The Tor Browser: comparison gfx/skia/trunk/src/pathops/SkPathOpsQuad.cpp

--1:000000000000
+:a860967b3780
+/*
+* Copyright 2012 Google Inc.
+*
+* Use of this source code is governed by a BSD-style license that can be
+* found in the LICENSE file.
+*/
+#include "SkIntersections.h"
+#include "SkLineParameters.h"
+#include "SkPathOpsCubic.h"
+#include "SkPathOpsQuad.h"
+#include "SkPathOpsTriangle.h"
+// from http://blog.gludion.com/2009/08/distance-to-quadratic-bezier-curve.html
+// (currently only used by testing)
+double SkDQuad::nearestT(const SkDPoint& pt) const {
+SkDVector pos = fPts[0] - pt;
+// search points P of bezier curve with PM.(dP / dt) = 0
+// a calculus leads to a 3d degree equation :
+SkDVector A = fPts[1] - fPts[0];
+SkDVector B = fPts[2] - fPts[1];
+B -= A;
+double a = B.dot(B);
+double b = 3 * A.dot(B);
+double c = 2 * A.dot(A) + pos.dot(B);
+double d = pos.dot(A);
+double ts[3];
+int roots = SkDCubic::RootsValidT(a, b, c, d, ts);
+double d0 = pt.distanceSquared(fPts[0]);
+double d2 = pt.distanceSquared(fPts[2]);
+double distMin = SkTMin(d0, d2);
+int bestIndex = -1;
+for (int index = 0; index < roots; ++index) {
+SkDPoint onQuad = ptAtT(ts[index]);
+double dist = pt.distanceSquared(onQuad);
+if (distMin > dist) {
+distMin = dist;
+bestIndex = index;
+}
+}
+if (bestIndex >= 0) {
+return ts[bestIndex];
+}
+return d0 < d2 ? 0 : 1;
+}
+bool SkDQuad::pointInHull(const SkDPoint& pt) const {
+return ((const SkDTriangle&) fPts).contains(pt);
+}
+SkDPoint SkDQuad::top(double startT, double endT) const {
+SkDQuad sub = subDivide(startT, endT);
+SkDPoint topPt = sub[0];
+if (topPt.fY > sub[2].fY || (topPt.fY == sub[2].fY && topPt.fX > sub[2].fX)) {
+topPt = sub[2];
+}
+if (!between(sub[0].fY, sub[1].fY, sub[2].fY)) {
+double extremeT;
+if (FindExtrema(sub[0].fY, sub[1].fY, sub[2].fY, &extremeT)) {
+extremeT = startT + (endT - startT) * extremeT;
+SkDPoint test = ptAtT(extremeT);
+if (topPt.fY > test.fY || (topPt.fY == test.fY && topPt.fX > test.fX)) {
+topPt = test;
+}
+}
+}
+return topPt;
+}
+int SkDQuad::AddValidTs(double s[], int realRoots, double* t) {
+int foundRoots = 0;
+for (int index = 0; index < realRoots; ++index) {
+double tValue = s[index];
+if (approximately_zero_or_more(tValue) && approximately_one_or_less(tValue)) {
+if (approximately_less_than_zero(tValue)) {
+tValue = 0;
+} else if (approximately_greater_than_one(tValue)) {
+tValue = 1;
+}
+for (int idx2 = 0; idx2 < foundRoots; ++idx2) {
+if (approximately_equal(t[idx2], tValue)) {
+goto nextRoot;
+}
+}
+t[foundRoots++] = tValue;
+}
+nextRoot:
+{}
+}
+return foundRoots;
+}
+// note: caller expects multiple results to be sorted smaller first
+// note: http://en.wikipedia.org/wiki/Loss_of_significance has an interesting
+//  analysis of the quadratic equation, suggesting why the following looks at
+//  the sign of B -- and further suggesting that the greatest loss of precision
+//  is in b squared less two a c
+int SkDQuad::RootsValidT(double A, double B, double C, double t[2]) {
+double s[2];
+int realRoots = RootsReal(A, B, C, s);
+int foundRoots = AddValidTs(s, realRoots, t);
+return foundRoots;
+}
+/*
+Numeric Solutions (5.6) suggests to solve the quadratic by computing
+Q = -1/2(B + sgn(B)Sqrt(B^2 - 4 A C))
+and using the roots
+t1 = Q / A
+t2 = C / Q
+*/
+// this does not discard real roots <= 0 or >= 1
+int SkDQuad::RootsReal(const double A, const double B, const double C, double s[2]) {
+const double p = B / (2 * A);
+const double q = C / A;
+if (approximately_zero(A) && (approximately_zero_inverse(p) || approximately_zero_inverse(q))) {
+if (approximately_zero(B)) {
+s[0] = 0;
+return C == 0;
+}
+s[0] = -C / B;
+return 1;
+}
+/* normal form: x^2 + px + q = 0 */
+const double p2 = p * p;
+if (!AlmostDequalUlps(p2, q) && p2 < q) {
+return 0;
+}
+double sqrt_D = 0;
+if (p2 > q) {
+sqrt_D = sqrt(p2 - q);
+}
+s[0] = sqrt_D - p;
+s[1] = -sqrt_D - p;
+return 1 + !AlmostDequalUlps(s[0], s[1]);
+}
+bool SkDQuad::isLinear(int startIndex, int endIndex) const {
+SkLineParameters lineParameters;
+lineParameters.quadEndPoints(*this, startIndex, endIndex);
+// FIXME: maybe it's possible to avoid this and compare non-normalized
+lineParameters.normalize();
+double distance = lineParameters.controlPtDistance(*this);
+return approximately_zero(distance);
+}
+SkDCubic SkDQuad::toCubic() const {
+SkDCubic cubic;
+cubic[0] = fPts[0];
+cubic[2] = fPts[1];
+cubic[3] = fPts[2];
+cubic[1].fX = (cubic[0].fX + cubic[2].fX * 2) / 3;
+cubic[1].fY = (cubic[0].fY + cubic[2].fY * 2) / 3;
+cubic[2].fX = (cubic[3].fX + cubic[2].fX * 2) / 3;
+cubic[2].fY = (cubic[3].fY + cubic[2].fY * 2) / 3;
+return cubic;
+}
+SkDVector SkDQuad::dxdyAtT(double t) const {
+double a = t - 1;
+double b = 1 - 2 * t;
+double c = t;
+SkDVector result = { a * fPts[0].fX + b * fPts[1].fX + c * fPts[2].fX,
+a * fPts[0].fY + b * fPts[1].fY + c * fPts[2].fY };
+return result;
+}
+// OPTIMIZE: assert if caller passes in t == 0 / t == 1 ?
+SkDPoint SkDQuad::ptAtT(double t) const {
+if (0 == t) {
+return fPts[0];
+}
+if (1 == t) {
+return fPts[2];
+}
+double one_t = 1 - t;
+double a = one_t * one_t;
+double b = 2 * one_t * t;
+double c = t * t;
+SkDPoint result = { a * fPts[0].fX + b * fPts[1].fX + c * fPts[2].fX,
+a * fPts[0].fY + b * fPts[1].fY + c * fPts[2].fY };
+return result;
+}
+/*
+Given a quadratic q, t1, and t2, find a small quadratic segment.
+The new quadratic is defined by A, B, and C, where
+A = c[0]*(1 - t1)*(1 - t1) + 2*c[1]*t1*(1 - t1) + c[2]*t1*t1
+C = c[3]*(1 - t1)*(1 - t1) + 2*c[2]*t1*(1 - t1) + c[1]*t1*t1
+To find B, compute the point halfway between t1 and t2:
+q(at (t1 + t2)/2) == D
+Next, compute where D must be if we know the value of B:
+_12 = A/2 + B/2
+12_ = B/2 + C/2
+123 = A/4 + B/2 + C/4
+= D
+Group the known values on one side:
+B   = D*2 - A/2 - C/2
+*/
+static double interp_quad_coords(const double* src, double t) {
+double ab = SkDInterp(src[0], src[2], t);
+double bc = SkDInterp(src[2], src[4], t);
+double abc = SkDInterp(ab, bc, t);
+return abc;
+}
+bool SkDQuad::monotonicInY() const {
+return between(fPts[0].fY, fPts[1].fY, fPts[2].fY);
+}
+SkDQuad SkDQuad::subDivide(double t1, double t2) const {
+SkDQuad dst;
+double ax = dst[0].fX = interp_quad_coords(&fPts[0].fX, t1);
+double ay = dst[0].fY = interp_quad_coords(&fPts[0].fY, t1);
+double dx = interp_quad_coords(&fPts[0].fX, (t1 + t2) / 2);
+double dy = interp_quad_coords(&fPts[0].fY, (t1 + t2) / 2);
+double cx = dst[2].fX = interp_quad_coords(&fPts[0].fX, t2);
+double cy = dst[2].fY = interp_quad_coords(&fPts[0].fY, t2);
+/* bx = */ dst[1].fX = 2*dx - (ax + cx)/2;
+/* by = */ dst[1].fY = 2*dy - (ay + cy)/2;
+return dst;
+}
+void SkDQuad::align(int endIndex, SkDPoint* dstPt) const {
+if (fPts[endIndex].fX == fPts[1].fX) {
+dstPt->fX = fPts[endIndex].fX;
+}
+if (fPts[endIndex].fY == fPts[1].fY) {
+dstPt->fY = fPts[endIndex].fY;
+}
+}
+SkDPoint SkDQuad::subDivide(const SkDPoint& a, const SkDPoint& c, double t1, double t2) const {
+SkASSERT(t1 != t2);
+SkDPoint b;
+#if 0
+// this approach assumes that the control point computed directly is accurate enough
+double dx = interp_quad_coords(&fPts[0].fX, (t1 + t2) / 2);
+double dy = interp_quad_coords(&fPts[0].fY, (t1 + t2) / 2);
+b.fX = 2 * dx - (a.fX + c.fX) / 2;
+b.fY = 2 * dy - (a.fY + c.fY) / 2;
+#else
+SkDQuad sub = subDivide(t1, t2);
+SkDLine b0 = {{a, sub[1] + (a - sub[0])}};
+SkDLine b1 = {{c, sub[1] + (c - sub[2])}};
+SkIntersections i;
+i.intersectRay(b0, b1);
+if (i.used() == 1) {
+b = i.pt(0);
+} else {
+SkASSERT(i.used() == 2 || i.used() == 0);
+b = SkDPoint::Mid(b0[1], b1[1]);
+}
+#endif
+if (t1 == 0 || t2 == 0) {
+align(0, &b);
+}
+if (t1 == 1 || t2 == 1) {
+align(2, &b);
+}
+if (precisely_subdivide_equal(b.fX, a.fX)) {
+b.fX = a.fX;
+} else if (precisely_subdivide_equal(b.fX, c.fX)) {
+b.fX = c.fX;
+}
+if (precisely_subdivide_equal(b.fY, a.fY)) {
+b.fY = a.fY;
+} else if (precisely_subdivide_equal(b.fY, c.fY)) {
+b.fY = c.fY;
+}
+return b;
+}
+/* classic one t subdivision */
+static void interp_quad_coords(const double* src, double* dst, double t) {
+double ab = SkDInterp(src[0], src[2], t);
+double bc = SkDInterp(src[2], src[4], t);
+dst[0] = src[0];
+dst[2] = ab;
+dst[4] = SkDInterp(ab, bc, t);
+dst[6] = bc;
+dst[8] = src[4];
+}
+SkDQuadPair SkDQuad::chopAt(double t) const
+{
+SkDQuadPair dst;
+interp_quad_coords(&fPts[0].fX, &dst.pts[0].fX, t);
+interp_quad_coords(&fPts[0].fY, &dst.pts[0].fY, t);
+return dst;
+}
+static int valid_unit_divide(double numer, double denom, double* ratio)
+{
+if (numer < 0) {
+numer = -numer;
+denom = -denom;
+}
+if (denom == 0 || numer == 0 || numer >= denom) {
+return 0;
+}
+double r = numer / denom;
+if (r == 0) {  // catch underflow if numer <<<< denom
+return 0;
+}
+*ratio = r;
+return 1;
+}
+/** Quad'(t) = At + B, where
+A = 2(a - 2b + c)
+B = 2(b - a)
+Solve for t, only if it fits between 0 < t < 1
+*/
+int SkDQuad::FindExtrema(double a, double b, double c, double tValue[1]) {
+/*  At + B == 0
+t = -B / A
+*/
+return valid_unit_divide(a - b, a - b - b + c, tValue);
+}
+/* Parameterization form, given A*t*t + 2*B*t*(1-t) + C*(1-t)*(1-t)
+*
+* a = A - 2*B +   C
+* b =     2*B - 2*C
+* c =             C
+*/
+void SkDQuad::SetABC(const double* quad, double* a, double* b, double* c) {
+*a = quad[0];      // a = A
+*b = 2 * quad[2];  // b =     2*B
+*c = quad[4];      // c =             C
+*b -= *c;          // b =     2*B -   C
+*a -= *b;          // a = A - 2*B +   C
+*b -= *c;          // b =     2*B - 2*C
+}
+#ifdef SK_DEBUG
+void SkDQuad::dump() {
+SkDebugf("{{");
+int index = 0;
+do {
+fPts[index].dump();
+SkDebugf(", ");
+} while (++index < 2);
+fPts[index].dump();
+SkDebugf("}}\n");
+}
+#endif

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comparison: gfx/skia/trunk/src/pathops/SkPathOpsQuad.cpp

gfx/skia/trunk/src/pathops/SkPathOpsQuad.cpp