12eea2b10d
git-svn-id: http://skia.googlecode.com/svn/trunk@7875 2bbb7eff-a529-9590-31e7-b0007b416f81
537 lines
23 KiB
C++
537 lines
23 KiB
C++
/*
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* Copyright 2012 Google Inc.
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*
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* Use of this source code is governed by a BSD-style license that can be
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* found in the LICENSE file.
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*/
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#include "CubicUtilities.h"
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#include "CurveIntersection.h"
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#include "Intersections.h"
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#include "IntersectionUtilities.h"
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#include "LineIntersection.h"
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#include "LineUtilities.h"
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#include "QuadraticUtilities.h"
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#if ONE_OFF_DEBUG
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static const double tLimits[2][2] = {{0.772784538, 0.77278492}, {0.999111748, 0.999112129}};
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#endif
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#define DEBUG_QUAD_PART 0
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#define SWAP_TOP_DEBUG 0
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static int quadPart(const Cubic& cubic, double tStart, double tEnd, Quadratic& simple) {
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Cubic part;
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sub_divide(cubic, tStart, tEnd, part);
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Quadratic quad;
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demote_cubic_to_quad(part, quad);
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// FIXME: should reduceOrder be looser in this use case if quartic is going to blow up on an
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// extremely shallow quadratic?
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int order = reduceOrder(quad, simple, kReduceOrder_TreatAsFill);
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#if DEBUG_QUAD_PART
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SkDebugf("%s cubic=(%1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g) t=(%1.17g,%1.17g)\n",
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__FUNCTION__, cubic[0].x, cubic[0].y, cubic[1].x, cubic[1].y, cubic[2].x, cubic[2].y,
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cubic[3].x, cubic[3].y, tStart, tEnd);
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SkDebugf("%s part=(%1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g)"
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" quad=(%1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g)\n", __FUNCTION__, part[0].x, part[0].y,
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part[1].x, part[1].y, part[2].x, part[2].y, part[3].x, part[3].y, quad[0].x, quad[0].y,
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quad[1].x, quad[1].y, quad[2].x, quad[2].y);
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SkDebugf("%s simple=(%1.17g,%1.17g", __FUNCTION__, simple[0].x, simple[0].y);
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if (order > 1) {
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SkDebugf(" %1.17g,%1.17g", simple[1].x, simple[1].y);
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}
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if (order > 2) {
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SkDebugf(" %1.17g,%1.17g", simple[2].x, simple[2].y);
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}
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SkDebugf(")\n");
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SkASSERT(order < 4 && order > 0);
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#endif
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return order;
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}
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static void intersectWithOrder(const Quadratic& simple1, int order1, const Quadratic& simple2,
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int order2, Intersections& i) {
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if (order1 == 3 && order2 == 3) {
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intersect2(simple1, simple2, i);
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} else if (order1 <= 2 && order2 <= 2) {
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intersect((const _Line&) simple1, (const _Line&) simple2, i);
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} else if (order1 == 3 && order2 <= 2) {
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intersect(simple1, (const _Line&) simple2, i);
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} else {
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SkASSERT(order1 <= 2 && order2 == 3);
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intersect(simple2, (const _Line&) simple1, i);
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for (int s = 0; s < i.fUsed; ++s) {
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SkTSwap(i.fT[0][s], i.fT[1][s]);
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}
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}
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}
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static double distanceFromEnd(double t) {
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return t > 0.5 ? 1 - t : t;
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}
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// OPTIMIZATION: this used to try to guess the value for delta, and that may still be worthwhile
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static void bumpForRetry(double t1, double t2, double& s1, double& e1, double& s2, double& e2) {
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double dt1 = distanceFromEnd(t1);
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double dt2 = distanceFromEnd(t2);
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double delta = 1.0 / gPrecisionUnit;
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if (dt1 < dt2) {
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if (t1 == dt1) {
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s1 = SkTMax(s1 - delta, 0.);
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} else {
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e1 = SkTMin(e1 + delta, 1.);
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}
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} else {
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if (t2 == dt2) {
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s2 = SkTMax(s2 - delta, 0.);
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} else {
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e2 = SkTMin(e2 + delta, 1.);
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}
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}
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}
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static bool doIntersect(const Cubic& cubic1, double t1s, double t1m, double t1e,
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const Cubic& cubic2, double t2s, double t2m, double t2e, Intersections& i) {
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bool result = false;
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i.upDepth();
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// divide the quadratics at the new t value and try again
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double p1s = t1s;
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double p1e = t1m;
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for (int p1 = 0; p1 < 2; ++p1) {
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Quadratic s1a;
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int o1a = quadPart(cubic1, p1s, p1e, s1a);
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double p2s = t2s;
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double p2e = t2m;
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for (int p2 = 0; p2 < 2; ++p2) {
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Quadratic s2a;
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int o2a = quadPart(cubic2, p2s, p2e, s2a);
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Intersections locals;
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#if ONE_OFF_DEBUG
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if (tLimits[0][0] >= p1s && tLimits[0][1] <= p1e
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&& tLimits[1][0] >= p2s && tLimits[1][1] <= p2e) {
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SkDebugf("t1=(%1.9g,%1.9g) o1=%d t2=(%1.9g,%1.9g) o2=%d\n",
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p1s, p1e, o1a, p2s, p2e, o2a);
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if (o1a == 2) {
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SkDebugf("{{%1.9g,%1.9g}, {%1.9g,%1.9g}},\n",
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s1a[0].x, s1a[0].y, s1a[1].x, s1a[1].y);
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} else {
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SkDebugf("{{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}},\n",
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s1a[0].x, s1a[0].y, s1a[1].x, s1a[1].y, s1a[2].x, s1a[2].y);
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}
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if (o2a == 2) {
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SkDebugf("{{%1.9g,%1.9g}, {%1.9g,%1.9g}},\n",
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s2a[0].x, s2a[0].y, s2a[1].x, s2a[1].y);
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} else {
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SkDebugf("{{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}},\n",
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s2a[0].x, s2a[0].y, s2a[1].x, s2a[1].y, s2a[2].x, s2a[2].y);
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}
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Intersections xlocals;
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intersectWithOrder(s1a, o1a, s2a, o2a, xlocals);
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SkDebugf("xlocals.fUsed=%d depth=%d\n", xlocals.used(), i.depth());
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}
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#endif
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intersectWithOrder(s1a, o1a, s2a, o2a, locals);
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for (int tIdx = 0; tIdx < locals.used(); ++tIdx) {
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double to1 = p1s + (p1e - p1s) * locals.fT[0][tIdx];
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double to2 = p2s + (p2e - p2s) * locals.fT[1][tIdx];
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// if the computed t is not sufficiently precise, iterate
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_Point p1, p2;
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xy_at_t(cubic1, to1, p1.x, p1.y);
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xy_at_t(cubic2, to2, p2.x, p2.y);
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#if ONE_OFF_DEBUG
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SkDebugf("to1=%1.9g p1=(%1.9g,%1.9g) to2=%1.9g p2=(%1.9g,%1.9g) d=%1.9g\n",
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to1, p1.x, p1.y, to2, p2.x, p2.y, p1.distance(p2));
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#endif
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if (p1.approximatelyEqualHalf(p2)) {
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i.insertSwap(to1, to2, p1);
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result = true;
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} else {
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result = doIntersect(cubic1, p1s, to1, p1e, cubic2, p2s, to2, p2e, i);
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if (!result && p1.approximatelyEqual(p2)) {
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i.insertSwap(to1, to2, p1);
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#if SWAP_TOP_DEBUG
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SkDebugf("!!!\n");
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#endif
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result = true;
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} else
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// if both cubics curve in the same direction, the quadratic intersection
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// may mark a range that does not contain the cubic intersection. If no
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// intersection is found, look again including the t distance of the
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// of the quadratic intersection nearest a quadratic end (which in turn is
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// nearest the actual cubic)
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if (!result) {
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double b1s = p1s;
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double b1e = p1e;
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double b2s = p2s;
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double b2e = p2e;
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bumpForRetry(locals.fT[0][tIdx], locals.fT[1][tIdx], b1s, b1e, b2s, b2e);
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result = doIntersect(cubic1, b1s, to1, b1e, cubic2, b2s, to2, b2e, i);
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}
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}
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}
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p2s = p2e;
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p2e = t2e;
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}
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p1s = p1e;
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p1e = t1e;
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}
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i.downDepth();
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return result;
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}
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// this flavor approximates the cubics with quads to find the intersecting ts
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// OPTIMIZE: if this strategy proves successful, the quad approximations, or the ts used
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// to create the approximations, could be stored in the cubic segment
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// FIXME: this strategy needs to intersect the convex hull on either end with the opposite to
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// account for inset quadratics that cause the endpoint intersection to avoid detection
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// the segments can be very short -- the length of the maximum quadratic error (precision)
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static bool intersect2(const Cubic& cubic1, double t1s, double t1e, const Cubic& cubic2,
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double t2s, double t2e, double precisionScale, Intersections& i) {
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Cubic c1, c2;
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sub_divide(cubic1, t1s, t1e, c1);
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sub_divide(cubic2, t2s, t2e, c2);
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SkTDArray<double> ts1;
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cubic_to_quadratics(c1, calcPrecision(c1) * precisionScale, ts1);
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SkTDArray<double> ts2;
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cubic_to_quadratics(c2, calcPrecision(c2) * precisionScale, ts2);
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double t1Start = t1s;
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int ts1Count = ts1.count();
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for (int i1 = 0; i1 <= ts1Count; ++i1) {
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const double tEnd1 = i1 < ts1Count ? ts1[i1] : 1;
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const double t1 = t1s + (t1e - t1s) * tEnd1;
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Quadratic s1;
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int o1 = quadPart(cubic1, t1Start, t1, s1);
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double t2Start = t2s;
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int ts2Count = ts2.count();
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for (int i2 = 0; i2 <= ts2Count; ++i2) {
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const double tEnd2 = i2 < ts2Count ? ts2[i2] : 1;
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const double t2 = t2s + (t2e - t2s) * tEnd2;
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Quadratic s2;
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int o2 = quadPart(cubic2, t2Start, t2, s2);
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#if ONE_OFF_DEBUG
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if (tLimits[0][0] >= t1Start && tLimits[0][1] <= t1
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&& tLimits[1][0] >= t2Start && tLimits[1][1] <= t2) {
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Cubic cSub1, cSub2;
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sub_divide(cubic1, t1Start, tEnd1, cSub1);
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sub_divide(cubic2, t2Start, tEnd2, cSub2);
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SkDebugf("t1=(%1.9g,%1.9g) t2=(%1.9g,%1.9g)\n",
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t1Start, t1, t2Start, t2);
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Intersections xlocals;
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intersectWithOrder(s1, o1, s2, o2, xlocals);
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SkDebugf("xlocals.fUsed=%d\n", xlocals.used());
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}
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#endif
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Intersections locals;
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intersectWithOrder(s1, o1, s2, o2, locals);
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for (int tIdx = 0; tIdx < locals.used(); ++tIdx) {
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double to1 = t1Start + (t1 - t1Start) * locals.fT[0][tIdx];
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double to2 = t2Start + (t2 - t2Start) * locals.fT[1][tIdx];
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// if the computed t is not sufficiently precise, iterate
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_Point p1, p2;
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xy_at_t(cubic1, to1, p1.x, p1.y);
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xy_at_t(cubic2, to2, p2.x, p2.y);
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if (p1.approximatelyEqual(p2)) {
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i.insert(to1, to2, p1);
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} else {
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#if ONE_OFF_DEBUG
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if (tLimits[0][0] >= t1Start && tLimits[0][1] <= t1
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&& tLimits[1][0] >= t2Start && tLimits[1][1] <= t2) {
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SkDebugf("t1=(%1.9g,%1.9g) t2=(%1.9g,%1.9g)\n",
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t1Start, t1, t2Start, t2);
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}
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#endif
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bool found = doIntersect(cubic1, t1Start, to1, t1, cubic2, t2Start, to2, t2, i);
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if (!found) {
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double b1s = t1Start;
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double b1e = t1;
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double b2s = t2Start;
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double b2e = t2;
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bumpForRetry(locals.fT[0][tIdx], locals.fT[1][tIdx], b1s, b1e, b2s, b2e);
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doIntersect(cubic1, b1s, to1, b1e, cubic2, b2s, to2, b2e, i);
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}
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}
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}
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int coincidentCount = locals.coincidentUsed();
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if (coincidentCount) {
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// FIXME: one day, we'll probably need to allow coincident + non-coincident pts
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SkASSERT(coincidentCount == locals.used());
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SkASSERT(coincidentCount == 2);
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double coTs[2][2];
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for (int tIdx = 0; tIdx < coincidentCount; ++tIdx) {
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if (locals.fIsCoincident[0] & (1 << tIdx)) {
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coTs[0][tIdx] = t1Start + (t1 - t1Start) * locals.fT[0][tIdx];
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}
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if (locals.fIsCoincident[1] & (1 << tIdx)) {
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coTs[1][tIdx] = t2Start + (t2 - t2Start) * locals.fT[1][tIdx];
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}
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}
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i.insertCoincidentPair(coTs[0][0], coTs[0][1], coTs[1][0], coTs[1][1],
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locals.fPt[0], locals.fPt[1]);
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}
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t2Start = t2;
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}
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t1Start = t1;
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}
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return i.intersected();
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}
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// this flavor centers potential intersections recursively. In contrast, '2' may inadvertently
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// chase intersections near quadratic ends, requiring odd hacks to find them.
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static bool intersect3(const Cubic& cubic1, double t1s, double t1e, const Cubic& cubic2,
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double t2s, double t2e, double precisionScale, Intersections& i) {
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i.upDepth();
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bool result = false;
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Cubic c1, c2;
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sub_divide(cubic1, t1s, t1e, c1);
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sub_divide(cubic2, t2s, t2e, c2);
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SkTDArray<double> ts1;
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cubic_to_quadratics(c1, calcPrecision(c1) * precisionScale, ts1);
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SkTDArray<double> ts2;
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cubic_to_quadratics(c2, calcPrecision(c2) * precisionScale, ts2);
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double t1Start = t1s;
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int ts1Count = ts1.count();
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for (int i1 = 0; i1 <= ts1Count; ++i1) {
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const double tEnd1 = i1 < ts1Count ? ts1[i1] : 1;
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const double t1 = t1s + (t1e - t1s) * tEnd1;
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Quadratic s1;
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int o1 = quadPart(cubic1, t1Start, t1, s1);
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double t2Start = t2s;
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int ts2Count = ts2.count();
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for (int i2 = 0; i2 <= ts2Count; ++i2) {
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const double tEnd2 = i2 < ts2Count ? ts2[i2] : 1;
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const double t2 = t2s + (t2e - t2s) * tEnd2;
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if (cubic1 == cubic2 && t1Start >= t2Start) {
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t2Start = t2;
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continue;
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}
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Quadratic s2;
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int o2 = quadPart(cubic2, t2Start, t2, s2);
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#if ONE_OFF_DEBUG
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if (tLimits[0][0] >= t1Start && tLimits[0][1] <= t1
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&& tLimits[1][0] >= t2Start && tLimits[1][1] <= t2) {
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Cubic cSub1, cSub2;
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sub_divide(cubic1, t1Start, tEnd1, cSub1);
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sub_divide(cubic2, t2Start, tEnd2, cSub2);
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SkDebugf("t1=(%1.9g,%1.9g) t2=(%1.9g,%1.9g)\n",
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t1Start, t1, t2Start, t2);
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Intersections xlocals;
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intersectWithOrder(s1, o1, s2, o2, xlocals);
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SkDebugf("xlocals.fUsed=%d\n", xlocals.used());
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}
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#endif
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Intersections locals;
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intersectWithOrder(s1, o1, s2, o2, locals);
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double coStart[2] = { -1 };
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_Point coPoint;
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for (int tIdx = 0; tIdx < locals.used(); ++tIdx) {
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double to1 = t1Start + (t1 - t1Start) * locals.fT[0][tIdx];
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double to2 = t2Start + (t2 - t2Start) * locals.fT[1][tIdx];
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// if the computed t is not sufficiently precise, iterate
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_Point p1, p2;
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xy_at_t(cubic1, to1, p1.x, p1.y);
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xy_at_t(cubic2, to2, p2.x, p2.y);
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if (p1.approximatelyEqual(p2)) {
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if (locals.fIsCoincident[0] & 1 << tIdx) {
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if (coStart[0] < 0) {
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coStart[0] = to1;
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coStart[1] = to2;
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coPoint = p1;
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} else {
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i.insertCoincidentPair(coStart[0], to1, coStart[1], to2, coPoint, p1);
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coStart[0] = -1;
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}
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result = true;
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} else if (cubic1 != cubic2 || !approximately_equal(to1, to2)) {
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if (i.swapped()) { // FIXME: insert should respect swap
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i.insert(to2, to1, p1);
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} else {
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i.insert(to1, to2, p1);
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}
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result = true;
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}
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} else {
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double offset = precisionScale / 16; // FIME: const is arbitrary -- test & refine
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double c1Min = SkTMax(0., to1 - offset);
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double c1Max = SkTMin(1., to1 + offset);
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double c2Min = SkTMax(0., to2 - offset);
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double c2Max = SkTMin(1., to2 + offset);
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bool found = intersect3(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i);
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if (false && !found) {
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// either offset was overagressive or cubics didn't really intersect
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// if they didn't intersect, then quad tangents ought to be nearly parallel
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offset = precisionScale / 2; // try much less agressive offset
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c1Min = SkTMax(0., to1 - offset);
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c1Max = SkTMin(1., to1 + offset);
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c2Min = SkTMax(0., to2 - offset);
|
|
c2Max = SkTMin(1., to2 + offset);
|
|
found = intersect3(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i);
|
|
if (found) {
|
|
SkDebugf("%s *** over-aggressive? offset=%1.9g depth=%d\n", __FUNCTION__,
|
|
offset, i.depth());
|
|
}
|
|
// try parallel measure
|
|
_Vector d1 = dxdy_at_t(cubic1, to1);
|
|
_Vector d2 = dxdy_at_t(cubic2, to2);
|
|
double shallow = d1.cross(d2);
|
|
#if 1 || ONE_OFF_DEBUG // not sure this is worth debugging
|
|
if (!approximately_zero(shallow)) {
|
|
SkDebugf("%s *** near-miss? shallow=%1.9g depth=%d\n", __FUNCTION__,
|
|
offset, i.depth());
|
|
}
|
|
#endif
|
|
if (i.depth() == 1 && shallow < 0.6) {
|
|
SkDebugf("%s !!! near-miss? shallow=%1.9g depth=%d\n", __FUNCTION__,
|
|
offset, i.depth());
|
|
}
|
|
}
|
|
}
|
|
}
|
|
SkASSERT(coStart[0] == -1);
|
|
t2Start = t2;
|
|
}
|
|
t1Start = t1;
|
|
}
|
|
i.downDepth();
|
|
return result;
|
|
}
|
|
|
|
// intersect the end of the cubic with the other. Try lines from the end to control and opposite
|
|
// end to determine range of t on opposite cubic.
|
|
static bool intersectEnd(const Cubic& cubic1, bool start, const Cubic& cubic2, const _Rect& bounds2,
|
|
Intersections& i) {
|
|
_Line line;
|
|
int t1Index = start ? 0 : 3;
|
|
line[0] = cubic1[t1Index];
|
|
// don't bother if the two cubics are connnected
|
|
if (line[0].approximatelyEqual(cubic2[0]) || line[0].approximatelyEqual(cubic2[3])) {
|
|
return false;
|
|
}
|
|
double tMin = 1, tMax = 0;
|
|
for (int index = 0; index < 4; ++index) {
|
|
if (index == t1Index) {
|
|
continue;
|
|
}
|
|
_Vector dxy1 = cubic1[index] - line[0];
|
|
dxy1 /= gPrecisionUnit;
|
|
line[1] = line[0] + dxy1;
|
|
_Rect lineBounds;
|
|
lineBounds.setBounds(line);
|
|
if (!bounds2.intersects(lineBounds)) {
|
|
continue;
|
|
}
|
|
Intersections local;
|
|
if (!intersect(cubic2, line, local)) {
|
|
continue;
|
|
}
|
|
for (int index = 0; index < local.fUsed; ++index) {
|
|
tMin = SkTMin(tMin, local.fT[0][index]);
|
|
tMax = SkTMax(tMax, local.fT[0][index]);
|
|
}
|
|
}
|
|
if (tMin > tMax) {
|
|
return false;
|
|
}
|
|
double tMin1 = start ? 0 : 1 - 1.0 / gPrecisionUnit;
|
|
double tMax1 = start ? 1.0 / gPrecisionUnit : 1;
|
|
double tMin2 = SkTMax(tMin - 1.0 / gPrecisionUnit, 0.0);
|
|
double tMax2 = SkTMin(tMax + 1.0 / gPrecisionUnit, 1.0);
|
|
return intersect3(cubic1, tMin1, tMax1, cubic2, tMin2, tMax2, 1, i);
|
|
}
|
|
|
|
// FIXME: add intersection of convex hull on cubics' ends with the opposite cubic. The hull line
|
|
// segments can be constructed to be only as long as the calculated precision suggests. If the hull
|
|
// line segments intersect the cubic, then use the intersections to construct a subdivision for
|
|
// quadratic curve fitting.
|
|
bool intersect2(const Cubic& c1, const Cubic& c2, Intersections& i) {
|
|
bool result = intersect2(c1, 0, 1, c2, 0, 1, 1, i);
|
|
// FIXME: pass in cached bounds from caller
|
|
_Rect c1Bounds, c2Bounds;
|
|
c1Bounds.setBounds(c1); // OPTIMIZE use setRawBounds ?
|
|
c2Bounds.setBounds(c2);
|
|
result |= intersectEnd(c1, false, c2, c2Bounds, i);
|
|
result |= intersectEnd(c1, true, c2, c2Bounds, i);
|
|
i.swap();
|
|
result |= intersectEnd(c2, false, c1, c1Bounds, i);
|
|
result |= intersectEnd(c2, true, c1, c1Bounds, i);
|
|
i.swap();
|
|
return result;
|
|
}
|
|
|
|
const double CLOSE_ENOUGH = 0.001;
|
|
|
|
static bool closeStart(const Cubic& cubic, int cubicIndex, Intersections& i, _Point& pt) {
|
|
if (i.fT[cubicIndex][0] != 0 || i.fT[cubicIndex][1] > CLOSE_ENOUGH) {
|
|
return false;
|
|
}
|
|
pt = xy_at_t(cubic, (i.fT[cubicIndex][0] + i.fT[cubicIndex][1]) / 2);
|
|
return true;
|
|
}
|
|
|
|
static bool closeEnd(const Cubic& cubic, int cubicIndex, Intersections& i, _Point& pt) {
|
|
int last = i.used() - 1;
|
|
if (i.fT[cubicIndex][last] != 1 || i.fT[cubicIndex][last - 1] < 1 - CLOSE_ENOUGH) {
|
|
return false;
|
|
}
|
|
pt = xy_at_t(cubic, (i.fT[cubicIndex][last] + i.fT[cubicIndex][last - 1]) / 2);
|
|
return true;
|
|
}
|
|
|
|
bool intersect3(const Cubic& c1, const Cubic& c2, Intersections& i) {
|
|
bool result = intersect3(c1, 0, 1, c2, 0, 1, 1, i);
|
|
// FIXME: pass in cached bounds from caller
|
|
_Rect c1Bounds, c2Bounds;
|
|
c1Bounds.setBounds(c1); // OPTIMIZE use setRawBounds ?
|
|
c2Bounds.setBounds(c2);
|
|
result |= intersectEnd(c1, false, c2, c2Bounds, i);
|
|
result |= intersectEnd(c1, true, c2, c2Bounds, i);
|
|
i.swap();
|
|
result |= intersectEnd(c2, false, c1, c1Bounds, i);
|
|
result |= intersectEnd(c2, true, c1, c1Bounds, i);
|
|
i.swap();
|
|
// If an end point and a second point very close to the end is returned, the second
|
|
// point may have been detected because the approximate quads
|
|
// intersected at the end and close to it. Verify that the second point is valid.
|
|
if (i.used() <= 1 || i.coincidentUsed()) {
|
|
return result;
|
|
}
|
|
_Point pt[2];
|
|
if (closeStart(c1, 0, i, pt[0]) && closeStart(c2, 1, i, pt[1])
|
|
&& pt[0].approximatelyEqual(pt[1])) {
|
|
i.removeOne(1);
|
|
}
|
|
if (closeEnd(c1, 0, i, pt[0]) && closeEnd(c2, 1, i, pt[1])
|
|
&& pt[0].approximatelyEqual(pt[1])) {
|
|
i.removeOne(i.used() - 2);
|
|
}
|
|
return result;
|
|
}
|
|
|
|
// Up promote the quad to a cubic.
|
|
// OPTIMIZATION If this is a common use case, optimize by duplicating
|
|
// the intersect 3 loop to avoid the promotion / demotion code
|
|
int intersect(const Cubic& cubic, const Quadratic& quad, Intersections& i) {
|
|
Cubic up;
|
|
toCubic(quad, up);
|
|
(void) intersect3(cubic, up, i);
|
|
return i.used();
|
|
}
|
|
|
|
/* http://www.ag.jku.at/compass/compasssample.pdf
|
|
( Self-Intersection Problems and Approximate Implicitization by Jan B. Thomassen
|
|
Centre of Mathematics for Applications, University of Oslo http://www.cma.uio.no janbth@math.uio.no
|
|
SINTEF Applied Mathematics http://www.sintef.no )
|
|
describes a method to find the self intersection of a cubic by taking the gradient of the implicit
|
|
form dotted with the normal, and solving for the roots. My math foo is too poor to implement this.*/
|
|
|
|
int intersect(const Cubic& c, Intersections& i) {
|
|
// check to see if x or y end points are the extrema. Are other quick rejects possible?
|
|
if ((between(c[0].x, c[1].x, c[3].x) && between(c[0].x, c[2].x, c[3].x))
|
|
|| (between(c[0].y, c[1].y, c[3].y) && between(c[0].y, c[2].y, c[3].y))) {
|
|
return false;
|
|
}
|
|
(void) intersect3(c, c, i);
|
|
return i.used();
|
|
}
|