When solving the cubic line intersection directly fails, use binary search as a fallback.

The cubic line intersection math empirically works 99.99% of the time (fails 3100 out of 1B random tests) but when it fails, an intersection may be missed altogether.

The binary search is may not find a solution if the cubic line failed to find any solutions at all, but so far that case hasn't arisen.

BUG=skia:2504
TBR=reed@google.com

Author: caryclark@google.com

Review URL: https://codereview.chromium.org/266063003

git-svn-id: http://skia.googlecode.com/svn/trunk@14614 2bbb7eff-a529-9590-31e7-b0007b416f81

1 files changed, 283 insertions, 0 deletions

diff --git a/tests/PathOpsCubicLineIntersectionIdeas.cpp b/tests/PathOpsCubicLineIntersectionIdeas.cpp
new file mode 100644
index 0000000000..2887a2ccfc
--- /dev/null
+++ b/tests/PathOpsCubicLineIntersectionIdeas.cpp
@@ -0,0 +1,283 @@
+/*
+ * Copyright 2014 Google Inc.
+ *
+ * Use of this source code is governed by a BSD-style license that can be
+ * found in the LICENSE file.
+ */
+#include "PathOpsTestCommon.h"
+#include "SkIntersections.h"
+#include "SkPathOpsCubic.h"
+#include "SkPathOpsLine.h"
+#include "SkPathOpsQuad.h"
+#include "SkRandom.h"
+#include "SkReduceOrder.h"
+#include "Test.h"
+
+static bool gPathOpsCubicLineIntersectionIdeasVerbose = false;
+
+static struct CubicLineFailures {
+    SkDCubic c;
+    double t;
+    SkDPoint p;
+} cubicLineFailures[] = {
+    {{{{-164.3726806640625, 36.826904296875}, {-189.045166015625, -953.2220458984375},
+        {926.505859375, -897.36175537109375}, {-139.33489990234375, 204.40771484375}}},
+        0.37329583, {107.54935269006289, -632.13736293162208}},
+    {{{{784.056884765625, -554.8350830078125}, {67.5489501953125, 509.0224609375},
+        {-447.713134765625, 751.375}, {415.7784423828125, 172.22265625}}},
+        0.660005242, {-32.973148967736151, 478.01341797403569}},
+    {{{{-580.6834716796875, -127.044921875}, {-872.8983154296875, -945.54302978515625},
+        {260.8092041015625, -909.34991455078125}, {-976.2125244140625, -18.46551513671875}}},
+        0.578826774, {-390.17910153915489, -687.21144412296007}},
+};
+
+int cubicLineFailuresCount = (int) SK_ARRAY_COUNT(cubicLineFailures);
+
+double measuredSteps[] = {
+    9.15910731e-007, 8.6600277e-007, 7.4122059e-007, 6.92087618e-007, 8.35290245e-007,
+    3.29763199e-007, 5.07547773e-007, 4.41294224e-007, 0, 0,
+    3.76879167e-006, 1.06126249e-006, 2.36873967e-006, 1.62421134e-005, 3.09103599e-005,
+    4.38917976e-005, 0.000112348938, 0.000243149242, 0.000433174114, 0.00170880232,
+    0.00272619724, 0.00518844604, 0.000352621078, 0.00175960064, 0.027875185,
+    0.0351329803, 0.103964925,
+};
+
+/* last output : errors=3121 
+    9.1796875e-007 8.59375e-007 7.5e-007 6.875e-007 8.4375e-007
+    3.125e-007 5e-007 4.375e-007 0 0
+    3.75e-006 1.09375e-006 2.1875e-006 1.640625e-005 3.0859375e-005
+    4.38964844e-005 0.000112304687 0.000243164063 0.000433181763 0.00170898437
+    0.00272619247 0.00518844604 0.000352621078 0.00175960064 0.027875185
+    0.0351329803 0.103964925
+*/
+
+static double binary_search(const SkDCubic& cubic, double step, const SkDPoint& pt, double t,
+        int* iters) {
+    double firstStep = step;
+    do {
+        *iters += 1;
+        SkDPoint cubicAtT = cubic.ptAtT(t);
+        if (cubicAtT.approximatelyEqual(pt)) {
+            break;
+        }
+        double calcX = cubicAtT.fX - pt.fX;
+        double calcY = cubicAtT.fY - pt.fY;
+        double calcDist = calcX * calcX + calcY * calcY;
+        if (step == 0) {
+            SkDebugf("binary search failed: step=%1.9g cubic=", firstStep);
+            cubic.dump();
+            SkDebugf(" t=%1.9g ", t);
+            pt.dump();
+            SkDebugf("\n");
+            return -1;
+        }
+        double lastStep = step;
+        step /= 2;
+        SkDPoint lessPt = cubic.ptAtT(t - lastStep);
+        double lessX = lessPt.fX - pt.fX;
+        double lessY = lessPt.fY - pt.fY;
+        double lessDist = lessX * lessX + lessY * lessY;
+        // use larger x/y difference to choose step
+        if (calcDist > lessDist) {
+            t -= step;
+            t = SkTMax(0., t);
+        } else {
+            SkDPoint morePt = cubic.ptAtT(t + lastStep);
+            double moreX = morePt.fX - pt.fX;
+            double moreY = morePt.fY - pt.fY;
+            double moreDist = moreX * moreX + moreY * moreY;
+            if (calcDist <= moreDist) {
+                continue;
+            }
+            t += step;
+            t = SkTMin(1., t);
+        }
+    } while (true);
+    return t;
+}
+
+#if 0
+static bool r2check(double A, double B, double C, double D, double* R2MinusQ3Ptr) {
+    if (approximately_zero(A)
+            && approximately_zero_when_compared_to(A, B)
+            && approximately_zero_when_compared_to(A, C)
+            && approximately_zero_when_compared_to(A, D)) {  // we're just a quadratic
+        return false;
+    }
+    if (approximately_zero_when_compared_to(D, A)
+            && approximately_zero_when_compared_to(D, B)
+            && approximately_zero_when_compared_to(D, C)) {  // 0 is one root
+        return false;
+    }
+    if (approximately_zero(A + B + C + D)) {  // 1 is one root
+        return false;
+    }
+    double a, b, c;
+    {
+        double invA = 1 / A;
+        a = B * invA;
+        b = C * invA;
+        c = D * invA;
+    }
+    double a2 = a * a;
+    double Q = (a2 - b * 3) / 9;
+    double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54;
+    double R2 = R * R;
+    double Q3 = Q * Q * Q;
+    double R2MinusQ3 = R2 - Q3;
+    *R2MinusQ3Ptr = R2MinusQ3;
+    return true;
+}
+#endif
+
+/* What is the relationship between the accuracy of the root in range and the magnitude of all
+   roots? To find out, create a bunch of cubics, and measure */
+
+DEF_TEST(PathOpsCubicLineRoots, reporter) {
+    if (!gPathOpsCubicLineIntersectionIdeasVerbose) {  // slow; exclude it by default
+        return;
+    }
+    SkRandom ran;
+    double worstStep[256] = {0};
+    int errors = 0;
+    int iters = 0;
+    double smallestR2 = 0;
+    double largestR2 = 0;
+    for (int index = 0; index < 1000000000; ++index) {
+        SkDPoint origin = {ran.nextRangeF(-1000, 1000), ran.nextRangeF(-1000, 1000)};
+        SkDCubic cubic = {{origin,
+                {ran.nextRangeF(-1000, 1000), ran.nextRangeF(-1000, 1000)},
+                {ran.nextRangeF(-1000, 1000), ran.nextRangeF(-1000, 1000)},
+                {ran.nextRangeF(-1000, 1000), ran.nextRangeF(-1000, 1000)}
+        }};
+        // construct a line at a known intersection
+        double t = ran.nextRangeF(0, 1);
+        SkDPoint pt = cubic.ptAtT(t);
+        // skip answers with no intersections (although note the bug!) or two, or more
+        // see if the line / cubic has a fun range of roots
+        double A, B, C, D;
+        SkDCubic::Coefficients(&cubic[0].fY, &A, &B, &C, &D);
+        D -= pt.fY;
+        double allRoots[3] = {0}, validRoots[3] = {0};
+        int realRoots = SkDCubic::RootsReal(A, B, C, D, allRoots);
+        int valid = SkDQuad::AddValidTs(allRoots, realRoots, validRoots);
+        if (valid != 1) {
+            continue;
+        }
+        if (realRoots == 1) {
+            continue;
+        }
+        t = validRoots[0];
+        SkDPoint calcPt = cubic.ptAtT(t);
+        if (calcPt.approximatelyEqual(pt)) {
+            continue;
+        }
+#if 0
+        double R2MinusQ3;
+        if (r2check(A, B, C, D, &R2MinusQ3)) {
+            smallestR2 = SkTMin(smallestR2, R2MinusQ3);
+            largestR2 = SkTMax(largestR2, R2MinusQ3);
+        }
+#endif
+        double largest = SkTMax(fabs(allRoots[0]), fabs(allRoots[1]));
+        if (realRoots == 3) {
+            largest = SkTMax(largest, fabs(allRoots[2]));
+        }
+        int largeBits;
+        if (largest <= 1) {
+#if 0
+            SkDebugf("realRoots=%d (%1.9g, %1.9g, %1.9g) valid=%d (%1.9g, %1.9g, %1.9g)\n",
+                realRoots, allRoots[0], allRoots[1], allRoots[2], valid, validRoots[0],
+                validRoots[1], validRoots[2]);
+#endif
+            double smallest = SkTMin(allRoots[0], allRoots[1]);
+            if (realRoots == 3) {
+                smallest = SkTMin(smallest, allRoots[2]);
+            }
+            SK_ALWAYSBREAK(smallest < 0);
+            SK_ALWAYSBREAK(smallest >= -1);
+            largeBits = 0;
+        } else {
+            frexp(largest, &largeBits);
+            SK_ALWAYSBREAK(largeBits >= 0);
+            SK_ALWAYSBREAK(largeBits < 256);
+        }
+        double step = 1e-6;
+        if (largeBits > 21) {
+            step = 1e-1;
+        } else if (largeBits > 18) {
+            step = 1e-2;
+        } else if (largeBits > 15) {
+            step = 1e-3;
+        } else if (largeBits > 12) {
+            step = 1e-4;
+        } else if (largeBits > 9) {
+            step = 1e-5;
+        }
+        double diff;
+        do {
+            double newT = binary_search(cubic, step, pt, t, &iters);
+            if (newT >= 0) {
+                diff = fabs(t - newT);
+                break;
+            }
+            step *= 1.5;
+            SK_ALWAYSBREAK(step < 1);
+        } while (true);
+        worstStep[largeBits] = SkTMax(worstStep[largeBits], diff);
+#if 0
+        {
+            cubic.dump();
+            SkDebugf("\n");
+            SkDLine line = {{{pt.fX - 1, pt.fY}, {pt.fX + 1, pt.fY}}};
+            line.dump();
+            SkDebugf("\n");
+        }
+#endif
+        ++errors;
+    }
+    SkDebugf("errors=%d avgIter=%1.9g", errors, (double) iters / errors);
+    SkDebugf(" steps: ");
+    int worstLimit = SK_ARRAY_COUNT(worstStep);
+    while (worstStep[--worstLimit] == 0) ;
+    for (int idx2 = 0; idx2 <= worstLimit; ++idx2) {
+        SkDebugf("%1.9g ", worstStep[idx2]);
+    }
+    SkDebugf("\n");
+    SkDebugf("smallestR2=%1.9g largestR2=%1.9g\n", smallestR2, largestR2);
+}
+
+static double testOneFailure(const CubicLineFailures& failure) {
+    const SkDCubic& cubic = failure.c;
+    const SkDPoint& pt = failure.p;
+    double A, B, C, D;
+    SkDCubic::Coefficients(&cubic[0].fY, &A, &B, &C, &D);
+    D -= pt.fY;
+    double allRoots[3] = {0}, validRoots[3] = {0};
+    int realRoots = SkDCubic::RootsReal(A, B, C, D, allRoots);
+    int valid = SkDQuad::AddValidTs(allRoots, realRoots, validRoots);
+    SK_ALWAYSBREAK(valid == 1);
+    SK_ALWAYSBREAK(realRoots != 1);
+    double t = validRoots[0];
+    SkDPoint calcPt = cubic.ptAtT(t);
+    SK_ALWAYSBREAK(!calcPt.approximatelyEqual(pt));
+    int iters = 0;
+    double newT = binary_search(cubic, 0.1, pt, t, &iters);
+    return newT;
+}
+
+DEF_TEST(PathOpsCubicLineFailures, reporter) {
+    return;  // disable for now
+    for (int index = 0; index < cubicLineFailuresCount; ++index) {
+        const CubicLineFailures& failure = cubicLineFailures[index];
+        double newT = testOneFailure(failure);
+        SK_ALWAYSBREAK(newT >= 0);
+    }
+}
+
+DEF_TEST(PathOpsCubicLineOneFailure, reporter) {
+    return;  // disable for now
+    const CubicLineFailures& failure = cubicLineFailures[1];
+    double newT = testOneFailure(failure);
+    SK_ALWAYSBREAK(newT >= 0);
+}