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 "SkPathOpsLine.h" 9 #include "SkPathOpsQuad.h" 10 11 /* 12 Find the interection of a line and quadratic by solving for valid t values. 13 14 From http://stackoverflow.com/questions/1853637/how-to-find-the-mathematical-function-defining-a-bezier-curve 15 16 "A Bezier curve is a parametric function. A quadratic Bezier curve (i.e. three 17 control points) can be expressed as: F(t) = A(1 - t)^2 + B(1 - t)t + Ct^2 where 18 A, B and C are points and t goes from zero to one. 19 20 This will give you two equations: 21 22 x = a(1 - t)^2 + b(1 - t)t + ct^2 23 y = d(1 - t)^2 + e(1 - t)t + ft^2 24 25 If you add for instance the line equation (y = kx + m) to that, you'll end up 26 with three equations and three unknowns (x, y and t)." 27 28 Similar to above, the quadratic is represented as 29 x = a(1-t)^2 + 2b(1-t)t + ct^2 30 y = d(1-t)^2 + 2e(1-t)t + ft^2 31 and the line as 32 y = g*x + h 33 34 Using Mathematica, solve for the values of t where the quadratic intersects the 35 line: 36 37 (in) t1 = Resultant[a*(1 - t)^2 + 2*b*(1 - t)*t + c*t^2 - x, 38 d*(1 - t)^2 + 2*e*(1 - t)*t + f*t^2 - g*x - h, x] 39 (out) -d + h + 2 d t - 2 e t - d t^2 + 2 e t^2 - f t^2 + 40 g (a - 2 a t + 2 b t + a t^2 - 2 b t^2 + c t^2) 41 (in) Solve[t1 == 0, t] 42 (out) { 43 {t -> (-2 d + 2 e + 2 a g - 2 b g - 44 Sqrt[(2 d - 2 e - 2 a g + 2 b g)^2 - 45 4 (-d + 2 e - f + a g - 2 b g + c g) (-d + a g + h)]) / 46 (2 (-d + 2 e - f + a g - 2 b g + c g)) 47 }, 48 {t -> (-2 d + 2 e + 2 a g - 2 b g + 49 Sqrt[(2 d - 2 e - 2 a g + 2 b g)^2 - 50 4 (-d + 2 e - f + a g - 2 b g + c g) (-d + a g + h)]) / 51 (2 (-d + 2 e - f + a g - 2 b g + c g)) 52 } 53 } 54 55 Using the results above (when the line tends towards horizontal) 56 A = (-(d - 2*e + f) + g*(a - 2*b + c) ) 57 B = 2*( (d - e ) - g*(a - b ) ) 58 C = (-(d ) + g*(a ) + h ) 59 60 If g goes to infinity, we can rewrite the line in terms of x. 61 x = g'*y + h' 62 63 And solve accordingly in Mathematica: 64 65 (in) t2 = Resultant[a*(1 - t)^2 + 2*b*(1 - t)*t + c*t^2 - g'*y - h', 66 d*(1 - t)^2 + 2*e*(1 - t)*t + f*t^2 - y, y] 67 (out) a - h' - 2 a t + 2 b t + a t^2 - 2 b t^2 + c t^2 - 68 g' (d - 2 d t + 2 e t + d t^2 - 2 e t^2 + f t^2) 69 (in) Solve[t2 == 0, t] 70 (out) { 71 {t -> (2 a - 2 b - 2 d g' + 2 e g' - 72 Sqrt[(-2 a + 2 b + 2 d g' - 2 e g')^2 - 73 4 (a - 2 b + c - d g' + 2 e g' - f g') (a - d g' - h')]) / 74 (2 (a - 2 b + c - d g' + 2 e g' - f g')) 75 }, 76 {t -> (2 a - 2 b - 2 d g' + 2 e g' + 77 Sqrt[(-2 a + 2 b + 2 d g' - 2 e g')^2 - 78 4 (a - 2 b + c - d g' + 2 e g' - f g') (a - d g' - h')])/ 79 (2 (a - 2 b + c - d g' + 2 e g' - f g')) 80 } 81 } 82 83 Thus, if the slope of the line tends towards vertical, we use: 84 A = ( (a - 2*b + c) - g'*(d - 2*e + f) ) 85 B = 2*(-(a - b ) + g'*(d - e ) ) 86 C = ( (a ) - g'*(d ) - h' ) 87 */ 88 89 class LineQuadraticIntersections { 90 public: 91 enum PinTPoint { 92 kPointUninitialized, 93 kPointInitialized 94 }; 95 96 LineQuadraticIntersections(const SkDQuad& q, const SkDLine& l, SkIntersections* i) 97 : fQuad(q) 98 , fLine(l) 99 , fIntersections(i) 100 , fAllowNear(true) { 101 i->setMax(3); // allow short partial coincidence plus discrete intersection 102 } 103 104 void allowNear(bool allow) { 105 fAllowNear = allow; 106 } 107 108 int intersectRay(double roots[2]) { 109 /* 110 solve by rotating line+quad so line is horizontal, then finding the roots 111 set up matrix to rotate quad to x-axis 112 |cos(a) -sin(a)| 113 |sin(a) cos(a)| 114 note that cos(a) = A(djacent) / Hypoteneuse 115 sin(a) = O(pposite) / Hypoteneuse 116 since we are computing Ts, we can ignore hypoteneuse, the scale factor: 117 | A -O | 118 | O A | 119 A = line[1].fX - line[0].fX (adjacent side of the right triangle) 120 O = line[1].fY - line[0].fY (opposite side of the right triangle) 121 for each of the three points (e.g. n = 0 to 2) 122 quad[n].fY' = (quad[n].fY - line[0].fY) * A - (quad[n].fX - line[0].fX) * O 123 */ 124 double adj = fLine[1].fX - fLine[0].fX; 125 double opp = fLine[1].fY - fLine[0].fY; 126 double r[3]; 127 for (int n = 0; n < 3; ++n) { 128 r[n] = (fQuad[n].fY - fLine[0].fY) * adj - (fQuad[n].fX - fLine[0].fX) * opp; 129 } 130 double A = r[2]; 131 double B = r[1]; 132 double C = r[0]; 133 A += C - 2 * B; // A = a - 2*b + c 134 B -= C; // B = -(b - c) 135 return SkDQuad::RootsValidT(A, 2 * B, C, roots); 136 } 137 138 int intersect() { 139 addExactEndPoints(); 140 if (fAllowNear) { 141 addNearEndPoints(); 142 } 143 if (fIntersections->used() == 2) { 144 // FIXME : need sharable code that turns spans into coincident if middle point is on 145 } else { 146 double rootVals[2]; 147 int roots = intersectRay(rootVals); 148 for (int index = 0; index < roots; ++index) { 149 double quadT = rootVals[index]; 150 double lineT = findLineT(quadT); 151 SkDPoint pt; 152 if (pinTs(&quadT, &lineT, &pt, kPointUninitialized)) { 153 fIntersections->insert(quadT, lineT, pt); 154 } 155 } 156 } 157 return fIntersections->used(); 158 } 159 160 int horizontalIntersect(double axisIntercept, double roots[2]) { 161 double D = fQuad[2].fY; // f 162 double E = fQuad[1].fY; // e 163 double F = fQuad[0].fY; // d 164 D += F - 2 * E; // D = d - 2*e + f 165 E -= F; // E = -(d - e) 166 F -= axisIntercept; 167 return SkDQuad::RootsValidT(D, 2 * E, F, roots); 168 } 169 170 int horizontalIntersect(double axisIntercept, double left, double right, bool flipped) { 171 addExactHorizontalEndPoints(left, right, axisIntercept); 172 if (fAllowNear) { 173 addNearHorizontalEndPoints(left, right, axisIntercept); 174 } 175 double rootVals[2]; 176 int roots = horizontalIntersect(axisIntercept, rootVals); 177 for (int index = 0; index < roots; ++index) { 178 double quadT = rootVals[index]; 179 SkDPoint pt = fQuad.ptAtT(quadT); 180 double lineT = (pt.fX - left) / (right - left); 181 if (pinTs(&quadT, &lineT, &pt, kPointInitialized)) { 182 fIntersections->insert(quadT, lineT, pt); 183 } 184 } 185 if (flipped) { 186 fIntersections->flip(); 187 } 188 return fIntersections->used(); 189 } 190 191 int verticalIntersect(double axisIntercept, double roots[2]) { 192 double D = fQuad[2].fX; // f 193 double E = fQuad[1].fX; // e 194 double F = fQuad[0].fX; // d 195 D += F - 2 * E; // D = d - 2*e + f 196 E -= F; // E = -(d - e) 197 F -= axisIntercept; 198 return SkDQuad::RootsValidT(D, 2 * E, F, roots); 199 } 200 201 int verticalIntersect(double axisIntercept, double top, double bottom, bool flipped) { 202 addExactVerticalEndPoints(top, bottom, axisIntercept); 203 if (fAllowNear) { 204 addNearVerticalEndPoints(top, bottom, axisIntercept); 205 } 206 double rootVals[2]; 207 int roots = verticalIntersect(axisIntercept, rootVals); 208 for (int index = 0; index < roots; ++index) { 209 double quadT = rootVals[index]; 210 SkDPoint pt = fQuad.ptAtT(quadT); 211 double lineT = (pt.fY - top) / (bottom - top); 212 if (pinTs(&quadT, &lineT, &pt, kPointInitialized)) { 213 fIntersections->insert(quadT, lineT, pt); 214 } 215 } 216 if (flipped) { 217 fIntersections->flip(); 218 } 219 return fIntersections->used(); 220 } 221 222 protected: 223 // add endpoints first to get zero and one t values exactly 224 void addExactEndPoints() { 225 for (int qIndex = 0; qIndex < 3; qIndex += 2) { 226 double lineT = fLine.exactPoint(fQuad[qIndex]); 227 if (lineT < 0) { 228 continue; 229 } 230 double quadT = (double) (qIndex >> 1); 231 fIntersections->insert(quadT, lineT, fQuad[qIndex]); 232 } 233 } 234 235 void addNearEndPoints() { 236 for (int qIndex = 0; qIndex < 3; qIndex += 2) { 237 double quadT = (double) (qIndex >> 1); 238 if (fIntersections->hasT(quadT)) { 239 continue; 240 } 241 double lineT = fLine.nearPoint(fQuad[qIndex], NULL); 242 if (lineT < 0) { 243 continue; 244 } 245 fIntersections->insert(quadT, lineT, fQuad[qIndex]); 246 } 247 // FIXME: see if line end is nearly on quad 248 } 249 250 void addExactHorizontalEndPoints(double left, double right, double y) { 251 for (int qIndex = 0; qIndex < 3; qIndex += 2) { 252 double lineT = SkDLine::ExactPointH(fQuad[qIndex], left, right, y); 253 if (lineT < 0) { 254 continue; 255 } 256 double quadT = (double) (qIndex >> 1); 257 fIntersections->insert(quadT, lineT, fQuad[qIndex]); 258 } 259 } 260 261 void addNearHorizontalEndPoints(double left, double right, double y) { 262 for (int qIndex = 0; qIndex < 3; qIndex += 2) { 263 double quadT = (double) (qIndex >> 1); 264 if (fIntersections->hasT(quadT)) { 265 continue; 266 } 267 double lineT = SkDLine::NearPointH(fQuad[qIndex], left, right, y); 268 if (lineT < 0) { 269 continue; 270 } 271 fIntersections->insert(quadT, lineT, fQuad[qIndex]); 272 } 273 // FIXME: see if line end is nearly on quad 274 } 275 276 void addExactVerticalEndPoints(double top, double bottom, double x) { 277 for (int qIndex = 0; qIndex < 3; qIndex += 2) { 278 double lineT = SkDLine::ExactPointV(fQuad[qIndex], top, bottom, x); 279 if (lineT < 0) { 280 continue; 281 } 282 double quadT = (double) (qIndex >> 1); 283 fIntersections->insert(quadT, lineT, fQuad[qIndex]); 284 } 285 } 286 287 void addNearVerticalEndPoints(double top, double bottom, double x) { 288 for (int qIndex = 0; qIndex < 3; qIndex += 2) { 289 double quadT = (double) (qIndex >> 1); 290 if (fIntersections->hasT(quadT)) { 291 continue; 292 } 293 double lineT = SkDLine::NearPointV(fQuad[qIndex], top, bottom, x); 294 if (lineT < 0) { 295 continue; 296 } 297 fIntersections->insert(quadT, lineT, fQuad[qIndex]); 298 } 299 // FIXME: see if line end is nearly on quad 300 } 301 302 double findLineT(double t) { 303 SkDPoint xy = fQuad.ptAtT(t); 304 double dx = fLine[1].fX - fLine[0].fX; 305 double dy = fLine[1].fY - fLine[0].fY; 306 if (fabs(dx) > fabs(dy)) { 307 return (xy.fX - fLine[0].fX) / dx; 308 } 309 return (xy.fY - fLine[0].fY) / dy; 310 } 311 312 bool pinTs(double* quadT, double* lineT, SkDPoint* pt, PinTPoint ptSet) { 313 if (!approximately_one_or_less_double(*lineT)) { 314 return false; 315 } 316 if (!approximately_zero_or_more_double(*lineT)) { 317 return false; 318 } 319 double qT = *quadT = SkPinT(*quadT); 320 double lT = *lineT = SkPinT(*lineT); 321 if (lT == 0 || lT == 1 || (ptSet == kPointUninitialized && qT != 0 && qT != 1)) { 322 *pt = fLine.ptAtT(lT); 323 } else if (ptSet == kPointUninitialized) { 324 *pt = fQuad.ptAtT(qT); 325 } 326 SkPoint gridPt = pt->asSkPoint(); 327 if (SkDPoint::ApproximatelyEqual(gridPt, fLine[0].asSkPoint())) { 328 *pt = fLine[0]; 329 *lineT = 0; 330 } else if (SkDPoint::ApproximatelyEqual(gridPt, fLine[1].asSkPoint())) { 331 *pt = fLine[1]; 332 *lineT = 1; 333 } 334 if (fIntersections->used() > 0 && approximately_equal((*fIntersections)[1][0], *lineT)) { 335 return false; 336 } 337 if (gridPt == fQuad[0].asSkPoint()) { 338 *pt = fQuad[0]; 339 *quadT = 0; 340 } else if (gridPt == fQuad[2].asSkPoint()) { 341 *pt = fQuad[2]; 342 *quadT = 1; 343 } 344 return true; 345 } 346 347 private: 348 const SkDQuad& fQuad; 349 const SkDLine& fLine; 350 SkIntersections* fIntersections; 351 bool fAllowNear; 352 }; 353 354 int SkIntersections::horizontal(const SkDQuad& quad, double left, double right, double y, 355 bool flipped) { 356 SkDLine line = {{{ left, y }, { right, y }}}; 357 LineQuadraticIntersections q(quad, line, this); 358 return q.horizontalIntersect(y, left, right, flipped); 359 } 360 361 int SkIntersections::vertical(const SkDQuad& quad, double top, double bottom, double x, 362 bool flipped) { 363 SkDLine line = {{{ x, top }, { x, bottom }}}; 364 LineQuadraticIntersections q(quad, line, this); 365 return q.verticalIntersect(x, top, bottom, flipped); 366 } 367 368 int SkIntersections::intersect(const SkDQuad& quad, const SkDLine& line) { 369 LineQuadraticIntersections q(quad, line, this); 370 q.allowNear(fAllowNear); 371 return q.intersect(); 372 } 373 374 int SkIntersections::intersectRay(const SkDQuad& quad, const SkDLine& line) { 375 LineQuadraticIntersections q(quad, line, this); 376 fUsed = q.intersectRay(fT[0]); 377 for (int index = 0; index < fUsed; ++index) { 378 fPt[index] = quad.ptAtT(fT[0][index]); 379 } 380 return fUsed; 381 } 382
