path ops work in progress

BUG=

Review URL: https://codereview.chromium.org/18058007

git-svn-id: http://skia.googlecode.com/svn/trunk@9908 2bbb7eff-a529-9590-31e7-b0007b416f81

19 files changed, 422 insertions, 332 deletions

diff --git a/src/pathops/SkDCubicLineIntersection.cpp b/src/pathops/SkDCubicLineIntersection.cpp
index 11876f8d73..418e107e41 100644
--- a/src/pathops/SkDCubicLineIntersection.cpp
+++ b/src/pathops/SkDCubicLineIntersection.cpp
@@ -105,6 +105,11 @@ int intersect() {
         double lineT = findLineT(cubicT);
         if (pinTs(&cubicT, &lineT)) {
             SkDPoint pt = line.xyAtT(lineT);
+#if ONE_OFF_DEBUG
+            SkDPoint cPt = cubic.xyAtT(cubicT);
+            SkDebugf("%s pt=(%1.9g,%1.9g) cPt=(%1.9g,%1.9g)\n", __FUNCTION__, pt.fX, pt.fY,
+                    cPt.fX, cPt.fY);
+#endif
             intersections.insert(cubicT, lineT, pt);
         }
     }
@@ -165,11 +170,34 @@ protected:
 
 void addEndPoints() {
     for (int cIndex = 0; cIndex < 4; cIndex += 3) {
+        bool foundEnd = false;
         for (int lIndex = 0; lIndex < 2; lIndex++) {
             if (cubic[cIndex] == line[lIndex]) {
                 intersections.insert(cIndex >> 1, lIndex, line[lIndex]);
+                foundEnd = true;
             }
         }
+        // for the test case this was written for, the dist / error ratio was 170.6667
+        // it looks like the cubic stops short of touching the line, but this fixed it.
+        if (foundEnd) {
+            continue;
+        }
+        // See if the cubic end touches the line. 
+        double dist = line.isLeft(cubic[cIndex]); // this distance isn't cartesian
+        SkDVector lineLen = line[1] - line[0]; // the x/y magnitudes of the line
+        // compute the ULPS of the larger of the x/y deltas
+        double larger = SkTMax(SkTAbs(lineLen.fX), SkTAbs(lineLen.fY));
+        if (!RoughlyEqualUlps(larger, larger + dist)) { // is the dist within ULPS tolerance?
+            continue;
+        }
+        double lineT = findLineT(cIndex >> 1);
+        if (!between(0, lineT, 1)) {
+            continue;
+        }
+        SkDPoint linePt = line.xyAtT(lineT);
+        if (linePt.approximatelyEqual(cubic[cIndex])) {
+            intersections.insert(cIndex >> 1, lineT, cubic[cIndex]);
+        }
     }
 }
 
diff --git a/src/pathops/SkDLineIntersection.cpp b/src/pathops/SkDLineIntersection.cpp
index 13c0dbbef4..3b88b88702 100644
--- a/src/pathops/SkDLineIntersection.cpp
+++ b/src/pathops/SkDLineIntersection.cpp
@@ -75,13 +75,51 @@ int SkIntersections::intersectRay(const SkDLine& a, const SkDLine& b) {
     return computePoints(a, used);
 }
 
-/*
-   Determine the intersection point of two line segments
-   Return FALSE if the lines don't intersect
-   from: http://paulbourke.net/geometry/lineline2d/
- */
+static bool checkEndPoint(double x, double y, const SkDLine& l, double* tPtr, int useX) {
+    if (!between(l[0].fX, x, l[1].fX) || !between(l[0].fY, y, l[1].fY)) {
+        return false;
+    }
+    double xLen = l[1].fX - l[0].fX;
+    double yLen = l[1].fY - l[0].fY;
+    if (useX < 0) {
+        useX = SkTAbs(xLen) > SkTAbs(yLen);
+    }
+    // OPTIMIZATION: do between test before divide
+    double t = useX ? (x - l[0].fX) / xLen : (y - l[0].fY) / yLen;
+    if (!between(0, t, 1)) {
+        return false;
+    }
+    double opp = useX ? (1 - t) * l[0].fY + t * l[1].fY : (1 - t) * l[0].fX + t * l[1].fX;
+    if (!AlmostEqualUlps(opp, useX ? y : x)) {
+        return false;
+    }
+    *tPtr = t;
+    return true;
+}
 
+// note that this only works if both lines are neither horizontal nor vertical
 int SkIntersections::intersect(const SkDLine& a, const SkDLine& b) {
+    // see if end points intersect the opposite line
+    double t;
+    for (int iA = 0; iA < 2; ++iA) {
+        if (!checkEndPoint(a[iA].fX, a[iA].fY, b, &t, -1)) {
+            continue;
+        }
+        insert(iA, t, a[iA]);
+    }
+    for (int iB = 0; iB < 2; ++iB) {
+        if (!checkEndPoint(b[iB].fX, b[iB].fY, a, &t, -1)) {
+            continue;
+        }
+        insert(t, iB, b[iB]);
+    }
+    if (used() > 0) {
+        SkASSERT(fUsed <= 2);
+        return used(); // coincident lines are returned here
+    }
+    /* Determine the intersection point of two line segments
+       Return FALSE if the lines don't intersect
+       from: http://paulbourke.net/geometry/lineline2d/ */
     double axLen = a[1].fX - a[0].fX;
     double ayLen = a[1].fY - a[0].fY;
     double bxLen = b[1].fX - b[0].fX;
@@ -105,64 +143,14 @@ int SkIntersections::intersect(const SkDLine& a, const SkDLine& b) {
             && !approximately_zero_inverse(numerB))) && !sk_double_isnan(numerA)
             && !sk_double_isnan(numerB)) {
         if (mayNotOverlap) {
-            return fUsed = 0;
+            return 0;
         }
         fT[0][0] = numerA;
         fT[1][0] = numerB;
         fPt[0] = a.xyAtT(numerA);
         return computePoints(a, 1);
     }
-   /* See if the axis intercepts match:
-              ay - ax * ayLen / axLen  ==          by - bx * ayLen / axLen
-     axLen * (ay - ax * ayLen / axLen) == axLen * (by - bx * ayLen / axLen)
-     axLen *  ay - ax * ayLen          == axLen *  by - bx * ayLen
-    */
-    if (!AlmostEqualUlps(axLen * a[0].fY - ayLen * a[0].fX,
-            axLen * b[0].fY - ayLen * b[0].fX)) {
-        return fUsed = 0;
-    }
-    const double* aPtr;
-    const double* bPtr;
-    if (fabs(axLen) > fabs(ayLen) || fabs(bxLen) > fabs(byLen)) {
-        aPtr = &a[0].fX;
-        bPtr = &b[0].fX;
-    } else {
-        aPtr = &a[0].fY;
-        bPtr = &b[0].fY;
-    }
-    double a0 = aPtr[0];
-    double a1 = aPtr[2];
-    double b0 = bPtr[0];
-    double b1 = bPtr[2];
-    // OPTIMIZATION: restructure to reject before the divide
-    // e.g., if ((a0 - b0) * (a0 - a1) < 0 || abs(a0 - b0) > abs(a0 - a1))
-    // (except efficient)
-    double aDenom = a0 - a1;
-    if (approximately_zero(aDenom)) {
-        if (!between(b0, a0, b1)) {
-            return fUsed = 0;
-        }
-        fT[0][0] = fT[0][1] = 0;
-    } else {
-        double at0 = (a0 - b0) / aDenom;
-        double at1 = (a0 - b1) / aDenom;
-        if ((at0 < 0 && at1 < 0) || (at0 > 1 && at1 > 1)) {
-            return fUsed = 0;
-        }
-        fT[0][0] = SkTMax(SkTMin(at0, 1.0), 0.0);
-        fT[0][1] = SkTMax(SkTMin(at1, 1.0), 0.0);
-    }
-    double bDenom = b0 - b1;
-    if (approximately_zero(bDenom)) {
-        fT[1][0] = fT[1][1] = 0;
-    } else {
-        int bIn = aDenom * bDenom < 0;
-        fT[1][bIn] = SkTMax(SkTMin((b0 - a0) / bDenom, 1.0), 0.0);
-        fT[1][!bIn] = SkTMax(SkTMin((b0 - a1) / bDenom, 1.0), 0.0);
-    }
-    bool second = fabs(fT[0][0] - fT[0][1]) > FLT_EPSILON;
-    SkASSERT((fabs(fT[1][0] - fT[1][1]) <= FLT_EPSILON) ^ second);
-    return computePoints(a, 1 + second);
+    return 0;
 }
 
 int SkIntersections::horizontal(const SkDLine& line, double y) {
@@ -174,7 +162,7 @@ int SkIntersections::horizontal(const SkDLine& line, double y) {
     if (min > y || max < y) {
         return fUsed = 0;
     }
-    if (AlmostEqualUlps(min, max)) {
+    if (AlmostEqualUlps(min, max) && max - min < fabs(line[0].fX - line[1].fX)) {
         fT[0][0] = 0;
         fT[0][1] = 1;
         return fUsed = 2;
@@ -183,42 +171,51 @@ int SkIntersections::horizontal(const SkDLine& line, double y) {
     return fUsed = 1;
 }
 
+static bool checkEndPointH(const SkDPoint& end, double left, double right,
+                           double y, bool flipped, double* tPtr) {
+    if (!between(left, end.fX, right) || !AlmostEqualUlps(y, end.fY)) {
+        return false;
+    }
+    double t = (end.fX - left) / (right - left);
+    SkASSERT(between(0, t, 1));
+    *tPtr = flipped ? 1 - t : t;
+    return true;
+}
+
 int SkIntersections::horizontal(const SkDLine& line, double left, double right,
                                 double y, bool flipped) {
-    int result = horizontal(line, y);
-    switch (result) {
-        case 0:
-            break;
-        case 1: {
-            double xIntercept = line[0].fX + fT[0][0] * (line[1].fX - line[0].fX);
-            if (!precisely_between(left, xIntercept, right)) {
-                return fUsed = 0;
-            }
-            fT[1][0] = (xIntercept - left) / (right - left);
-            break;
+    // see if end points intersect the opposite line
+    double t;
+    if (checkEndPoint(left, y, line, &t, true)) {
+        insert(t, flipped, left, y);
+    }
+    if (left != right) {
+        if (checkEndPoint(right, y, line, &t, true)) {
+            insert(t, !flipped, right, y);
         }
-        case 2:
-            double a0 = line[0].fX;
-            double a1 = line[1].fX;
-            double b0 = flipped ? right : left;
-            double b1 = flipped ? left : right;
-            // FIXME: share common code below
-            double at0 = (a0 - b0) / (a0 - a1);
-            double at1 = (a0 - b1) / (a0 - a1);
-            if ((at0 < 0 && at1 < 0) || (at0 > 1 && at1 > 1)) {
-                return fUsed = 0;
+        for (int index = 0; index < 2; ++index) {
+            if (!checkEndPointH(line[index], left, right, y, flipped, &t)) {
+                continue;
             }
-            fT[0][0] = SkTMax(SkTMin(at0, 1.0), 0.0);
-            fT[0][1] = SkTMax(SkTMin(at1, 1.0), 0.0);
-            int bIn = (a0 - a1) * (b0 - b1) < 0;
-            fT[1][bIn] = SkTMax(SkTMin((b0 - a0) / (b0 - b1), 1.0), 0.0);
-            fT[1][!bIn] = SkTMax(SkTMin((b0 - a1) / (b0 - b1), 1.0), 0.0);
-            bool second = fabs(fT[0][0] - fT[0][1]) > FLT_EPSILON;
-            SkASSERT((fabs(fT[1][0] - fT[1][1]) <= FLT_EPSILON) ^ second);
-            return computePoints(line, 1 + second);
+            insert(index, t, line[index]);
+        }
+    }
+    if (used() > 0) {
+        SkASSERT(fUsed <= 2);
+        return used(); // coincident lines are returned here
+    }
+    int result = horizontal(line, y);
+    if (!result) {
+        return 0;
+    }
+    SkASSERT(result == 1);
+    double xIntercept = line[0].fX + fT[0][0] * (line[1].fX - line[0].fX);
+    if (!precisely_between(left, xIntercept, right)) {
+        return fUsed = 0;
     }
+    fT[1][0] = (xIntercept - left) / (right - left);
     if (flipped) {
-        // OPTIMIZATION: instead of swapping, pass original line, use [1].fX - [0].fX
+        // OPTIMIZATION: ? instead of swapping, pass original line, use [1].fX - [0].fX
         for (int index = 0; index < result; ++index) {
             fT[1][index] = 1 - fT[1][index];
         }
@@ -244,40 +241,49 @@ int SkIntersections::vertical(const SkDLine& line, double x) {
     return fUsed = 1;
 }
 
+static bool checkEndPointV(const SkDPoint& end, double top, double bottom,
+                           double x, bool flipped, double* tPtr) {
+    if (!between(top, end.fY, bottom) || !AlmostEqualUlps(x, end.fX)) {
+        return false;
+    }
+    double t = (end.fY - top) / (bottom - top);
+    SkASSERT(between(0, t, 1));
+    *tPtr = flipped ? 1 - t : t;
+    return true;
+}
+
 int SkIntersections::vertical(const SkDLine& line, double top, double bottom,
-                              double x, bool flipped) {
-    int result = vertical(line, x);
-    switch (result) {
-        case 0:
-            break;
-        case 1: {
-            double yIntercept = line[0].fY + fT[0][0] * (line[1].fY - line[0].fY);
-            if (!precisely_between(top, yIntercept, bottom)) {
-                return fUsed = 0;
-            }
-            fT[1][0] = (yIntercept - top) / (bottom - top);
-            break;
+                                double x, bool flipped) {
+    // see if end points intersect the opposite line
+    double t;
+    if (checkEndPoint(x, top, line, &t, false)) {
+        insert(t, flipped, x, top);
+    }
+    if (top != bottom) {
+        if (checkEndPoint(x, bottom,line, &t, false)) {
+            insert(t, !flipped, x, bottom);
         }
-        case 2:
-            double a0 = line[0].fY;
-            double a1 = line[1].fY;
-            double b0 = flipped ? bottom : top;
-            double b1 = flipped ? top : bottom;
-            // FIXME: share common code above
-            double at0 = (a0 - b0) / (a0 - a1);
-            double at1 = (a0 - b1) / (a0 - a1);
-            if ((at0 < 0 && at1 < 0) || (at0 > 1 && at1 > 1)) {
-                return fUsed = 0;
+        for (int index = 0; index < 2; ++index) {
+            if (!checkEndPointV(line[index], top, bottom, x, flipped, &t)) {
+                continue;
             }
-            fT[0][0] = SkTMax(SkTMin(at0, 1.0), 0.0);
-            fT[0][1] = SkTMax(SkTMin(at1, 1.0), 0.0);
-            int bIn = (a0 - a1) * (b0 - b1) < 0;
-            fT[1][bIn] = SkTMax(SkTMin((b0 - a0) / (b0 - b1), 1.0), 0.0);
-            fT[1][!bIn] = SkTMax(SkTMin((b0 - a1) / (b0 - b1), 1.0), 0.0);
-            bool second = fabs(fT[0][0] - fT[0][1]) > FLT_EPSILON;
-            SkASSERT((fabs(fT[1][0] - fT[1][1]) <= FLT_EPSILON) ^ second);
-            return computePoints(line, 1 + second);
+            insert( index, t, line[index]);
+        }
+    }
+    if (used() > 0) {
+        SkASSERT(fUsed <= 2);
+        return used(); // coincident lines are returned here
+    }
+    int result = vertical(line, x);
+    if (!result) {
+        return 0;
+    }
+    SkASSERT(result == 1);
+    double yIntercept = line[0].fY + fT[0][0] * (line[1].fY - line[0].fY);
+    if (!precisely_between(top, yIntercept, bottom)) {
+        return fUsed = 0;
     }
+    fT[1][0] = (yIntercept - top) / (bottom - top);
     if (flipped) {
         // OPTIMIZATION: instead of swapping, pass original line, use [1].fY - [0].fY
         for (int index = 0; index < result; ++index) {
diff --git a/src/pathops/SkDQuadLineIntersection.cpp b/src/pathops/SkDQuadLineIntersection.cpp
index afaa1552e4..216e31f285 100644
--- a/src/pathops/SkDQuadLineIntersection.cpp
+++ b/src/pathops/SkDQuadLineIntersection.cpp
@@ -200,38 +200,57 @@ protected:
     // add endpoints first to get zero and one t values exactly
     void addEndPoints() {
         for (int qIndex = 0; qIndex < 3; qIndex += 2) {
+            bool foundEnd = false;
             for (int lIndex = 0; lIndex < 2; lIndex++) {
                 if (quad[qIndex] == line[lIndex]) {
                     intersections->insert(qIndex >> 1, lIndex, line[lIndex]);
+                    foundEnd = true;
                 }
             }
+            if (foundEnd) {
+                continue;
+            }
+            // See if the quad end touches the line. 
+            double dist = line.isLeft(quad[qIndex]); // this distance isn't cartesian
+            SkDVector lineLen = line[1] - line[0]; // the x/y magnitudes of the line
+            // compute the ULPS of the larger of the x/y deltas
+            double larger = SkTMax(SkTAbs(lineLen.fX), SkTAbs(lineLen.fY));
+            if (!RoughlyEqualUlps(larger, larger + dist)) { // is the dist within ULPS tolerance?
+                continue;
+            }
+            double lineT = findLineT(qIndex >> 1);
+            if (!between(0, lineT, 1)) {
+                continue;
+            }
+            SkDPoint linePt = line.xyAtT(lineT);
+            if (linePt.approximatelyEqual(quad[qIndex])) {
+                intersections->insert(qIndex >> 1, lineT, quad[qIndex]);
+            }
         }
     }
 
     void addHorizontalEndPoints(double left, double right, double y) {
         for (int qIndex = 0; qIndex < 3; qIndex += 2) {
-            if (quad[qIndex].fY != y) {
+            if (!AlmostEqualUlps(quad[qIndex].fY, y)) {
                 continue;
             }
-            if (quad[qIndex].fX == left) {
-                intersections->insert(qIndex >> 1, 0, quad[qIndex]);
-            }
-            if (quad[qIndex].fX == right) {
-                intersections->insert(qIndex >> 1, 1, quad[qIndex]);
+            double x = quad[qIndex].fX;
+            if (between(left, x, right)) {
+                double t = (x - left) / (right - left);
+                intersections->insert(qIndex >> 1, t, quad[qIndex]);
             }
         }
     }
 
     void addVerticalEndPoints(double top, double bottom, double x) {
         for (int qIndex = 0; qIndex < 3; qIndex += 2) {
-            if (quad[qIndex].fX != x) {
+            if (!AlmostEqualUlps(quad[qIndex].fX, x)) {
                 continue;
             }
-            if (quad[qIndex].fY == top) {
-                intersections->insert(qIndex >> 1, 0, quad[qIndex]);
-            }
-            if (quad[qIndex].fY == bottom) {
-                intersections->insert(qIndex >> 1, 1, quad[qIndex]);
+            double y = quad[qIndex].fY;
+            if (between(top, y, bottom)) {
+                double t = (y - top) / (bottom - top);
+                intersections->insert(qIndex >> 1, t, quad[qIndex]);
             }
         }
     }
@@ -240,10 +259,22 @@ protected:
         SkDPoint xy = quad.xyAtT(t);
         double dx = line[1].fX - line[0].fX;
         double dy = line[1].fY - line[0].fY;
+#if 0
         if (fabs(dx) > fabs(dy)) {
             return (xy.fX - line[0].fX) / dx;
         }
         return (xy.fY - line[0].fY) / dy;
+#else
+        double dxT = (xy.fX - line[0].fX) / dx;
+        double dyT = (xy.fY - line[0].fY) / dy;
+        if (!between(FLT_EPSILON, dxT, 1 - FLT_EPSILON) && between(0, dyT, 1)) {
+            return dyT;
+        }
+        if (!between(FLT_EPSILON, dyT, 1 - FLT_EPSILON) && between(0, dxT, 1)) {
+            return dxT;
+        }
+        return fabs(dx) > fabs(dy) ? dxT : dyT;
+#endif
     }
 
     static bool PinTs(double* quadT, double* lineT) {
diff --git a/src/pathops/SkIntersections.cpp b/src/pathops/SkIntersections.cpp
index ace8474d1d..af6cc080ef 100644
--- a/src/pathops/SkIntersections.cpp
+++ b/src/pathops/SkIntersections.cpp
@@ -56,10 +56,31 @@ void SkIntersections::flip() {
 
 void SkIntersections::insertCoincidentPair(double s1, double e1, double s2, double e2,
         const SkDPoint& startPt, const SkDPoint& endPt) {
-    if (fSwap) {
-        remove(s2, e2, startPt, endPt);
-    } else {
-        remove(s1, e1, startPt, endPt);
+    SkASSERT(s1 < e1);
+    SkASSERT(s2 != e2);
+    if (coincidentUsed() != fUsed) { // if the curve is partially coincident, treat it as fully so
+        for (int index = fUsed - 1; index >= 0; --index) {
+            if (fIsCoincident[0] & (1 << index)) {
+                continue;
+            }
+            double nonCoinT = fT[0][index];
+            if (!between(s1, nonCoinT, e1)) {
+                if (s1 > nonCoinT) {
+                    s1 = nonCoinT;
+                } else {
+                    e1 = nonCoinT;
+                }
+            }
+            nonCoinT = fT[1][index];
+            if (!between(s2, nonCoinT, e2)) {
+                if ((s2 > nonCoinT) ^ (s2 > e2)) {
+                    s2 = nonCoinT;
+                } else {
+                    e2 = nonCoinT;
+                }
+            }
+            removeOne(index);
+        }
     }
     SkASSERT(coincidentUsed() == fUsed);
     SkASSERT((coincidentUsed() & 1) != 1);
@@ -135,7 +156,7 @@ void SkIntersections::insertCoincidentPair(double s1, double e1, double s2, doub
     insertCoincident(e1, e2, endPt);
 }
 
-int SkIntersections::insert(double one, double two, const SkDPoint& pt) {
+int SkIntersections::insert(double one, double two, double x, double y) {
     if (fIsCoincident[0] == 3 && between(fT[0][0], one, fT[0][1])) {
         // For now, don't allow a mix of coincident and non-coincident intersections
         return -1;
@@ -152,7 +173,8 @@ int SkIntersections::insert(double one, double two, const SkDPoint& pt) {
                     || (precisely_equal(two, 1) && !precisely_equal(oldTwo, 1))) {
                 fT[0][index] = one;
                 fT[1][index] = two;
-                fPt[index] = pt;
+                fPt[index].fX = x;
+                fPt[index].fY = y;
             }
             return -1;
         }
@@ -174,13 +196,18 @@ int SkIntersections::insert(double one, double two, const SkDPoint& pt) {
         fIsCoincident[0] += fIsCoincident[0] & ~((1 << index) - 1);
         fIsCoincident[1] += fIsCoincident[1] & ~((1 << index) - 1);
     }
-    fPt[index] = pt;
+    fPt[index].fX = x;
+    fPt[index].fY = y;
     fT[0][index] = one;
     fT[1][index] = two;
     ++fUsed;
     return index;
 }
 
+int SkIntersections::insert(double one, double two, const SkDPoint& pt) {
+    return insert(one, two, pt.fX, pt.fY);
+}
+
 void SkIntersections::insertCoincident(double one, double two, const SkDPoint& pt) {
     int index = insertSwap(one, two, pt);
     int bit = 1 << index;
@@ -209,6 +236,7 @@ void SkIntersections::quickRemoveOne(int index, int replace) {
     }
 }
 
+#if 0
 void SkIntersections::remove(double one, double two, const SkDPoint& startPt,
         const SkDPoint& endPt) {
     for (int index = fUsed - 1; index >= 0; --index) {
@@ -220,6 +248,7 @@ void SkIntersections::remove(double one, double two, const SkDPoint& startPt,
         }
     }
 }
+#endif
 
 void SkIntersections::removeOne(int index) {
     int remaining = --fUsed - index;
diff --git a/src/pathops/SkIntersections.h b/src/pathops/SkIntersections.h
index 23470d7884..c0bb61fef0 100644
--- a/src/pathops/SkIntersections.h
+++ b/src/pathops/SkIntersections.h
@@ -96,6 +96,14 @@ public:
         }
     }
 
+    int insertSwap(double one, double two, double x, double y) {
+        if (fSwap) {
+            return insert(two, one, x, y);
+        } else {
+            return insert(one, two, x, y);
+        }
+    }
+
     bool isCoincident(int index) {
         return (fIsCoincident[0] & 1 << index) != 0;
     }
@@ -196,6 +204,7 @@ public:
     int horizontal(const SkDCubic&, double left, double right, double y, double tRange[3]);
     // FIXME : does not respect swap
     int insert(double one, double two, const SkDPoint& pt);
+    int insert(double one, double two, double x, double y);
     // start if index == 0 : end if index == 1
     void insertCoincident(double one, double two, const SkDPoint& pt);
     void insertCoincidentPair(double s1, double e1, double s2, double e2,
@@ -233,7 +242,7 @@ public:
 private:
     int computePoints(const SkDLine& line, int used);
     // used by addCoincident to remove ordinary intersections in range
-    void remove(double one, double two, const SkDPoint& startPt, const SkDPoint& endPt);
+ //   void remove(double one, double two, const SkDPoint& startPt, const SkDPoint& endPt);
 
     SkDPoint fPt[9];
     double fT[2][9];
diff --git a/src/pathops/SkOpAngle.cpp b/src/pathops/SkOpAngle.cpp
index 0ac5a3da78..0dd0d65f79 100644
--- a/src/pathops/SkOpAngle.cpp
+++ b/src/pathops/SkOpAngle.cpp
@@ -22,7 +22,7 @@ static const int bugChar = strlen(funcName) + 1;
 
 #if DEBUG_ANGLE
     static bool CompareResult(SkString* bugOut, const char* append, bool compare) {
-        bugOut->appendf(append);
+        bugOut->appendf("%s", append);
         bugOut->writable_str()[bugChar] = "><"[compare];
         SkDebugf("%s\n", bugOut->c_str());
         return compare;
@@ -141,7 +141,7 @@ bool SkOpAngle::operator<(const SkOpAngle& rh) const {  // this/lh: left-hand; r
         }
     }
     if (fSide * rh.fSide == 0) {
-        SkASSERT(fSide + rh.fSide != 0);
+        SkASSERT(fSide + rh.fSide != 0); // hitting this assert means coincidence was undetected
         return COMPARE_RESULT("9 fSide * rh.fSide == 0 ...", fSide < rh.fSide);
     }
     // at this point, the initial tangent line is nearly coincident
diff --git a/src/pathops/SkOpEdgeBuilder.cpp b/src/pathops/SkOpEdgeBuilder.cpp
index d7f52752bf..5187b5f1e6 100644
--- a/src/pathops/SkOpEdgeBuilder.cpp
+++ b/src/pathops/SkOpEdgeBuilder.cpp
@@ -4,6 +4,7 @@
  * Use of this source code is governed by a BSD-style license that can be
  * found in the LICENSE file.
  */
+#include "SkGeometry.h"
 #include "SkOpEdgeBuilder.h"
 #include "SkReduceOrder.h"
 
@@ -37,70 +38,114 @@ bool SkOpEdgeBuilder::finish() {
     if (fCurrentContour && !fCurrentContour->segments().count()) {
         fContours.pop_back();
     }
-    // correct pointers in contours since fReducePts may have moved as it grew
-    int cIndex = 0;
-    int extraCount = fExtra.count();
-    SkASSERT(extraCount == 0 || fExtra[0] == -1);
-    int eIndex = 0;
-    int rIndex = 0;
-    while (++eIndex < extraCount) {
-        int offset = fExtra[eIndex];
-        if (offset < 0) {
-            ++cIndex;
-            continue;
+    return true;
+}
+
+void SkOpEdgeBuilder::closeContour(const SkPoint& curveEnd, const SkPoint& curveStart) {
+    if ((!AlmostEqualUlps(curveEnd.fX, curveStart.fX)
+            || !AlmostEqualUlps(curveEnd.fY, curveStart.fY))) {
+        fPathVerbs.push_back(SkPath::kLine_Verb);
+        fPathPts.push_back_n(1, &curveStart);    
+    } else {
+        if (curveEnd.fX != curveStart.fX || curveEnd.fY != curveStart.fY) {
+            fPathPts[fPathPts.count() - 1] = curveStart;
+        } else {
+            fPathPts[fPathPts.count() - 1] = curveStart;
         }
-        fCurrentContour = &fContours[cIndex];
-        rIndex += fCurrentContour->updateSegment(offset - 1,
-                &fReducePts[rIndex]);
     }
-    fExtra.reset();  // we're done with this
-    return true;
+    fPathVerbs.push_back(SkPath::kClose_Verb);
 }
 
-// Note that copying the points here avoids copying the resulting path later.
-// To allow Op() to take one of the input paths as an output parameter, either the source data
-// must be copied (as implemented below) or the result must be copied.
-// OPTIMIZATION: This copies both sets of input points every time. If the input data was read
-// directly, the output path would only need to be copied if it was also one of the input paths.
 int SkOpEdgeBuilder::preFetch() {
     if (!fPath->isFinite()) {
         fUnparseable = true;
         return 0;
     }
+    SkAutoConicToQuads quadder;
+    const SkScalar quadderTol = SK_Scalar1 / 16;
     SkPath::RawIter iter(*fPath);
+    SkPoint curveStart;
+    SkPoint curve[4];
     SkPoint pts[4];
     SkPath::Verb verb;
+    bool lastCurve = false;
     do {
         verb = iter.next(pts);
-        fPathVerbs.push_back(verb);
-        if (verb == SkPath::kMove_Verb) {
-            fPathPts.push_back(pts[0]);
-        } else if (verb >= SkPath::kLine_Verb && verb <= SkPath::kCubic_Verb) {
-            fPathPts.push_back_n(SkPathOpsVerbToPoints(verb), &pts[1]);
+        switch (verb) {
+            case SkPath::kMove_Verb:
+                if (!fAllowOpenContours && lastCurve) {
+                    closeContour(curve[0], curveStart);
+                }
+                fPathVerbs.push_back(verb);
+                fPathPts.push_back(pts[0]);
+                curveStart = curve[0] = pts[0];
+                lastCurve = false;
+                continue;
+            case SkPath::kLine_Verb:
+                if (AlmostEqualUlps(curve[0].fX, pts[1].fX)
+                        && AlmostEqualUlps(curve[0].fY, pts[1].fY)) {
+                    continue;  // skip degenerate points
+                }
+                break;
+            case SkPath::kQuad_Verb:
+                curve[1] = pts[1];
+                curve[2] = pts[2];
+                verb = SkReduceOrder::Quad(curve, pts);
+                if (verb == SkPath::kMove_Verb) {
+                    continue;  // skip degenerate points
+                }
+                break;
+            case SkPath::kConic_Verb: {
+                    const SkPoint* quadPts = quadder.computeQuads(pts, iter.conicWeight(),
+                            quadderTol);
+                    const int nQuads = quadder.countQuads();
+                    for (int i = 0; i < nQuads; ++i) {
+                       fPathVerbs.push_back(SkPath::kQuad_Verb);
+                    }
+                    fPathPts.push_back_n(nQuads * 2, quadPts);
+                    curve[0] = quadPts[nQuads * 2 - 1];
+                    lastCurve = true;
+                }
+                continue;
+            case SkPath::kCubic_Verb:
+                curve[1] = pts[1];
+                curve[2] = pts[2];
+                curve[3] = pts[3];
+                verb = SkReduceOrder::Cubic(curve, pts);
+                if (verb == SkPath::kMove_Verb) {
+                    continue;  // skip degenerate points
+                }
+                break;
+            case SkPath::kClose_Verb:
+                closeContour(curve[0], curveStart);
+                lastCurve = false;
+                continue;
+            case SkPath::kDone_Verb:
+                continue;
         }
+        fPathVerbs.push_back(verb);
+        int ptCount = SkPathOpsVerbToPoints(verb);
+        fPathPts.push_back_n(ptCount, &pts[1]);
+        curve[0] = pts[ptCount];
+        lastCurve = true;
     } while (verb != SkPath::kDone_Verb);
+    if (!fAllowOpenContours && lastCurve) {
+        closeContour(curve[0], curveStart);
+    }
+    fPathVerbs.push_back(SkPath::kDone_Verb);
     return fPathVerbs.count() - 1;
 }
 
 bool SkOpEdgeBuilder::close() {
-    if (fFinalCurveStart && fFinalCurveEnd && *fFinalCurveStart != *fFinalCurveEnd) {
-        fReducePts.push_back(*fFinalCurveStart);
-        fReducePts.push_back(*fFinalCurveEnd);
-        const SkPoint* lineStart = fReducePts.end() - 2;
-        fExtra.push_back(fCurrentContour->addLine(lineStart));
-    }
     complete();
     return true;
 }
 
 bool SkOpEdgeBuilder::walk() {
-    SkPath::Verb reducedVerb;
     uint8_t* verbPtr = fPathVerbs.begin();
     uint8_t* endOfFirstHalf = &verbPtr[fSecondHalf];
-    const SkPoint* pointsPtr = fPathPts.begin();
+    const SkPoint* pointsPtr = fPathPts.begin() - 1;
     SkPath::Verb verb;
-    fFinalCurveStart = NULL;
-    fFinalCurveEnd = NULL;
     while ((verb = (SkPath::Verb) *verbPtr) != SkPath::kDone_Verb) {
         if (verbPtr == endOfFirstHalf) {
             fOperand = true;
@@ -119,49 +164,18 @@ bool SkOpEdgeBuilder::walk() {
                     fCurrentContour = fContours.push_back_n(1);
                     fCurrentContour->setOperand(fOperand);
                     fCurrentContour->setXor(fXorMask[fOperand] == kEvenOdd_PathOpsMask);
-                    fExtra.push_back(-1);  // start new contour
                 }
-                fFinalCurveEnd = pointsPtr++;
+                pointsPtr += 1;
                 continue;
-            case SkPath::kLine_Verb: {
-                const SkPoint& lineEnd = pointsPtr[0];
-                const SkPoint& lineStart = pointsPtr[-1];
-                // skip degenerate points
-                if (lineStart.fX != lineEnd.fX || lineStart.fY != lineEnd.fY) {
-                    fCurrentContour->addLine(&lineStart);
-                }
-                } break;
-            case SkPath::kQuad_Verb: {
-                const SkPoint* quadStart = &pointsPtr[-1];
-                reducedVerb = SkReduceOrder::Quad(quadStart, &fReducePts);
-                if (reducedVerb == 0) {
-                    break;  // skip degenerate points
-                }
-                if (reducedVerb == SkPath::kLine_Verb) {
-                    const SkPoint* lineStart = fReducePts.end() - 2;
-                    fExtra.push_back(fCurrentContour->addLine(lineStart));
-                    break;
-                }
-                fCurrentContour->addQuad(quadStart);
-                } break;
-            case SkPath::kCubic_Verb: {
-                const SkPoint* cubicStart = &pointsPtr[-1];
-                reducedVerb = SkReduceOrder::Cubic(cubicStart, &fReducePts);
-                if (reducedVerb == 0) {
-                    break;  // skip degenerate points
-                }
-                if (reducedVerb == SkPath::kLine_Verb) {
-                    const SkPoint* lineStart = fReducePts.end() - 2;
-                    fExtra.push_back(fCurrentContour->addLine(lineStart));
-                    break;
-                }
-                if (reducedVerb == SkPath::kQuad_Verb) {
-                    const SkPoint* quadStart = fReducePts.end() - 3;
-                    fExtra.push_back(fCurrentContour->addQuad(quadStart));
-                    break;
-                }
-                fCurrentContour->addCubic(cubicStart);
-                } break;
+            case SkPath::kLine_Verb:
+                fCurrentContour->addLine(pointsPtr);
+                break;
+            case SkPath::kQuad_Verb:
+                fCurrentContour->addQuad(pointsPtr);
+                break;
+            case SkPath::kCubic_Verb:
+                fCurrentContour->addCubic(pointsPtr);
+                break;
             case SkPath::kClose_Verb:
                 SkASSERT(fCurrentContour);
                 if (!close()) {
@@ -172,7 +186,6 @@ bool SkOpEdgeBuilder::walk() {
                 SkDEBUGFAIL("bad verb");
                 return false;
         }
-        fFinalCurveStart = &pointsPtr[SkPathOpsVerbToPoints(verb) - 1];
         pointsPtr += SkPathOpsVerbToPoints(verb);
         SkASSERT(fCurrentContour);
     }
diff --git a/src/pathops/SkOpEdgeBuilder.h b/src/pathops/SkOpEdgeBuilder.h
index 2a2bf034e4..df0795b0c8 100644
--- a/src/pathops/SkOpEdgeBuilder.h
+++ b/src/pathops/SkOpEdgeBuilder.h
@@ -43,6 +43,7 @@ public:
     void init();
 
 private:
+    void closeContour(const SkPoint& curveEnd, const SkPoint& curveStart);
     bool close();
     int preFetch();
     bool walk();
@@ -52,11 +53,7 @@ private:
     SkTArray<uint8_t, true> fPathVerbs;
     SkOpContour* fCurrentContour;
     SkTArray<SkOpContour>& fContours;
-    SkTArray<SkPoint, true> fReducePts;  // segments created on the fly
-    SkTArray<int, true> fExtra;  // -1 marks new contour, > 0 offsets into contour
     SkPathOpsMask fXorMask[2];
-    const SkPoint* fFinalCurveStart;
-    const SkPoint* fFinalCurveEnd;
     int fSecondHalf;
     bool fOperand;
     bool fAllowOpenContours;
diff --git a/src/pathops/SkOpSegment.cpp b/src/pathops/SkOpSegment.cpp
index 08f4f7eace..54e44904f4 100644
--- a/src/pathops/SkOpSegment.cpp
+++ b/src/pathops/SkOpSegment.cpp
@@ -861,7 +861,7 @@ int SkOpSegment::computeSum(int startIndex, int endIndex, bool binary) {
     // FIXME?: Not sure if this sort must be ordered or if the relaxed ordering is OK ...
     bool sortable = SortAngles(angles, &sorted, SkOpSegment::kMustBeOrdered_SortAngleKind);
 #if DEBUG_SORT
-    sorted[0]->segment()->debugShowSort(__FUNCTION__, sorted, 0, 0, 0);
+    sorted[0]->segment()->debugShowSort(__FUNCTION__, sorted, 0, 0, 0, sortable);
 #endif
     if (!sortable) {
         return SK_MinS32;
@@ -896,7 +896,7 @@ int SkOpSegment::computeSum(int startIndex, int endIndex, bool binary) {
         winding += spanWinding;
     }
 #if DEBUG_SORT
-    base->debugShowSort(__FUNCTION__, sorted, firstIndex, winding, oWinding);
+    base->debugShowSort(__FUNCTION__, sorted, firstIndex, winding, oWinding, sortable);
 #endif
     int nextIndex = firstIndex + 1;
     int lastIndex = firstIndex != 0 ? firstIndex : angleCount;
@@ -1134,6 +1134,7 @@ SkOpSegment* SkOpSegment::findNextOp(SkTDArray<SkOpSpan*>* chase, int* nextStart
         while (precisely_zero(startT - other->fTs[*nextEnd].fT));
         SkASSERT(step < 0 ? *nextEnd >= 0 : *nextEnd < other->fTs.count());
         if (other->isTiny(SkMin32(*nextStart, *nextEnd))) {
+            *unsortable = true;
             return NULL;
         }
         return other;
@@ -1150,7 +1151,7 @@ SkOpSegment* SkOpSegment::findNextOp(SkTDArray<SkOpSpan*>* chase, int* nextStart
     int firstIndex = findStartingEdge(sorted, startIndex, end);
     SkASSERT(firstIndex >= 0);
 #if DEBUG_SORT
-    debugShowSort(__FUNCTION__, sorted, firstIndex);
+    debugShowSort(__FUNCTION__, sorted, firstIndex, sortable);
 #endif
     if (!sortable) {
         *unsortable = true;
@@ -1272,7 +1273,7 @@ SkOpSegment* SkOpSegment::findNextWinding(SkTDArray<SkOpSpan*>* chase, int* next
     int firstIndex = findStartingEdge(sorted, startIndex, end);
     SkASSERT(firstIndex >= 0);
 #if DEBUG_SORT
-    debugShowSort(__FUNCTION__, sorted, firstIndex);
+    debugShowSort(__FUNCTION__, sorted, firstIndex, sortable);
 #endif
     if (!sortable) {
         *unsortable = true;
@@ -1400,7 +1401,8 @@ SkOpSegment* SkOpSegment::findNextXor(int* nextStart, int* nextEnd, bool* unsort
     if (!sortable) {
         *unsortable = true;
 #if DEBUG_SORT
-        debugShowSort(__FUNCTION__, sorted, findStartingEdge(sorted, startIndex, end), 0, 0);
+        debugShowSort(__FUNCTION__, sorted, findStartingEdge(sorted, startIndex, end), 0, 0,
+                sortable);
 #endif
         return NULL;
     }
@@ -1408,7 +1410,7 @@ SkOpSegment* SkOpSegment::findNextXor(int* nextStart, int* nextEnd, bool* unsort
     int firstIndex = findStartingEdge(sorted, startIndex, end);
     SkASSERT(firstIndex >= 0);
 #if DEBUG_SORT
-    debugShowSort(__FUNCTION__, sorted, firstIndex, 0, 0);
+    debugShowSort(__FUNCTION__, sorted, firstIndex, 0, 0, sortable);
 #endif
     SkASSERT(sorted[firstIndex]->segment() == this);
     int nextIndex = firstIndex + 1;
@@ -1654,7 +1656,7 @@ SkOpSegment* SkOpSegment::findTop(int* tIndexPtr, int* endIndexPtr, bool* unsort
     }
     SkASSERT(first < SK_MaxS32);
 #if DEBUG_SORT  // || DEBUG_SWAP_TOP
-    sorted[first]->segment()->debugShowSort(__FUNCTION__, sorted, first, 0, 0);
+    sorted[first]->segment()->debugShowSort(__FUNCTION__, sorted, first, 0, 0, sortable);
 #endif
     if (onlySortable && !sortable) {
         *unsortable = true;
@@ -2565,6 +2567,9 @@ int SkOpSegment::windingAtT(double tHit, int tIndex, bool crossOpp, SkScalar* dx
 #endif
         return SK_MinS32;
     }
+    if (windVal < 0) {  // reverse sign if opp contour traveled in reverse 
+            *dx = -*dx;
+    }
     if (winding * *dx > 0) {  // if same signs, result is negative
         winding += *dx > 0 ? -windVal : windVal;
     }
@@ -2769,12 +2774,12 @@ void SkOpSegment::debugShowNewWinding(const char* fun, const SkOpSpan& span, int
 #if DEBUG_SORT || DEBUG_SWAP_TOP
 void SkOpSegment::debugShowSort(const char* fun, const SkTArray<SkOpAngle*, true>& angles,
                                 int first, const int contourWinding,
-                                const int oppContourWinding) const {
+                                const int oppContourWinding, bool sortable) const {
     if (--gDebugSortCount < 0) {
         return;
     }
     SkASSERT(angles[first]->segment() == this);
-    SkASSERT(angles.count() > 1);
+    SkASSERT(!sortable || angles.count() > 1);
     int lastSum = contourWinding;
     int oppLastSum = oppContourWinding;
     const SkOpAngle* firstAngle = angles[first];
@@ -2878,12 +2883,12 @@ void SkOpSegment::debugShowSort(const char* fun, const SkTArray<SkOpAngle*, true
 }
 
 void SkOpSegment::debugShowSort(const char* fun, const SkTArray<SkOpAngle*, true>& angles,
-                                int first) {
+                                int first, bool sortable) {
     const SkOpAngle* firstAngle = angles[first];
     const SkOpSegment* segment = firstAngle->segment();
     int winding = segment->updateWinding(firstAngle);
     int oppWinding = segment->updateOppWinding(firstAngle);
-    debugShowSort(fun, angles, first, winding, oppWinding);
+    debugShowSort(fun, angles, first, winding, oppWinding, sortable);
 }
 
 #endif
diff --git a/src/pathops/SkOpSegment.h b/src/pathops/SkOpSegment.h
index 94efcb53fe..3cbd29e77e 100644
--- a/src/pathops/SkOpSegment.h
+++ b/src/pathops/SkOpSegment.h
@@ -318,8 +318,9 @@ public:
 #endif
 #if DEBUG_SORT || DEBUG_SWAP_TOP
     void debugShowSort(const char* fun, const SkTArray<SkOpAngle*, true>& angles, int first,
-            const int contourWinding, const int oppContourWinding) const;
-    void debugShowSort(const char* fun, const SkTArray<SkOpAngle*, true>& angles, int first);
+            const int contourWinding, const int oppContourWinding, bool sortable) const;
+    void debugShowSort(const char* fun, const SkTArray<SkOpAngle*, true>& angles, int first,
+            bool sortable);
 #endif
 #if DEBUG_CONCIDENT
     void debugShowTs() const;
diff --git a/src/pathops/SkPathOpsCommon.cpp b/src/pathops/SkPathOpsCommon.cpp
index 0fa5ce0c92..5a30c3a98e 100644
--- a/src/pathops/SkPathOpsCommon.cpp
+++ b/src/pathops/SkPathOpsCommon.cpp
@@ -138,7 +138,7 @@ SkOpSegment* FindChase(SkTDArray<SkOpSpan*>& chase, int& tIndex, int& endIndex)
                 SkOpSegment::kMayBeUnordered_SortAngleKind);
         int angleCount = sorted.count();
 #if DEBUG_SORT
-        sorted[0]->segment()->debugShowSort(__FUNCTION__, sorted, 0, 0, 0);
+        sorted[0]->segment()->debugShowSort(__FUNCTION__, sorted, 0, 0, 0, sortable);
 #endif
         if (!sortable) {
             continue;
@@ -162,7 +162,7 @@ SkOpSegment* FindChase(SkTDArray<SkOpSpan*>& chase, int& tIndex, int& endIndex)
             winding += spanWinding;
         }
     #if DEBUG_SORT
-        segment->debugShowSort(__FUNCTION__, sorted, firstIndex, winding, 0);
+        segment->debugShowSort(__FUNCTION__, sorted, firstIndex, winding, 0, sortable);
     #endif
         // we care about first sign and whether wind sum indicates this
         // edge is inside or outside. Maybe need to pass span winding
diff --git a/src/pathops/SkPathOpsDebug.cpp b/src/pathops/SkPathOpsDebug.cpp
index 6c8ca954f1..1071b52050 100644
--- a/src/pathops/SkPathOpsDebug.cpp
+++ b/src/pathops/SkPathOpsDebug.cpp
@@ -61,68 +61,29 @@ int gDebugSortCount;
 const char* kPathOpStr[] = {"diff", "sect", "union", "xor"};
 #endif
 
-#if DEBUG_SHOW_PATH
-static void showPathContours(SkPath::Iter& iter, const char* pathName) {
-    uint8_t verb;
-    SkPoint pts[4];
-    while ((verb = iter.next(pts)) != SkPath::kDone_Verb) {
-        switch (verb) {
-            case SkPath::kMove_Verb:
-                SkDebugf("    %s.moveTo(%#1.9gf, %#1.9gf);\n", pathName, pts[0].fX, pts[0].fY);
-                continue;
-            case SkPath::kLine_Verb:
-                SkDebugf("    %s.lineTo(%#1.9gf, %#1.9gf);\n", pathName, pts[1].fX, pts[1].fY);
-                break;
-            case SkPath::kQuad_Verb:
-                SkDebugf("    %s.quadTo(%#1.9gf, %#1.9gf, %#1.9gf, %#1.9gf);\n", pathName,
-                    pts[1].fX, pts[1].fY, pts[2].fX, pts[2].fY);
-                break;
-            case SkPath::kCubic_Verb:
-                SkDebugf("    %s.cubicTo(%#1.9gf, %#1.9gf, %#1.9gf, %#1.9gf, %#1.9gf, %#1.9gf);\n",
-                    pathName, pts[1].fX, pts[1].fY, pts[2].fX, pts[2].fY, pts[3].fX, pts[3].fY);
-                break;
-            case SkPath::kClose_Verb:
-                SkDebugf("    %s.close();\n", pathName);
-                break;
-            default:
-                SkDEBUGFAIL("bad verb");
-                return;
-        }
-    }
-}
-
-static const char* gFillTypeStr[] = {
-    "kWinding_FillType",
-    "kEvenOdd_FillType",
-    "kInverseWinding_FillType",
-    "kInverseEvenOdd_FillType"
-};
-
-
-void ShowFunctionHeader() {
-    SkDebugf("\nstatic void test#(skiatest::Reporter* reporter) {\n");
+#if DEBUG_SHOW_TEST_NAME
+void* PathOpsDebugCreateNameStr() {
+    return SkNEW_ARRAY(char, DEBUG_FILENAME_STRING_LENGTH);
 }
 
-void ShowPath(const SkPath& path, const char* pathName) {
-    SkPath::Iter iter(path, true);
-    SkPath::FillType fillType = path.getFillType();
-    SkASSERT(fillType >= SkPath::kWinding_FillType && fillType <= SkPath::kInverseEvenOdd_FillType);
-    SkDebugf("    SkPath %s;\n", pathName);
-    SkDebugf("    %s.setFillType(SkPath::%s);\n", pathName, gFillTypeStr[fillType]);
-    iter.setPath(path, true);
-    showPathContours(iter, pathName);
+void PathOpsDebugDeleteNameStr(void* v) {
+    SkDELETE_ARRAY(reinterpret_cast<char* >(v));
 }
 
-static const char* gOpStrs[] = {
-    "kDifference_PathOp",
-    "kIntersect_PathOp",
-    "kUnion_PathOp",
-    "kXor_PathOp",
-    "kReverseDifference_PathOp",
-};
-
-void ShowOp(SkPathOp op, const char* pathOne, const char* pathTwo) {
-    SkDebugf("    testPathOp(reporter, %s, %s, %s);\n", pathOne, pathTwo, gOpStrs[op]);
-    SkDebugf("}\n");
+void DebugBumpTestName(char* test) {
+    char* num = test + strlen(test);
+    while (num[-1] >= '0' && num[-1] <= '9') {
+        --num;
+    }
+    if (num[0] == '\0') {
+        return;
+    }
+    int dec = atoi(num);
+    if (dec == 0) {
+        return;
+    }
+    ++dec;
+    SK_SNPRINTF(num, DEBUG_FILENAME_STRING_LENGTH - (num - test), "%d", dec);
 }
 #endif
+
diff --git a/src/pathops/SkPathOpsDebug.h b/src/pathops/SkPathOpsDebug.h
index cc1b8ead95..5484147c3a 100644
--- a/src/pathops/SkPathOpsDebug.h
+++ b/src/pathops/SkPathOpsDebug.h
@@ -56,7 +56,6 @@ extern int gDebugMaxWindValue;
 #define DEBUG_FLOW 0
 #define DEBUG_MARK_DONE 0
 #define DEBUG_PATH_CONSTRUCTION 0
-#define DEBUG_SHOW_PATH 0
 #define DEBUG_SHOW_TEST_NAME 0
 #define DEBUG_SHOW_TEST_PROGRESS 0
 #define DEBUG_SHOW_WINDING 0
@@ -86,7 +85,6 @@ extern int gDebugMaxWindValue;
 #define DEBUG_FLOW 1
 #define DEBUG_MARK_DONE 1
 #define DEBUG_PATH_CONSTRUCTION 1
-#define DEBUG_SHOW_PATH 0
 #define DEBUG_SHOW_TEST_NAME 1
 #define DEBUG_SHOW_TEST_PROGRESS 1
 #define DEBUG_SHOW_WINDING 0
@@ -141,14 +139,20 @@ void winding_printf(int winding);
 extern const char* kPathOpStr[];
 #endif
 
-#ifndef DEBUG_TEST
-#define DEBUG_TEST 0
+#if DEBUG_SHOW_TEST_NAME
+#include "SkTLS.h"
+
+extern void* PathOpsDebugCreateNameStr();
+extern void PathOpsDebugDeleteNameStr(void* v);
+#define DEBUG_FILENAME_STRING_LENGTH 64
+#define DEBUG_FILENAME_STRING \
+    (reinterpret_cast<char* >(SkTLS::Get(PathOpsDebugCreateNameStr, PathOpsDebugDeleteNameStr)))
+extern void DebugBumpTestName(char* );
+extern void DebugShowPath(const SkPath& one, const SkPath& two, SkPathOp op, const char* name);
 #endif
 
-#if DEBUG_SHOW_PATH
-void ShowFunctionHeader();
-void ShowPath(const SkPath& path, const char* pathName);
-void ShowOp(SkPathOp op, const char* pathOne, const char* pathTwo);
+#ifndef DEBUG_TEST
+#define DEBUG_TEST 0
 #endif
 
 #endif
diff --git a/src/pathops/SkPathOpsOp.cpp b/src/pathops/SkPathOpsOp.cpp
index 7e1c772893..0df4859cdf 100644
--- a/src/pathops/SkPathOpsOp.cpp
+++ b/src/pathops/SkPathOpsOp.cpp
@@ -41,7 +41,7 @@ static SkOpSegment* findChaseOp(SkTDArray<SkOpSpan*>& chase, int& nextStart, int
                 SkOpSegment::kMayBeUnordered_SortAngleKind);
         int angleCount = sorted.count();
 #if DEBUG_SORT
-        sorted[0]->segment()->debugShowSort(__FUNCTION__, sorted, 0);
+        sorted[0]->segment()->debugShowSort(__FUNCTION__, sorted, 0, sortable);
 #endif
         if (!sortable) {
             continue;
@@ -54,7 +54,7 @@ static SkOpSegment* findChaseOp(SkTDArray<SkOpSpan*>& chase, int& nextStart, int
             segment = angle->segment();
         } while (segment->windSum(angle) == SK_MinS32);
     #if DEBUG_SORT
-        segment->debugShowSort(__FUNCTION__, sorted, firstIndex);
+        segment->debugShowSort(__FUNCTION__, sorted, firstIndex, sortable);
     #endif
         int sumMiWinding = segment->updateWindingReverse(angle);
         int sumSuWinding = segment->updateOppWindingReverse(angle);
@@ -232,11 +232,12 @@ static const bool gOutInverse[kReverseDifference_PathOp + 1][2][2] = {
 };
 
 bool Op(const SkPath& one, const SkPath& two, SkPathOp op, SkPath* result) {
-#if DEBUG_SHOW_PATH
-    ShowFunctionHeader();
-    ShowPath(one, "path");
-    ShowPath(two, "pathB");
-    ShowOp(op, "path", "pathB");
+#if DEBUG_SHOW_TEST_NAME
+    char* debugName = DEBUG_FILENAME_STRING;
+    if (debugName && debugName[0]) {
+        DebugBumpTestName(debugName);
+        DebugShowPath(one, two, op, debugName);
+    }
 #endif
     op = gOpInverse[op][one.isInverseFillType()][two.isInverseFillType()];
     SkPath::FillType fillType = gOutInverse[op][one.isInverseFillType()][two.isInverseFillType()]
diff --git a/src/pathops/SkPathOpsPoint.h b/src/pathops/SkPathOpsPoint.h
index 534154f199..ad959b6669 100644
--- a/src/pathops/SkPathOpsPoint.h
+++ b/src/pathops/SkPathOpsPoint.h
@@ -111,14 +111,8 @@ struct SkDPoint {
     }
 
     bool approximatelyEqual(const SkPoint& a) const {
-        double denom = SkTMax(fabs(fX), SkTMax(fabs(fY),
-                SkScalarToDouble(SkTMax(fabsf(a.fX), fabsf(a.fY)))));
-        if (denom == 0) {
-            return true;
-        }
-        double inv = 1 / denom;
-        return approximately_equal_double(fX * inv, a.fX * inv)
-                && approximately_equal_double(fY * inv, a.fY * inv);
+        return AlmostEqualUlps(SkDoubleToScalar(fX), a.fX)
+                && AlmostEqualUlps(SkDoubleToScalar(fY), a.fY);
     }
 
     bool approximatelyEqualHalf(const SkDPoint& a) const {
diff --git a/src/pathops/SkPathOpsTypes.cpp b/src/pathops/SkPathOpsTypes.cpp
index b071ca5371..999e1b215d 100644
--- a/src/pathops/SkPathOpsTypes.cpp
+++ b/src/pathops/SkPathOpsTypes.cpp
@@ -7,11 +7,11 @@
 #include "SkFloatBits.h"
 #include "SkPathOpsTypes.h"
 
-const int UlpsEpsilon = 16;
+
 
 // from http://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/
 // FIXME: move to SkFloatBits.h
-bool AlmostEqualUlps(float A, float B) {
+static bool equal_ulps(float A, float B, int epsilon) {
     SkFloatIntUnion floatIntA, floatIntB;
     floatIntA.fFloat = A;
     floatIntB.fFloat = B;
@@ -23,7 +23,17 @@ bool AlmostEqualUlps(float A, float B) {
     }
     // Find the difference in ULPs.
     int ulpsDiff = abs(floatIntA.fSignBitInt - floatIntB.fSignBitInt);
-    return ulpsDiff <= UlpsEpsilon;
+    return ulpsDiff <= epsilon;
+}
+
+bool AlmostEqualUlps(float A, float B) {
+    const int UlpsEpsilon = 16;
+    return equal_ulps(A, B, UlpsEpsilon);
+}
+
+bool RoughlyEqualUlps(float A, float B) {
+    const int UlpsEpsilon = 256;
+    return equal_ulps(A, B, UlpsEpsilon);
 }
 
 // cube root approximation using bit hack for 64-bit float
diff --git a/src/pathops/SkPathOpsTypes.h b/src/pathops/SkPathOpsTypes.h
index 6ead79455c..20641d3345 100644
--- a/src/pathops/SkPathOpsTypes.h
+++ b/src/pathops/SkPathOpsTypes.h
@@ -28,6 +28,11 @@ inline bool AlmostEqualUlps(double A, double B) {
     return AlmostEqualUlps(SkDoubleToScalar(A), SkDoubleToScalar(B));
 }
 
+bool RoughlyEqualUlps(float A, float B);
+inline bool RoughlyEqualUlps(double A, double B) {
+    return RoughlyEqualUlps(SkDoubleToScalar(A), SkDoubleToScalar(B));
+}
+
 // FLT_EPSILON == 1.19209290E-07 == 1 / (2 ^ 23)
 // DBL_EPSILON == 2.22045e-16
 const double FLT_EPSILON_CUBED = FLT_EPSILON * FLT_EPSILON * FLT_EPSILON;
diff --git a/src/pathops/SkReduceOrder.cpp b/src/pathops/SkReduceOrder.cpp
index ab85f3dd3e..3dfdc9daee 100644
--- a/src/pathops/SkReduceOrder.cpp
+++ b/src/pathops/SkReduceOrder.cpp
@@ -425,31 +425,27 @@ int SkReduceOrder::reduce(const SkDCubic& cubic, Quadratics allowQuadratics,
     return 4;
 }
 
-SkPath::Verb SkReduceOrder::Quad(const SkPoint a[3], SkTArray<SkPoint, true>* reducePts) {
+SkPath::Verb SkReduceOrder::Quad(const SkPoint a[3], SkPoint* reducePts) {
     SkDQuad quad;
     quad.set(a);
     SkReduceOrder reducer;
     int order = reducer.reduce(quad, kFill_Style);
     if (order == 2) {  // quad became line
         for (int index = 0; index < order; ++index) {
-            SkPoint& pt = reducePts->push_back();
-            pt.fX = SkDoubleToScalar(reducer.fLine[index].fX);
-            pt.fY = SkDoubleToScalar(reducer.fLine[index].fY);
+            *reducePts++ = reducer.fLine[index].asSkPoint();
         }
     }
     return SkPathOpsPointsToVerb(order - 1);
 }
 
-SkPath::Verb SkReduceOrder::Cubic(const SkPoint a[4], SkTArray<SkPoint, true>* reducePts) {
+SkPath::Verb SkReduceOrder::Cubic(const SkPoint a[4], SkPoint* reducePts) {
     SkDCubic cubic;
     cubic.set(a);
     SkReduceOrder reducer;
     int order = reducer.reduce(cubic, kAllow_Quadratics, kFill_Style);
     if (order == 2 || order == 3) {  // cubic became line or quad
         for (int index = 0; index < order; ++index) {
-            SkPoint& pt = reducePts->push_back();
-            pt.fX = SkDoubleToScalar(reducer.fQuad[index].fX);
-            pt.fY = SkDoubleToScalar(reducer.fQuad[index].fY);
+            *reducePts++ = reducer.fQuad[index].asSkPoint();
         }
     }
     return SkPathOpsPointsToVerb(order - 1);
diff --git a/src/pathops/SkReduceOrder.h b/src/pathops/SkReduceOrder.h
index 82f8ffb143..c167951f8b 100644
--- a/src/pathops/SkReduceOrder.h
+++ b/src/pathops/SkReduceOrder.h
@@ -27,8 +27,8 @@ union SkReduceOrder {
     int reduce(const SkDLine& line);
     int reduce(const SkDQuad& quad, Style);
 
-    static SkPath::Verb Cubic(const SkPoint pts[4], SkTArray<SkPoint, true>* reducePts);
-    static SkPath::Verb Quad(const SkPoint pts[3], SkTArray<SkPoint, true>* reducePts);
+    static SkPath::Verb Cubic(const SkPoint pts[4], SkPoint* reducePts);
+    static SkPath::Verb Quad(const SkPoint pts[3], SkPoint* reducePts);
 
     SkDLine fLine;
     SkDQuad fQuad;