SkDCubic::RootsValidT(A, B, C, D, roots); 107 } 108 109 int intersect() { 110 addExactEndPoints(); 111 if (fAllowNear) { 112 addNearEndPoints(); 113 } 114 double rootVals[3]; 115 int roots = intersectRay(rootVals); 116 for (int index = 0; index < roots; ++index) { 117 double cubicT = rootVals[index]; 118 double lineT = findLineT(cubicT); 119 SkDPoint pt; 120 if (pinTs(&cubicT, &lineT, &pt, kPointUninitialized)) { 121 #if ONE_OFF_DEBUG 122 SkDPoint cPt = fCubic.ptAtT(cubicT); 123 SkDebugf("%s pt=(%1.9g,%1.9g) cPt=(%1.9g,%1.9g)\n", __FUNCTION__, pt.fX, pt.fY, 124 cPt.fX, cPt.fY); 125 #endif 126 for (int inner = 0; inner < fIntersections->used(); ++inner) { 127 if (fIntersections->pt(inner) != pt) { 128 continue; 129 } 130 double existingCubicT = (*fIntersections)[0][inner]; 131 if (cubicT == existingCubicT) { 132 goto skipInsert; 133 } 134 // check if midway on cubic is also same point. If so, discard this 135 double cubicMidT = (existingCubicT + cubicT) / 2; 136 SkDPoint cubicMidPt = fCubic.ptAtT(cubicMidT); 137 if (cubicMidPt.approximatelyEqual(pt)) { 138 goto skipInsert; 139 } 140 } 141 fIntersections->insert(cubicT, lineT, pt); 142 skipInsert: 143 ; 144 } 145 } 146 return fIntersections->used(); 147 } 148 149 int horizontalIntersect(double axisIntercept, double roots[3]) { 150 double A, B, C, D; 151 SkDCubic::Coefficients(&fCubic[0].fY, &A, &B, &C, &D); 152 D -= axisIntercept; 153 return SkDCubic::RootsValidT(A, B, C, D, roots); 154 } 155 156 int horizontalIntersect(double axisIntercept, double left, double right, bool flipped) { 157 addExactHorizontalEndPoints(left, right, axisIntercept); 158 if (fAllowNear) { 159 addNearHorizontalEndPoints(left, right, axisIntercept); 160 } 161 double rootVals[3]; 162 int roots = horizontalIntersect(axisIntercept, rootVals); 163 for (int index = 0; index < roots; ++index) { 164 double cubicT = rootVals[index]; 165 SkDPoint pt = fCubic.ptAtT(cubicT); 166 double lineT = (pt.fX - left) / (right - left); 167 if (pinTs(&cubicT, &lineT, &pt, kPointInitialized)) { 168 fIntersections->insert(cubicT, lineT, pt); 169 } 170 } 171 if (flipped) { 172 fIntersections->flip(); 173 } 174 return fIntersections->used(); 175 } 176 177 int verticalIntersect(double axisIntercept, double roots[3]) { 178 double A, B, C, D; 179 SkDCubic::Coefficients(&fCubic[0].fX, &A, &B, &C, &D); 180 D -= axisIntercept; 181 return SkDCubic::RootsValidT(A, B, C, D, roots); 182 } 183 184 int verticalIntersect(double axisIntercept, double top, double bottom, bool flipped) { 185 addExactVerticalEndPoints(top, bottom, axisIntercept); 186 if (fAllowNear) { 187 addNearVerticalEndPoints(top, bottom, axisIntercept); 188 } 189 double rootVals[3]; 190 int roots = verticalIntersect(axisIntercept, rootVals); 191 for (int index = 0; index < roots; ++index) { 192 double cubicT = rootVals[index]; 193 SkDPoint pt = fCubic.ptAtT(cubicT); 194 double lineT = (pt.fY - top) / (bottom - top); 195 if (pinTs(&cubicT, &lineT, &pt, kPointInitialized)) { 196 fIntersections->insert(cubicT, lineT, pt); 197 } 198 } 199 if (flipped) { 200 fIntersections->flip(); 201 } 202 return fIntersections->used(); 203 } 204 205 protected: 206 207 void addExactEndPoints() { 208 for (int cIndex = 0; cIndex < 4; cIndex += 3) { 209 double lineT = fLine.exactPoint(fCubic[cIndex]); 210 if (lineT < 0) { 211 continue; 212 } 213 double cubicT = (double) (cIndex >> 1); 214 fIntersections->insert(cubicT, lineT, fCubic[cIndex]); 215 } 216 } 217 218 /* Note that this does not look for endpoints of the line that are near the cubic. 219 These points are found later when check ends looks for missing points */ 220 void addNearEndPoints() { 221 for (int cIndex = 0; cIndex < 4; cIndex += 3) { 222 double cubicT = (double) (cIndex >> 1); 223 if (fIntersections->hasT(cubicT)) { 224 continue; 225 } 226 double lineT = fLine.nearPoint(fCubic[cIndex]); 227 if (lineT < 0) { 228 continue; 229 } 230 fIntersections->insert(cubicT, lineT, fCubic[cIndex]); 231 } 232 } 233 234 void addExactHorizontalEndPoints(double left, double right, double y) { 235 for (int cIndex = 0; cIndex < 4; cIndex += 3) { 236 double lineT = SkDLine::ExactPointH(fCubic[cIndex], left, right, y); 237 if (lineT < 0) { 238 continue; 239 } 240 double cubicT = (double) (cIndex >> 1); 241 fIntersections->insert(cubicT, lineT, fCubic[cIndex]); 242 } 243 } 244 245 void addNearHorizontalEndPoints(double left, double right, double y) { 246 for (int cIndex = 0; cIndex < 4; cIndex += 3) { 247 double cubicT = (double) (cIndex >> 1); 248 if (fIntersections->hasT(cubicT)) { 249 continue; 250 } 251 double lineT = SkDLine::NearPointH(fCubic[cIndex], left, right, y); 252 if (lineT < 0) { 253 continue; 254 } 255 fIntersections->insert(cubicT, lineT, fCubic[cIndex]); 256 } 257 // FIXME: see if line end is nearly on cubic 258 } 259 260 void addExactVerticalEndPoints(double top, double bottom, double x) { 261 for (int cIndex = 0; cIndex < 4; cIndex += 3) { 262 double lineT = SkDLine::ExactPointV(fCubic[cIndex], top, bottom, x); 263 if (lineT < 0) { 264 continue; 265 } 266 double cubicT = (double) (cIndex >> 1); 267 fIntersections->insert(cubicT, lineT, fCubic[cIndex]); 268 } 269 } 270 271 void addNearVerticalEndPoints(double top, double bottom, double x) { 272 for (int cIndex = 0; cIndex < 4; cIndex += 3) { 273 double cubicT = (double) (cIndex >> 1); 274 if (fIntersections->hasT(cubicT)) { 275 continue; 276 } 277 double lineT = SkDLine::NearPointV(fCubic[cIndex], top, bottom, x); 278 if (lineT < 0) { 279 continue; 280 } 281 fIntersections->insert(cubicT, lineT, fCubic[cIndex]); 282 } 283 // FIXME: see if line end is nearly on cubic 284 } 285 286 double findLineT(double t) { 287 SkDPoint xy = fCubic.ptAtT(t); 288 double dx = fLine[1].fX - fLine[0].fX; 289 double dy = fLine[1].fY - fLine[0].fY; 290 if (fabs(dx) > fabs(dy)) { 291 return (xy.fX - fLine[0].fX) / dx; 292 } 293 return (xy.fY - fLine[0].fY) / dy; 294 } 295 296 bool pinTs(double* cubicT, double* lineT, SkDPoint* pt, PinTPoint ptSet) { 297 if (!approximately_one_or_less(*lineT)) { 298 return false; 299 } 300 if (!approximately_zero_or_more(*lineT)) { 301 return false; 302 } 303 double cT = *cubicT = SkPinT(*cubicT); 304 double lT = *lineT = SkPinT(*lineT); 305 if (lT == 0 || lT == 1 || (ptSet == kPointUninitialized && cT != 0 && cT != 1)) { 306 *pt = fLine.ptAtT(lT); 307 } else if (ptSet == kPointUninitialized) { 308 *pt = fCubic.ptAtT(cT); 309 } 310 SkPoint gridPt = pt->asSkPoint(); 311 if (gridPt == fLine[0].asSkPoint()) { 312 *lineT = 0; 313 } else if (gridPt == fLine[1].asSkPoint()) { 314 *lineT = 1; 315 } 316 if (gridPt == fCubic[0].asSkPoint() && approximately_equal(*cubicT, 0)) { 317 *cubicT = 0; 318 } else if (gridPt == fCubic[3].asSkPoint() && approximately_equal(*cubicT, 1)) { 319 *cubicT = 1; 320 } 321 return true; 322 } 323 324private: 325 const SkDCubic& fCubic; 326 const SkDLine& fLine; 327 SkIntersections* fIntersections; 328 bool fAllowNear; 329}; 330 331int SkIntersections::horizontal(const SkDCubic& cubic, double left, double right, double y, 332 bool flipped) { 333 SkDLine line = {{{ left, y }, { right, y }}}; 334 LineCubicIntersections c(cubic, line, this); 335 return c.horizontalIntersect(y, left, right, flipped); 336} 337 338int SkIntersections::vertical(const SkDCubic& cubic, double top, double bottom, double x, 339 bool flipped) { 340 SkDLine line = {{{ x, top }, { x, bottom }}}; 341 LineCubicIntersections c(cubic, line, this); 342 return c.verticalIntersect(x, top, bottom, flipped); 343} 344 345int SkIntersections::intersect(const SkDCubic& cubic, const SkDLine& line) { 346 LineCubicIntersections c(cubic, line, this); 347 c.allowNear(fAllowNear); 348 return c.intersect(); 349} 350 351int SkIntersections::intersectRay(const SkDCubic& cubic, const SkDLine& line) { 352 LineCubicIntersections c(cubic, line, this); 353 fUsed = c.intersectRay(fT[0]); 354 for (int index = 0; index < fUsed; ++index) { 355 fPt[index] = cubic.ptAtT(fT[0][index]); 356 } 357 return fUsed; 358} 359