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 | */ |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 7 | #include "SkIntersections.h" |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 8 | #include "SkLineParameters.h" |
| 9 | #include "SkPathOpsCubic.h" |
| 10 | #include "SkPathOpsQuad.h" |
| 11 | #include "SkPathOpsTriangle.h" |
| 12 | |
| 13 | // from http://blog.gludion.com/2009/08/distance-to-quadratic-bezier-curve.html |
| 14 | // (currently only used by testing) |
| 15 | double SkDQuad::nearestT(const SkDPoint& pt) const { |
| 16 | SkDVector pos = fPts[0] - pt; |
| 17 | // search points P of bezier curve with PM.(dP / dt) = 0 |
| 18 | // a calculus leads to a 3d degree equation : |
| 19 | SkDVector A = fPts[1] - fPts[0]; |
| 20 | SkDVector B = fPts[2] - fPts[1]; |
| 21 | B -= A; |
| 22 | double a = B.dot(B); |
| 23 | double b = 3 * A.dot(B); |
| 24 | double c = 2 * A.dot(A) + pos.dot(B); |
| 25 | double d = pos.dot(A); |
| 26 | double ts[3]; |
| 27 | int roots = SkDCubic::RootsValidT(a, b, c, d, ts); |
| 28 | double d0 = pt.distanceSquared(fPts[0]); |
| 29 | double d2 = pt.distanceSquared(fPts[2]); |
caryclark@google.com | 3b97af5 | 2013-04-23 11:56:44 +0000 | [diff] [blame] | 30 | double distMin = SkTMin(d0, d2); |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 31 | int bestIndex = -1; |
| 32 | for (int index = 0; index < roots; ++index) { |
caryclark@google.com | 4fdbb22 | 2013-07-23 15:27:41 +0000 | [diff] [blame] | 33 | SkDPoint onQuad = ptAtT(ts[index]); |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 34 | double dist = pt.distanceSquared(onQuad); |
| 35 | if (distMin > dist) { |
| 36 | distMin = dist; |
| 37 | bestIndex = index; |
| 38 | } |
| 39 | } |
| 40 | if (bestIndex >= 0) { |
| 41 | return ts[bestIndex]; |
| 42 | } |
| 43 | return d0 < d2 ? 0 : 1; |
| 44 | } |
| 45 | |
| 46 | bool SkDQuad::pointInHull(const SkDPoint& pt) const { |
| 47 | return ((const SkDTriangle&) fPts).contains(pt); |
| 48 | } |
| 49 | |
| 50 | SkDPoint SkDQuad::top(double startT, double endT) const { |
| 51 | SkDQuad sub = subDivide(startT, endT); |
| 52 | SkDPoint topPt = sub[0]; |
| 53 | if (topPt.fY > sub[2].fY || (topPt.fY == sub[2].fY && topPt.fX > sub[2].fX)) { |
| 54 | topPt = sub[2]; |
| 55 | } |
| 56 | if (!between(sub[0].fY, sub[1].fY, sub[2].fY)) { |
| 57 | double extremeT; |
| 58 | if (FindExtrema(sub[0].fY, sub[1].fY, sub[2].fY, &extremeT)) { |
| 59 | extremeT = startT + (endT - startT) * extremeT; |
caryclark@google.com | 4fdbb22 | 2013-07-23 15:27:41 +0000 | [diff] [blame] | 60 | SkDPoint test = ptAtT(extremeT); |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 61 | if (topPt.fY > test.fY || (topPt.fY == test.fY && topPt.fX > test.fX)) { |
| 62 | topPt = test; |
| 63 | } |
| 64 | } |
| 65 | } |
| 66 | return topPt; |
| 67 | } |
| 68 | |
| 69 | int SkDQuad::AddValidTs(double s[], int realRoots, double* t) { |
| 70 | int foundRoots = 0; |
| 71 | for (int index = 0; index < realRoots; ++index) { |
| 72 | double tValue = s[index]; |
| 73 | if (approximately_zero_or_more(tValue) && approximately_one_or_less(tValue)) { |
| 74 | if (approximately_less_than_zero(tValue)) { |
| 75 | tValue = 0; |
| 76 | } else if (approximately_greater_than_one(tValue)) { |
| 77 | tValue = 1; |
| 78 | } |
| 79 | for (int idx2 = 0; idx2 < foundRoots; ++idx2) { |
| 80 | if (approximately_equal(t[idx2], tValue)) { |
| 81 | goto nextRoot; |
| 82 | } |
| 83 | } |
| 84 | t[foundRoots++] = tValue; |
| 85 | } |
| 86 | nextRoot: |
| 87 | {} |
| 88 | } |
| 89 | return foundRoots; |
| 90 | } |
| 91 | |
| 92 | // note: caller expects multiple results to be sorted smaller first |
| 93 | // note: http://en.wikipedia.org/wiki/Loss_of_significance has an interesting |
| 94 | // analysis of the quadratic equation, suggesting why the following looks at |
| 95 | // the sign of B -- and further suggesting that the greatest loss of precision |
| 96 | // is in b squared less two a c |
| 97 | int SkDQuad::RootsValidT(double A, double B, double C, double t[2]) { |
| 98 | double s[2]; |
| 99 | int realRoots = RootsReal(A, B, C, s); |
| 100 | int foundRoots = AddValidTs(s, realRoots, t); |
| 101 | return foundRoots; |
| 102 | } |
| 103 | |
| 104 | /* |
| 105 | Numeric Solutions (5.6) suggests to solve the quadratic by computing |
| 106 | Q = -1/2(B + sgn(B)Sqrt(B^2 - 4 A C)) |
| 107 | and using the roots |
| 108 | t1 = Q / A |
| 109 | t2 = C / Q |
| 110 | */ |
| 111 | // this does not discard real roots <= 0 or >= 1 |
| 112 | int SkDQuad::RootsReal(const double A, const double B, const double C, double s[2]) { |
| 113 | const double p = B / (2 * A); |
| 114 | const double q = C / A; |
| 115 | if (approximately_zero(A) && (approximately_zero_inverse(p) || approximately_zero_inverse(q))) { |
| 116 | if (approximately_zero(B)) { |
| 117 | s[0] = 0; |
| 118 | return C == 0; |
| 119 | } |
| 120 | s[0] = -C / B; |
| 121 | return 1; |
| 122 | } |
| 123 | /* normal form: x^2 + px + q = 0 */ |
| 124 | const double p2 = p * p; |
caryclark@google.com | 7eaa53d | 2013-10-02 14:49:34 +0000 | [diff] [blame] | 125 | if (!AlmostDequalUlps(p2, q) && p2 < q) { |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 126 | return 0; |
| 127 | } |
| 128 | double sqrt_D = 0; |
| 129 | if (p2 > q) { |
| 130 | sqrt_D = sqrt(p2 - q); |
| 131 | } |
| 132 | s[0] = sqrt_D - p; |
| 133 | s[1] = -sqrt_D - p; |
caryclark@google.com | 7eaa53d | 2013-10-02 14:49:34 +0000 | [diff] [blame] | 134 | return 1 + !AlmostDequalUlps(s[0], s[1]); |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 135 | } |
| 136 | |
| 137 | bool SkDQuad::isLinear(int startIndex, int endIndex) const { |
| 138 | SkLineParameters lineParameters; |
| 139 | lineParameters.quadEndPoints(*this, startIndex, endIndex); |
| 140 | // FIXME: maybe it's possible to avoid this and compare non-normalized |
| 141 | lineParameters.normalize(); |
| 142 | double distance = lineParameters.controlPtDistance(*this); |
| 143 | return approximately_zero(distance); |
| 144 | } |
| 145 | |
| 146 | SkDCubic SkDQuad::toCubic() const { |
| 147 | SkDCubic cubic; |
| 148 | cubic[0] = fPts[0]; |
| 149 | cubic[2] = fPts[1]; |
| 150 | cubic[3] = fPts[2]; |
| 151 | cubic[1].fX = (cubic[0].fX + cubic[2].fX * 2) / 3; |
| 152 | cubic[1].fY = (cubic[0].fY + cubic[2].fY * 2) / 3; |
| 153 | cubic[2].fX = (cubic[3].fX + cubic[2].fX * 2) / 3; |
| 154 | cubic[2].fY = (cubic[3].fY + cubic[2].fY * 2) / 3; |
| 155 | return cubic; |
| 156 | } |
| 157 | |
| 158 | SkDVector SkDQuad::dxdyAtT(double t) const { |
| 159 | double a = t - 1; |
| 160 | double b = 1 - 2 * t; |
| 161 | double c = t; |
| 162 | SkDVector result = { a * fPts[0].fX + b * fPts[1].fX + c * fPts[2].fX, |
| 163 | a * fPts[0].fY + b * fPts[1].fY + c * fPts[2].fY }; |
| 164 | return result; |
| 165 | } |
| 166 | |
caryclark@google.com | fa2aeee | 2013-07-15 13:29:13 +0000 | [diff] [blame] | 167 | // OPTIMIZE: assert if caller passes in t == 0 / t == 1 ? |
caryclark@google.com | 4fdbb22 | 2013-07-23 15:27:41 +0000 | [diff] [blame] | 168 | SkDPoint SkDQuad::ptAtT(double t) const { |
| 169 | if (0 == t) { |
| 170 | return fPts[0]; |
| 171 | } |
| 172 | if (1 == t) { |
| 173 | return fPts[2]; |
| 174 | } |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 175 | double one_t = 1 - t; |
| 176 | double a = one_t * one_t; |
| 177 | double b = 2 * one_t * t; |
| 178 | double c = t * t; |
| 179 | SkDPoint result = { a * fPts[0].fX + b * fPts[1].fX + c * fPts[2].fX, |
| 180 | a * fPts[0].fY + b * fPts[1].fY + c * fPts[2].fY }; |
| 181 | return result; |
| 182 | } |
| 183 | |
| 184 | /* |
| 185 | Given a quadratic q, t1, and t2, find a small quadratic segment. |
| 186 | |
| 187 | The new quadratic is defined by A, B, and C, where |
| 188 | A = c[0]*(1 - t1)*(1 - t1) + 2*c[1]*t1*(1 - t1) + c[2]*t1*t1 |
| 189 | C = c[3]*(1 - t1)*(1 - t1) + 2*c[2]*t1*(1 - t1) + c[1]*t1*t1 |
| 190 | |
| 191 | To find B, compute the point halfway between t1 and t2: |
| 192 | |
| 193 | q(at (t1 + t2)/2) == D |
| 194 | |
| 195 | Next, compute where D must be if we know the value of B: |
| 196 | |
| 197 | _12 = A/2 + B/2 |
| 198 | 12_ = B/2 + C/2 |
| 199 | 123 = A/4 + B/2 + C/4 |
| 200 | = D |
| 201 | |
| 202 | Group the known values on one side: |
| 203 | |
| 204 | B = D*2 - A/2 - C/2 |
| 205 | */ |
| 206 | |
| 207 | static double interp_quad_coords(const double* src, double t) { |
| 208 | double ab = SkDInterp(src[0], src[2], t); |
| 209 | double bc = SkDInterp(src[2], src[4], t); |
| 210 | double abc = SkDInterp(ab, bc, t); |
| 211 | return abc; |
| 212 | } |
| 213 | |
| 214 | bool SkDQuad::monotonicInY() const { |
| 215 | return between(fPts[0].fY, fPts[1].fY, fPts[2].fY); |
| 216 | } |
| 217 | |
| 218 | SkDQuad SkDQuad::subDivide(double t1, double t2) const { |
| 219 | SkDQuad dst; |
| 220 | double ax = dst[0].fX = interp_quad_coords(&fPts[0].fX, t1); |
| 221 | double ay = dst[0].fY = interp_quad_coords(&fPts[0].fY, t1); |
| 222 | double dx = interp_quad_coords(&fPts[0].fX, (t1 + t2) / 2); |
| 223 | double dy = interp_quad_coords(&fPts[0].fY, (t1 + t2) / 2); |
| 224 | double cx = dst[2].fX = interp_quad_coords(&fPts[0].fX, t2); |
| 225 | double cy = dst[2].fY = interp_quad_coords(&fPts[0].fY, t2); |
| 226 | /* bx = */ dst[1].fX = 2*dx - (ax + cx)/2; |
| 227 | /* by = */ dst[1].fY = 2*dy - (ay + cy)/2; |
| 228 | return dst; |
| 229 | } |
| 230 | |
skia.committer@gmail.com | 8f6ef40 | 2013-06-05 07:01:06 +0000 | [diff] [blame] | 231 | void SkDQuad::align(int endIndex, SkDPoint* dstPt) const { |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 232 | if (fPts[endIndex].fX == fPts[1].fX) { |
| 233 | dstPt->fX = fPts[endIndex].fX; |
| 234 | } |
| 235 | if (fPts[endIndex].fY == fPts[1].fY) { |
| 236 | dstPt->fY = fPts[endIndex].fY; |
| 237 | } |
| 238 | } |
| 239 | |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 240 | SkDPoint SkDQuad::subDivide(const SkDPoint& a, const SkDPoint& c, double t1, double t2) const { |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 241 | SkASSERT(t1 != t2); |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 242 | SkDPoint b; |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 243 | #if 0 |
| 244 | // this approach assumes that the control point computed directly is accurate enough |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 245 | double dx = interp_quad_coords(&fPts[0].fX, (t1 + t2) / 2); |
| 246 | double dy = interp_quad_coords(&fPts[0].fY, (t1 + t2) / 2); |
| 247 | b.fX = 2 * dx - (a.fX + c.fX) / 2; |
| 248 | b.fY = 2 * dy - (a.fY + c.fY) / 2; |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 249 | #else |
| 250 | SkDQuad sub = subDivide(t1, t2); |
| 251 | SkDLine b0 = {{a, sub[1] + (a - sub[0])}}; |
| 252 | SkDLine b1 = {{c, sub[1] + (c - sub[2])}}; |
| 253 | SkIntersections i; |
| 254 | i.intersectRay(b0, b1); |
commit-bot@chromium.org | 4431e77 | 2014-04-14 17:08:59 +0000 | [diff] [blame] | 255 | if (i.used() == 1 && i[0][0] >= 0 && i[1][0] >= 0) { |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 256 | b = i.pt(0); |
| 257 | } else { |
commit-bot@chromium.org | 4431e77 | 2014-04-14 17:08:59 +0000 | [diff] [blame] | 258 | SkASSERT(i.used() <= 2); |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 259 | b = SkDPoint::Mid(b0[1], b1[1]); |
| 260 | } |
| 261 | #endif |
| 262 | if (t1 == 0 || t2 == 0) { |
| 263 | align(0, &b); |
| 264 | } |
| 265 | if (t1 == 1 || t2 == 1) { |
| 266 | align(2, &b); |
| 267 | } |
commit-bot@chromium.org | 4431e77 | 2014-04-14 17:08:59 +0000 | [diff] [blame] | 268 | if (AlmostBequalUlps(b.fX, a.fX)) { |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 269 | b.fX = a.fX; |
commit-bot@chromium.org | 4431e77 | 2014-04-14 17:08:59 +0000 | [diff] [blame] | 270 | } else if (AlmostBequalUlps(b.fX, c.fX)) { |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 271 | b.fX = c.fX; |
| 272 | } |
commit-bot@chromium.org | 4431e77 | 2014-04-14 17:08:59 +0000 | [diff] [blame] | 273 | if (AlmostBequalUlps(b.fY, a.fY)) { |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 274 | b.fY = a.fY; |
commit-bot@chromium.org | 4431e77 | 2014-04-14 17:08:59 +0000 | [diff] [blame] | 275 | } else if (AlmostBequalUlps(b.fY, c.fY)) { |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 276 | b.fY = c.fY; |
| 277 | } |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 278 | return b; |
| 279 | } |
| 280 | |
| 281 | /* classic one t subdivision */ |
| 282 | static void interp_quad_coords(const double* src, double* dst, double t) { |
| 283 | double ab = SkDInterp(src[0], src[2], t); |
| 284 | double bc = SkDInterp(src[2], src[4], t); |
| 285 | dst[0] = src[0]; |
| 286 | dst[2] = ab; |
| 287 | dst[4] = SkDInterp(ab, bc, t); |
| 288 | dst[6] = bc; |
| 289 | dst[8] = src[4]; |
| 290 | } |
| 291 | |
| 292 | SkDQuadPair SkDQuad::chopAt(double t) const |
| 293 | { |
| 294 | SkDQuadPair dst; |
| 295 | interp_quad_coords(&fPts[0].fX, &dst.pts[0].fX, t); |
| 296 | interp_quad_coords(&fPts[0].fY, &dst.pts[0].fY, t); |
| 297 | return dst; |
| 298 | } |
| 299 | |
| 300 | static int valid_unit_divide(double numer, double denom, double* ratio) |
| 301 | { |
| 302 | if (numer < 0) { |
| 303 | numer = -numer; |
| 304 | denom = -denom; |
| 305 | } |
| 306 | if (denom == 0 || numer == 0 || numer >= denom) { |
| 307 | return 0; |
| 308 | } |
| 309 | double r = numer / denom; |
| 310 | if (r == 0) { // catch underflow if numer <<<< denom |
| 311 | return 0; |
| 312 | } |
| 313 | *ratio = r; |
| 314 | return 1; |
| 315 | } |
| 316 | |
| 317 | /** Quad'(t) = At + B, where |
| 318 | A = 2(a - 2b + c) |
| 319 | B = 2(b - a) |
| 320 | Solve for t, only if it fits between 0 < t < 1 |
| 321 | */ |
| 322 | int SkDQuad::FindExtrema(double a, double b, double c, double tValue[1]) { |
| 323 | /* At + B == 0 |
| 324 | t = -B / A |
| 325 | */ |
| 326 | return valid_unit_divide(a - b, a - b - b + c, tValue); |
| 327 | } |
| 328 | |
| 329 | /* Parameterization form, given A*t*t + 2*B*t*(1-t) + C*(1-t)*(1-t) |
| 330 | * |
| 331 | * a = A - 2*B + C |
| 332 | * b = 2*B - 2*C |
| 333 | * c = C |
| 334 | */ |
| 335 | void SkDQuad::SetABC(const double* quad, double* a, double* b, double* c) { |
| 336 | *a = quad[0]; // a = A |
| 337 | *b = 2 * quad[2]; // b = 2*B |
| 338 | *c = quad[4]; // c = C |
| 339 | *b -= *c; // b = 2*B - C |
| 340 | *a -= *b; // a = A - 2*B + C |
| 341 | *b -= *c; // b = 2*B - 2*C |
| 342 | } |