hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 1 | /* |
| 2 | * Copyright 2016 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 | |
| 8 | #include "SkCurveMeasure.h" |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 9 | #include "SkGeometry.h" |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 10 | |
| 11 | // for abs |
| 12 | #include <cmath> |
| 13 | |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 14 | #define UNIMPLEMENTED SkDEBUGF(("%s:%d unimplemented\n", __FILE__, __LINE__)) |
| 15 | |
| 16 | /// Used inside SkCurveMeasure::getTime's Newton's iteration |
| 17 | static inline SkPoint evaluate(const SkPoint pts[4], SkSegType segType, |
| 18 | SkScalar t) { |
| 19 | SkPoint pos; |
| 20 | switch (segType) { |
| 21 | case kQuad_SegType: |
| 22 | pos = SkEvalQuadAt(pts, t); |
| 23 | break; |
| 24 | case kLine_SegType: |
| 25 | pos = SkPoint::Make(SkScalarInterp(pts[0].x(), pts[1].x(), t), |
| 26 | SkScalarInterp(pts[0].y(), pts[1].y(), t)); |
| 27 | break; |
| 28 | case kCubic_SegType: |
| 29 | SkEvalCubicAt(pts, t, &pos, nullptr, nullptr); |
| 30 | break; |
| 31 | case kConic_SegType: { |
| 32 | SkConic conic(pts, pts[3].x()); |
| 33 | conic.evalAt(t, &pos); |
| 34 | } |
| 35 | break; |
| 36 | default: |
| 37 | UNIMPLEMENTED; |
| 38 | } |
| 39 | |
| 40 | return pos; |
| 41 | } |
| 42 | |
| 43 | /// Used inside SkCurveMeasure::getTime's Newton's iteration |
| 44 | static inline SkVector evaluateDerivative(const SkPoint pts[4], |
| 45 | SkSegType segType, SkScalar t) { |
| 46 | SkVector tan; |
| 47 | switch (segType) { |
| 48 | case kQuad_SegType: |
| 49 | tan = SkEvalQuadTangentAt(pts, t); |
| 50 | break; |
| 51 | case kLine_SegType: |
| 52 | tan = pts[1] - pts[0]; |
| 53 | break; |
| 54 | case kCubic_SegType: |
| 55 | SkEvalCubicAt(pts, t, nullptr, &tan, nullptr); |
| 56 | break; |
| 57 | case kConic_SegType: { |
| 58 | SkConic conic(pts, pts[3].x()); |
| 59 | conic.evalAt(t, nullptr, &tan); |
| 60 | } |
| 61 | break; |
| 62 | default: |
| 63 | UNIMPLEMENTED; |
| 64 | } |
| 65 | |
| 66 | return tan; |
| 67 | } |
| 68 | /// Used in ArcLengthIntegrator::computeLength |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 69 | static inline Sk8f evaluateDerivativeLength(const Sk8f& ts, |
Mike Klein | 1e76464 | 2016-10-14 17:09:03 -0400 | [diff] [blame] | 70 | const float (&xCoeff)[3][8], |
| 71 | const float (&yCoeff)[3][8], |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 72 | const SkSegType segType) { |
| 73 | Sk8f x; |
| 74 | Sk8f y; |
Mike Klein | 1e76464 | 2016-10-14 17:09:03 -0400 | [diff] [blame] | 75 | |
| 76 | Sk8f x0 = Sk8f::Load(&xCoeff[0]), |
| 77 | x1 = Sk8f::Load(&xCoeff[1]), |
| 78 | x2 = Sk8f::Load(&xCoeff[2]); |
| 79 | |
| 80 | Sk8f y0 = Sk8f::Load(&yCoeff[0]), |
| 81 | y1 = Sk8f::Load(&yCoeff[1]), |
| 82 | y2 = Sk8f::Load(&yCoeff[2]); |
| 83 | |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 84 | switch (segType) { |
| 85 | case kQuad_SegType: |
Mike Klein | 1e76464 | 2016-10-14 17:09:03 -0400 | [diff] [blame] | 86 | x = x0*ts + x1; |
| 87 | y = y0*ts + y1; |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 88 | break; |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 89 | case kCubic_SegType: |
Mike Klein | 1e76464 | 2016-10-14 17:09:03 -0400 | [diff] [blame] | 90 | x = (x0*ts + x1)*ts + x2; |
| 91 | y = (y0*ts + y1)*ts + y2; |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 92 | break; |
| 93 | case kConic_SegType: |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 94 | UNIMPLEMENTED; |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 95 | break; |
| 96 | default: |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 97 | UNIMPLEMENTED; |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 98 | } |
| 99 | |
| 100 | x = x * x; |
| 101 | y = y * y; |
| 102 | |
| 103 | return (x + y).sqrt(); |
| 104 | } |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 105 | |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 106 | ArcLengthIntegrator::ArcLengthIntegrator(const SkPoint* pts, SkSegType segType) |
| 107 | : fSegType(segType) { |
| 108 | switch (fSegType) { |
| 109 | case kQuad_SegType: { |
| 110 | float Ax = pts[0].x(); |
| 111 | float Bx = pts[1].x(); |
| 112 | float Cx = pts[2].x(); |
| 113 | float Ay = pts[0].y(); |
| 114 | float By = pts[1].y(); |
| 115 | float Cy = pts[2].y(); |
| 116 | |
| 117 | // precompute coefficients for derivative |
Mike Klein | 1e76464 | 2016-10-14 17:09:03 -0400 | [diff] [blame] | 118 | Sk8f(2*(Ax - 2*Bx + Cx)).store(&xCoeff[0]); |
| 119 | Sk8f(2*(Bx - Ax)).store(&xCoeff[1]); |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 120 | |
Mike Klein | 1e76464 | 2016-10-14 17:09:03 -0400 | [diff] [blame] | 121 | Sk8f(2*(Ay - 2*By + Cy)).store(&yCoeff[0]); |
| 122 | Sk8f(2*(By - Ay)).store(&yCoeff[1]); |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 123 | } |
| 124 | break; |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 125 | case kCubic_SegType: |
| 126 | { |
| 127 | float Ax = pts[0].x(); |
| 128 | float Bx = pts[1].x(); |
| 129 | float Cx = pts[2].x(); |
| 130 | float Dx = pts[3].x(); |
| 131 | float Ay = pts[0].y(); |
| 132 | float By = pts[1].y(); |
| 133 | float Cy = pts[2].y(); |
| 134 | float Dy = pts[3].y(); |
| 135 | |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 136 | // precompute coefficients for derivative |
Mike Klein | 1e76464 | 2016-10-14 17:09:03 -0400 | [diff] [blame] | 137 | Sk8f(3*(-Ax + 3*(Bx - Cx) + Dx)).store(&xCoeff[0]); |
| 138 | Sk8f(6*(Ax - 2*Bx + Cx)).store(&xCoeff[1]); |
| 139 | Sk8f(3*(-Ax + Bx)).store(&xCoeff[2]); |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 140 | |
Mike Klein | 1e76464 | 2016-10-14 17:09:03 -0400 | [diff] [blame] | 141 | Sk8f(3*(-Ay + 3*(By - Cy) + Dy)).store(&yCoeff[0]); |
| 142 | Sk8f(6*(Ay - 2*By + Cy)).store(&yCoeff[1]); |
| 143 | Sk8f(3*(-Ay + By)).store(&yCoeff[2]); |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 144 | } |
| 145 | break; |
| 146 | case kConic_SegType: |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 147 | UNIMPLEMENTED; |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 148 | break; |
| 149 | default: |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 150 | UNIMPLEMENTED; |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 151 | } |
| 152 | } |
| 153 | |
| 154 | // We use Gaussian quadrature |
| 155 | // (https://en.wikipedia.org/wiki/Gaussian_quadrature) |
| 156 | // to approximate the arc length integral here, because it is amenable to SIMD. |
| 157 | SkScalar ArcLengthIntegrator::computeLength(SkScalar t) { |
| 158 | SkScalar length = 0.0f; |
| 159 | |
| 160 | Sk8f lengths = evaluateDerivativeLength(absc*t, xCoeff, yCoeff, fSegType); |
| 161 | lengths = weights*lengths; |
| 162 | // is it faster or more accurate to sum and then multiply or vice versa? |
| 163 | lengths = lengths*(t*0.5f); |
| 164 | |
| 165 | // Why does SkNx index with ints? does negative index mean something? |
| 166 | for (int i = 0; i < 8; i++) { |
| 167 | length += lengths[i]; |
| 168 | } |
| 169 | return length; |
| 170 | } |
| 171 | |
| 172 | SkCurveMeasure::SkCurveMeasure(const SkPoint* pts, SkSegType segType) |
| 173 | : fSegType(segType) { |
| 174 | switch (fSegType) { |
| 175 | case SkSegType::kQuad_SegType: |
| 176 | for (size_t i = 0; i < 3; i++) { |
| 177 | fPts[i] = pts[i]; |
| 178 | } |
| 179 | break; |
| 180 | case SkSegType::kLine_SegType: |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 181 | fPts[0] = pts[0]; |
| 182 | fPts[1] = pts[1]; |
| 183 | fLength = (fPts[1] - fPts[0]).length(); |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 184 | break; |
| 185 | case SkSegType::kCubic_SegType: |
| 186 | for (size_t i = 0; i < 4; i++) { |
| 187 | fPts[i] = pts[i]; |
| 188 | } |
| 189 | break; |
| 190 | case SkSegType::kConic_SegType: |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 191 | for (size_t i = 0; i < 4; i++) { |
| 192 | fPts[i] = pts[i]; |
| 193 | } |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 194 | break; |
| 195 | default: |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 196 | UNIMPLEMENTED; |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 197 | break; |
| 198 | } |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 199 | if (kLine_SegType != segType) { |
| 200 | fIntegrator = ArcLengthIntegrator(fPts, fSegType); |
| 201 | } |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 202 | } |
| 203 | |
| 204 | SkScalar SkCurveMeasure::getLength() { |
| 205 | if (-1.0f == fLength) { |
| 206 | fLength = fIntegrator.computeLength(1.0f); |
| 207 | } |
| 208 | return fLength; |
| 209 | } |
| 210 | |
| 211 | // Given an arc length targetLength, we want to determine what t |
| 212 | // gives us the corresponding arc length along the curve. |
| 213 | // We do this by letting the arc length integral := f(t) and |
| 214 | // solving for the root of the equation f(t) - targetLength = 0 |
| 215 | // using Newton's method and lerp-bisection. |
| 216 | // The computationally expensive parts are the integral approximation |
| 217 | // at each step, and computing the derivative of the arc length integral, |
| 218 | // which is equal to the length of the tangent (so we have to do a sqrt). |
| 219 | |
| 220 | SkScalar SkCurveMeasure::getTime(SkScalar targetLength) { |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 221 | if (targetLength <= 0.0f) { |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 222 | return 0.0f; |
| 223 | } |
| 224 | |
| 225 | SkScalar currentLength = getLength(); |
| 226 | |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 227 | if (targetLength > currentLength || (SkScalarNearlyEqual(targetLength, currentLength))) { |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 228 | return 1.0f; |
| 229 | } |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 230 | if (kLine_SegType == fSegType) { |
| 231 | return targetLength / currentLength; |
| 232 | } |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 233 | |
| 234 | // initial estimate of t is percentage of total length |
| 235 | SkScalar currentT = targetLength / currentLength; |
| 236 | SkScalar prevT = -1.0f; |
| 237 | SkScalar newT; |
| 238 | |
| 239 | SkScalar minT = 0.0f; |
| 240 | SkScalar maxT = 1.0f; |
| 241 | |
| 242 | int iterations = 0; |
| 243 | while (iterations < kNewtonIters + kBisectIters) { |
| 244 | currentLength = fIntegrator.computeLength(currentT); |
| 245 | SkScalar lengthDiff = currentLength - targetLength; |
| 246 | |
| 247 | // Update root bounds. |
| 248 | // If lengthDiff is positive, we have overshot the target, so |
| 249 | // we know the current t is an upper bound, and similarly |
| 250 | // for the lower bound. |
| 251 | if (lengthDiff > 0.0f) { |
| 252 | if (currentT < maxT) { |
| 253 | maxT = currentT; |
| 254 | } |
| 255 | } else { |
| 256 | if (currentT > minT) { |
| 257 | minT = currentT; |
| 258 | } |
| 259 | } |
| 260 | |
| 261 | // We have a tolerance on both the absolute value of the difference and |
| 262 | // on the t value |
| 263 | // because we may not have enough precision in the t to get close enough |
| 264 | // in the length. |
| 265 | if ((std::abs(lengthDiff) < kTolerance) || |
| 266 | (std::abs(prevT - currentT) < kTolerance)) { |
| 267 | break; |
| 268 | } |
| 269 | |
| 270 | prevT = currentT; |
| 271 | if (iterations < kNewtonIters) { |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 272 | // This is just newton's formula. |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 273 | SkScalar dt = evaluateDerivative(fPts, fSegType, currentT).length(); |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 274 | newT = currentT - (lengthDiff / dt); |
| 275 | |
| 276 | // If newT is out of bounds, bisect inside newton. |
| 277 | if ((newT < 0.0f) || (newT > 1.0f)) { |
| 278 | newT = (minT + maxT) * 0.5f; |
| 279 | } |
| 280 | } else if (iterations < kNewtonIters + kBisectIters) { |
| 281 | if (lengthDiff > 0.0f) { |
| 282 | maxT = currentT; |
| 283 | } else { |
| 284 | minT = currentT; |
| 285 | } |
| 286 | // TODO(hstern) do a lerp here instead of a bisection |
| 287 | newT = (minT + maxT) * 0.5f; |
| 288 | } else { |
| 289 | SkDEBUGF(("%.7f %.7f didn't get close enough after bisection.\n", |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 290 | currentT, currentLength)); |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 291 | break; |
| 292 | } |
| 293 | currentT = newT; |
| 294 | |
| 295 | SkASSERT(minT <= maxT); |
| 296 | |
| 297 | iterations++; |
| 298 | } |
| 299 | |
| 300 | // debug. is there an SKDEBUG or something for ifdefs? |
| 301 | fIters = iterations; |
| 302 | |
| 303 | return currentT; |
| 304 | } |
| 305 | |
hstern | 80ac591 | 2016-08-10 07:45:31 -0700 | [diff] [blame] | 306 | void SkCurveMeasure::getPosTanTime(SkScalar targetLength, SkPoint* pos, |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 307 | SkVector* tan, SkScalar* time) { |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 308 | SkScalar t = getTime(targetLength); |
| 309 | |
hstern | 80ac591 | 2016-08-10 07:45:31 -0700 | [diff] [blame] | 310 | if (time) { |
| 311 | *time = t; |
| 312 | } |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 313 | if (pos) { |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 314 | *pos = evaluate(fPts, fSegType, t); |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 315 | } |
| 316 | if (tan) { |
hstern | 5a4b18c | 2016-08-10 16:31:10 -0700 | [diff] [blame] | 317 | *tan = evaluateDerivative(fPts, fSegType, t); |
hstern | 0446a3c | 2016-08-08 12:28:13 -0700 | [diff] [blame] | 318 | } |
| 319 | } |