shape ops work in progress

mostly working on cubic/cubic intersect

git-svn-id: http://skia.googlecode.com/svn/trunk@7266 2bbb7eff-a529-9590-31e7-b0007b416f81

36 files changed, 3674 insertions, 455 deletions

diff --git a/experimental/Intersection/CubicBezierClip.cpp b/experimental/Intersection/CubicBezierClip.cpp
index d0141e9413..825a7b961e 100644
--- a/experimental/Intersection/CubicBezierClip.cpp
+++ b/experimental/Intersection/CubicBezierClip.cpp
@@ -26,7 +26,8 @@ bool bezier_clip(const Cubic& cubic1, const Cubic& cubic2, double& minT, double&
     }
 
     double distance[2];
-    endLine.controlPtDistance(cubic1, distance);
+    distance[0] = endLine.controlPtDistance(cubic1, 1);
+    distance[1] = endLine.controlPtDistance(cubic1, 2);
 
     // find fat line
     double top = distance[0];
@@ -87,4 +88,3 @@ bool bezier_clip(const Cubic& cubic1, const Cubic& cubic2, double& minT, double&
 
     return minT < maxT; // returns false if distance shows no intersection
 }
-
diff --git a/experimental/Intersection/CubicIntersection.cpp b/experimental/Intersection/CubicIntersection.cpp
index fcd4119097..d5a5bdef78 100644
--- a/experimental/Intersection/CubicIntersection.cpp
+++ b/experimental/Intersection/CubicIntersection.cpp
@@ -159,3 +159,133 @@ bool intersect(const Cubic& c1, const Cubic& c2, Intersections& i) {
     return c.intersect();
 }
 
+#include "CubicUtilities.h"
+
+// this flavor approximates the cubics with quads to find the intersecting ts
+// OPTIMIZE: if this strategy proves successful, the quad approximations, or the ts used
+// to create the approximations, could be stored in the cubic segment
+// fixme: this strategy needs to add short line segments on either end, or similarly extend the 
+// initial and final quadratics
+bool intersect2(const Cubic& c1, const Cubic& c2, Intersections& i) {
+    SkTDArray<double> ts1;
+    double precision1 = calcPrecision(c1);
+    cubic_to_quadratics(c1, precision1, ts1);
+    SkTDArray<double> ts2;
+    double precision2 = calcPrecision(c2);
+    cubic_to_quadratics(c2, precision2, ts2);
+    double t1Start = 0;
+    int ts1Count = ts1.count();
+    for (int i1 = 0; i1 <= ts1Count; ++i1) {
+        const double t1 = i1 < ts1Count ? ts1[i1] : 1;
+        Cubic part1;
+        sub_divide(c1, t1Start, t1, part1);
+        Quadratic q1;
+        demote_cubic_to_quad(part1, q1);
+  //      start here;
+        // should reduceOrder be looser in this use case if quartic is going to blow up on an
+        // extremely shallow quadratic?
+        // maybe quadratics to lines need the same sort of recursive solution that I hope to find
+        // for cubics to quadratics ...
+        Quadratic s1;
+        int o1 = reduceOrder(q1, s1);
+        double t2Start = 0;
+        int ts2Count = ts2.count();
+        for (int i2 = 0; i2 <= ts2Count; ++i2) {
+            const double t2 = i2 < ts2Count ? ts2[i2] : 1;
+            Cubic part2;
+            sub_divide(c2, t2Start, t2, part2);
+            Quadratic q2;
+            demote_cubic_to_quad(part2, q2);
+            Quadratic s2;
+            double o2 = reduceOrder(q2, s2);
+            Intersections locals;
+            if (o1 == 3 && o2 == 3) {
+                intersect2(q1, q2, locals);
+            } else if (o1 <= 2 && o2 <= 2) {
+                i.fUsed = intersect((const _Line&) s1, (const _Line&) s2, i.fT[0], i.fT[1]);
+            } else if (o1 == 3 && o2 <= 2) {
+                intersect(q1, (const _Line&) s2, i);
+            } else {
+                SkASSERT(o1 <= 2 && o2 == 3);
+                intersect(q2, (const _Line&) s1, i);
+                for (int s = 0; s < i.fUsed; ++s) {
+                    SkTSwap(i.fT[0][s], i.fT[1][s]);
+                }
+            }
+            for (int tIdx = 0; tIdx < locals.used(); ++tIdx) {
+                double to1 = t1Start + (t1 - t1Start) * locals.fT[0][tIdx];
+                double to2 = t2Start + (t2 - t2Start) * locals.fT[1][tIdx];
+                i.insert(to1, to2);
+            }
+            t2Start = t2;
+        }
+        t1Start = t1;
+    }
+    return i.intersected();
+}
+
+int intersect(const Cubic& cubic, const Quadratic& quad, Intersections& i) {
+    SkTDArray<double> ts;
+    double precision = calcPrecision(cubic);
+    cubic_to_quadratics(cubic, precision, ts);
+    double tStart = 0;
+    Cubic part;
+    int tsCount = ts.count();
+    for (int idx = 0; idx <= tsCount; ++idx) {
+        double t = idx < tsCount ? ts[idx] : 1;
+        Quadratic q1;
+        sub_divide(cubic, tStart, t, part);
+        demote_cubic_to_quad(part, q1);
+        Intersections locals;
+        intersect2(q1, quad, locals);
+        for (int tIdx = 0; tIdx < locals.used(); ++tIdx) {
+            double globalT = tStart + (t - tStart) * locals.fT[0][tIdx];
+            i.insertOne(globalT, 0);
+            globalT = locals.fT[1][tIdx];
+            i.insertOne(globalT, 1);
+        }
+        tStart = t;
+    }
+    return i.used();
+}
+
+bool intersect(const Cubic& cubic, Intersections& i) {
+    SkTDArray<double> ts;
+    double precision = calcPrecision(cubic);
+    cubic_to_quadratics(cubic, precision, ts);
+    int tsCount = ts.count();
+    if (tsCount == 1) {
+        return false;
+    }
+    double t1Start = 0;
+    Cubic part;
+    for (int idx = 0; idx < tsCount; ++idx) {
+        double t1 = ts[idx];
+        Quadratic q1;
+        sub_divide(cubic, t1Start, t1, part);
+        demote_cubic_to_quad(part, q1);
+        double t2Start = t1;
+        for (int i2 = idx + 1; i2 <= tsCount; ++i2) {
+            const double t2 = i2 < tsCount ? ts[i2] : 1;
+            Quadratic q2;
+            sub_divide(cubic, t2Start, t2, part);
+            demote_cubic_to_quad(part, q2);
+            Intersections locals;
+            intersect2(q1, q2, locals);
+            for (int tIdx = 0; tIdx < locals.used(); ++tIdx) {
+            // discard intersections at cusp? (maximum curvature)
+                double t1sect = locals.fT[0][tIdx];
+                double t2sect = locals.fT[1][tIdx];
+                if (idx + 1 == i2 && t1sect == 1 && t2sect == 0) {
+                    continue;
+                }
+                double to1 = t1Start + (t1 - t1Start) * t1sect;
+                double to2 = t2Start + (t2 - t2Start) * t2sect;
+                i.insert(to1, to2);
+            }
+            t2Start = t2;
+        }
+        t1Start = t1;
+    }
+    return i.intersected();
+}
diff --git a/experimental/Intersection/CubicIntersection_Test.cpp b/experimental/Intersection/CubicIntersection_Test.cpp
index 256529c939..7d27bf40b4 100644
--- a/experimental/Intersection/CubicIntersection_Test.cpp
+++ b/experimental/Intersection/CubicIntersection_Test.cpp
@@ -56,3 +56,212 @@ void CubicIntersection_Test() {
         }
     }
 }
+
+static void oneOff(const Cubic& cubic1, const Cubic& cubic2) {
+    SkTDArray<Quadratic> quads1;
+    cubic_to_quadratics(cubic1, calcPrecision(cubic1), quads1);
+    for (int index = 0; index < quads1.count(); ++index) {
+        const Quadratic& q = quads1[index];
+        SkDebugf("{{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}},\n", q[0].x, q[0].y, 
+                 q[1].x, q[1].y,  q[2].x, q[2].y);
+    }
+    SkDebugf("\n");
+    SkTDArray<Quadratic> quads2;
+    cubic_to_quadratics(cubic2, calcPrecision(cubic2), quads2);
+    for (int index = 0; index < quads2.count(); ++index) {
+        const Quadratic& q = quads2[index];
+        SkDebugf("{{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}},\n", q[0].x, q[0].y, 
+                 q[1].x, q[1].y,  q[2].x, q[2].y);
+    }
+    SkDebugf("\n");
+    Intersections intersections2;
+    intersect2(cubic1, cubic2, intersections2);
+    for (int pt = 0; pt < intersections2.used(); ++pt) {
+        double tt1 = intersections2.fT[0][pt];
+        double tx1, ty1;
+        xy_at_t(cubic1, tt1, tx1, ty1);
+        int pt2 = intersections2.fFlip ? intersections2.used() - pt - 1 : pt;
+        double tt2 = intersections2.fT[1][pt2];
+        double tx2, ty2;
+        xy_at_t(cubic2, tt2, tx2, ty2);
+        SkDebugf("%s t1=%1.9g (%1.9g, %1.9g) (%1.9g, %1.9g) t2=%1.9g\n", __FUNCTION__,
+            tt1, tx1, ty1, tx2, ty2, tt2);
+    }
+}
+
+static const Cubic testSet[] = {
+{{67.426548091427676, 37.993772624988935}, {23.483695892376684, 90.476863174921306}, {35.597065061143162, 79.872482633158796}, {75.38634169631932, 18.244890038969412}},
+{{61.336508189019057, 82.693132843213675}, {44.639380902349664, 54.074825790745592}, {16.815615499771951, 20.049704667203923}, {41.866884958868326, 56.735503699973002}},
+
+{{67.4265481, 37.9937726}, {23.4836959, 90.4768632}, {35.5970651, 79.8724826}, {75.3863417, 18.24489}},
+{{61.3365082, 82.6931328}, {44.6393809, 54.0748258}, {16.8156155, 20.0497047}, {41.866885, 56.7355037}},
+
+{{18.1312339, 31.6473732}, {95.5711034, 63.5350219}, {92.3283165, 62.0158945}, {18.5656052, 32.1268808}},
+{{97.402018, 35.7169972}, {33.1127443, 25.8935163}, {1.13970027, 54.9424981}, {56.4860195, 60.529264}},
+};
+
+const size_t testSetCount = sizeof(testSet) / sizeof(testSet[0]);
+
+void CubicIntersection_OneOffTest() {
+    for (size_t outer = 0; outer < testSetCount - 1; ++outer) {
+        SkDebugf("%s quads1[%d]\n", __FUNCTION__, outer);
+        const Cubic& cubic1 = testSet[outer];
+        for (size_t inner = outer + 1; inner < testSetCount; ++inner) {
+        SkDebugf("%s quads2[%d]\n", __FUNCTION__, inner);
+            const Cubic& cubic2 = testSet[inner];
+            oneOff(cubic1, cubic2);
+        }
+    }
+}
+
+#define DEBUG_CRASH 1
+
+class CubicChopper {
+public:
+
+// only finds one intersection
+CubicChopper(const Cubic& c1, const Cubic& c2)
+    : cubic1(c1)
+    , cubic2(c2)
+    , depth(0) {
+}
+
+bool intersect(double minT1, double maxT1, double minT2, double maxT2) {
+    Cubic sub1, sub2;
+    // FIXME: carry last subdivide and reduceOrder result with cubic
+    sub_divide(cubic1, minT1, maxT1, sub1);
+    sub_divide(cubic2, minT2, maxT2, sub2);
+    Intersections i;
+    intersect2(sub1, sub2, i);
+    if (i.used() == 0) {
+        return false;
+    }
+    double x1, y1, x2, y2;
+    t1 = minT1 + i.fT[0][0] * (maxT1 - minT1);
+    t2 = minT2 + i.fT[1][0] * (maxT2 - minT2);
+    xy_at_t(cubic1, t1, x1, y1);
+    xy_at_t(cubic2, t2, x2, y2);
+    if (AlmostEqualUlps(x1, x2) && AlmostEqualUlps(y1, y2)) {
+        return true;
+    }
+    double half1 = (minT1 + maxT1) / 2;
+    double half2 = (minT2 + maxT2) / 2;
+    ++depth;
+    bool result;
+    if (depth & 1) {
+        result = intersect(minT1, half1, minT2, maxT2) || intersect(half1, maxT1, minT2, maxT2)
+            || intersect(minT1, maxT1, minT2, half2) || intersect(minT1, maxT1, half2, maxT2);
+    } else {
+        result = intersect(minT1, maxT1, minT2, half2) || intersect(minT1, maxT1, half2, maxT2)
+            || intersect(minT1, half1, minT2, maxT2) || intersect(half1, maxT1, minT2, maxT2);
+    }
+    --depth;
+    return result;
+}
+
+const Cubic& cubic1;
+const Cubic& cubic2;
+double t1;
+double t2;
+int depth;
+};
+
+#define TRY_OLD 0 // old way fails on test == 1
+
+void CubicIntersection_RandTest() {
+    srand(0);
+    const int tests = 1000000; // 10000000;
+    double largestFactor = DBL_MAX;
+    for (int test = 0; test < tests; ++test) {
+        Cubic cubic1, cubic2;
+        for (int i = 0; i < 4; ++i) {
+            cubic1[i].x = (double) rand() / RAND_MAX * 100;
+            cubic1[i].y = (double) rand() / RAND_MAX * 100;
+            cubic2[i].x = (double) rand() / RAND_MAX * 100;
+            cubic2[i].y = (double) rand() / RAND_MAX * 100;
+        }
+        if (test == 2513) { // the pair crosses three times, but the quadratic approximation
+            continue; // only sees one -- should be OK to ignore the other two?
+        }
+        if (test == 12932) { // this exposes a weakness when one cubic touches the other but
+            continue; // does not touch the quad approximation. Captured in qc.htm as cubic15
+        }
+    #if DEBUG_CRASH
+        char str[1024];
+        sprintf(str, "{{%1.9g, %1.9g}, {%1.9g, %1.9g}, {%1.9g, %1.9g}, {%1.9g, %1.9g}},\n"
+            "{{%1.9g, %1.9g}, {%1.9g, %1.9g}, {%1.9g, %1.9g}, {%1.9g, %1.9g}},\n",
+                cubic1[0].x, cubic1[0].y,  cubic1[1].x, cubic1[1].y, cubic1[2].x, cubic1[2].y,
+                cubic1[3].x, cubic1[3].y,
+                cubic2[0].x, cubic2[0].y,  cubic2[1].x, cubic2[1].y, cubic2[2].x, cubic2[2].y,
+                cubic2[3].x, cubic2[3].y);
+    #endif
+        _Rect rect1, rect2;
+        rect1.setBounds(cubic1);
+        rect2.setBounds(cubic2);
+        bool boundsIntersect = rect1.left <= rect2.right && rect2.left <= rect2.right
+                && rect1.top <= rect2.bottom && rect2.top <= rect1.bottom;
+        Intersections i1, i2;
+    #if TRY_OLD
+        bool oldIntersects = intersect(cubic1, cubic2, i1);
+    #else
+        bool oldIntersects = false;
+    #endif
+        if (test == -1) {
+            SkDebugf("ready...\n");
+        }
+        bool newIntersects = intersect2(cubic1, cubic2, i2);
+        if (!boundsIntersect && (oldIntersects || newIntersects)) {
+            SkDebugf("%s %d unexpected intersection boundsIntersect=%d oldIntersects=%d"
+                    " newIntersects=%d\n%s %s\n", __FUNCTION__, test, boundsIntersect,
+                    oldIntersects, newIntersects, __FUNCTION__, str);
+            assert(0);
+        }
+        if (oldIntersects && !newIntersects) {
+            SkDebugf("%s %d missing intersection oldIntersects=%d newIntersects=%d\n%s %s\n",
+                    __FUNCTION__, test, oldIntersects, newIntersects, __FUNCTION__, str);
+            assert(0);
+        }
+        if (!oldIntersects && !newIntersects) {
+            continue;
+        }
+        if (i2.used() > 1) {
+            continue;
+            // just look at single intercepts for simplicity
+        }
+        Intersections self1, self2; // self-intersect checks
+        if (intersect(cubic1, self1)) {
+            continue;
+        }
+        if (intersect(cubic2, self2)) {
+            continue;
+        }
+        // binary search for range necessary to enclose real intersection
+        CubicChopper c(cubic1, cubic2);
+        bool result = c.intersect(0, 1, 0, 1);
+        if (!result) {
+            // FIXME: a failure here probably means that a core routine used by CubicChopper is failing
+            continue;
+        }
+        double delta1 = fabs(c.t1 - i2.fT[0][0]);
+        double delta2 = fabs(c.t2 - i2.fT[1][0]);
+        double calc1 = calcPrecision(cubic1);
+        double calc2 = calcPrecision(cubic2);
+        double factor1 = calc1 / delta1;
+        double factor2 = calc2 / delta2;
+        SkDebugf("%s %d calc1=%1.9g delta1=%1.9g factor1=%1.9g calc2=%1.9g delta2=%1.9g"
+                " factor2=%1.9g\n", __FUNCTION__, test,
+                calc1, delta1, factor1, calc2, delta2, factor2);
+        if (factor1 < largestFactor) {
+            SkDebugf("WE HAVE A WINNER! %1.9g\n", factor1);
+            SkDebugf("%s\n", str);
+            oneOff(cubic1, cubic2);
+            largestFactor = factor1;
+        }
+        if (factor2 < largestFactor) {
+            SkDebugf("WE HAVE A WINNER! %1.9g\n", factor2);
+            SkDebugf("%s\n", str);
+            oneOff(cubic1, cubic2);
+            largestFactor = factor2;
+        }
+    }
+}
diff --git a/experimental/Intersection/CubicIntersection_TestData.cpp b/experimental/Intersection/CubicIntersection_TestData.cpp
index 645d7b04b9..c138a672da 100644
--- a/experimental/Intersection/CubicIntersection_TestData.cpp
+++ b/experimental/Intersection/CubicIntersection_TestData.cpp
@@ -4,7 +4,6 @@
  * Use of this source code is governed by a BSD-style license that can be
  * found in the LICENSE file.
  */
-#define IN_TEST 1
 #include "CubicIntersection_TestData.h"
 #include <limits>
 
diff --git a/experimental/Intersection/CubicIntersection_TestData.h b/experimental/Intersection/CubicIntersection_TestData.h
index 27436f8dc7..9893b3c1ac 100644
--- a/experimental/Intersection/CubicIntersection_TestData.h
+++ b/experimental/Intersection/CubicIntersection_TestData.h
@@ -4,11 +4,8 @@
  * Use of this source code is governed by a BSD-style license that can be
  * found in the LICENSE file.
  */
-#if !defined(IN_TEST)
-    #define IN_TEST 1
-#endif
-
 #include "DataTypes.h"
+#include "DataTypes_Test.h"
 
 extern const Cubic pointDegenerates[];
 extern const Cubic notPointDegenerates[];
diff --git a/experimental/Intersection/CubicReduceOrder.cpp b/experimental/Intersection/CubicReduceOrder.cpp
index d1775d89f2..fdc7265923 100644
--- a/experimental/Intersection/CubicReduceOrder.cpp
+++ b/experimental/Intersection/CubicReduceOrder.cpp
@@ -149,22 +149,17 @@ static int check_linear(const Cubic& cubic, Cubic& reduction,
 bool isLinear(const Cubic& cubic, int startIndex, int endIndex) {
     LineParameters lineParameters;
     lineParameters.cubicEndPoints(cubic, startIndex, endIndex);
-    double normalSquared = lineParameters.normalSquared();
-    double distance[2]; // distance is not normalized
+    // FIXME: maybe it's possible to avoid this and compare non-normalized
+    lineParameters.normalize();
     int mask = other_two(startIndex, endIndex);
     int inner1 = startIndex ^ mask;
     int inner2 = endIndex ^ mask;
-    lineParameters.controlPtDistance(cubic, inner1, inner2, distance);
-    double limit = normalSquared;
-    int index;
-    for (index = 0; index < 2; ++index) {
-        double distSq = distance[index];
-        distSq *= distSq;
-        if (approximately_greater(distSq, limit)) {
-            return false;
-        }
+    double distance = lineParameters.controlPtDistance(cubic, inner1);
+    if (!approximately_zero(distance)) {
+        return false;
     }
-    return true;
+    distance = lineParameters.controlPtDistance(cubic, inner2);
+    return approximately_zero(distance);
 }
 
 /* food for thought:
diff --git a/experimental/Intersection/CubicToQuadratics.cpp b/experimental/Intersection/CubicToQuadratics.cpp
index 424836c633..ff6d7a228b 100644
--- a/experimental/Intersection/CubicToQuadratics.cpp
+++ b/experimental/Intersection/CubicToQuadratics.cpp
@@ -40,11 +40,18 @@ Tdiv>=1 - the entire cubic can be approximated by the mid-point approximation
 
 confirmed by (maybe stolen from)
 http://www.caffeineowl.com/graphics/2d/vectorial/cubic2quad01.html
+// maybe in turn derived from  http://www.cccg.ca/proceedings/2004/36.pdf
+// also stored at http://www.cis.usouthal.edu/~hain/general/Publications/Bezier/bezier%20cccg04%20paper.pdf
 
 */
 
 #include "CubicUtilities.h"
 #include "CurveIntersection.h"
+#include "LineIntersection.h"
+
+const bool AVERAGE_END_POINTS = true; // results in better fitting curves
+
+#define USE_CUBIC_END_POINTS 1
 
 static double calcTDiv(const Cubic& cubic, double precision, double start) {
     const double adjust = sqrt(3) / 36;
@@ -62,54 +69,105 @@ static double calcTDiv(const Cubic& cubic, double precision, double start) {
     double dy = c[3].y - 3 * (c[2].y - c[1].y) - c[0].y;
     double dist = sqrt(dx * dx + dy * dy);
     double tDiv3 = precision / (adjust * dist);
-    return cube_root(tDiv3);
+    double t = cube_root(tDiv3);
+    if (start > 0) {
+        t = start + (1 - start) * t;
+    }
+    return t;
 }
 
-static void demote(const Cubic& cubic, Quadratic& quad) {
+void demote_cubic_to_quad(const Cubic& cubic, Quadratic& quad) {
     quad[0] = cubic[0];
-    quad[1].x = (cubic[3].x - 3 * (cubic[2].x - cubic[1].x) - cubic[0].x) / 4;
-    quad[1].y = (cubic[3].y - 3 * (cubic[2].y - cubic[1].y) - cubic[0].y) / 4;
+if (AVERAGE_END_POINTS) {
+    const _Point fromC1 = { (3 * cubic[1].x - cubic[0].x) / 2, (3 * cubic[1].y - cubic[0].y) / 2 };
+    const _Point fromC2 = { (3 * cubic[2].x - cubic[3].x) / 2, (3 * cubic[2].y - cubic[3].y) / 2 };
+    quad[1].x = (fromC1.x + fromC2.x) / 2;
+    quad[1].y = (fromC1.y + fromC2.y) / 2;
+} else {
+    lineIntersect((const _Line&) cubic[0], (const _Line&) cubic[2], quad[1]);
+}
     quad[2] = cubic[3];
 }
 
 int cubic_to_quadratics(const Cubic& cubic, double precision, SkTDArray<Quadratic>& quadratics) {
-    quadratics.setCount(1); // FIXME: every place I have setCount(), I also want setReserve()
+    SkTDArray<double> ts;
+    cubic_to_quadratics(cubic, precision, ts);
+    int tsCount = ts.count();
+    double t1Start = 0;
+    int order = 0;
+    for (int idx = 0; idx <= tsCount; ++idx) {
+        double t1 = idx < tsCount ? ts[idx] : 1;
+        Cubic part;
+        sub_divide(cubic, t1Start, t1, part);
+        Quadratic q1;
+        demote_cubic_to_quad(part, q1);
+        Quadratic s1;
+        int o1 = reduceOrder(q1, s1);
+        if (order < o1) {
+            order = o1;
+        }
+        memcpy(quadratics.append(), o1 < 2 ? s1 : q1, sizeof(Quadratic));
+        t1Start = t1;
+    }
+    return order;
+}
+
+static bool addSimpleTs(const Cubic& cubic, double precision, SkTDArray<double>& ts) {
+    double tDiv = calcTDiv(cubic, precision, 0);
+    if (tDiv >= 1) {
+        return true;
+    }
+    if (tDiv >= 0.5) {
+        *ts.append() = 0.5;
+        return true;
+    }
+    return false;
+}
+
+static void addTs(const Cubic& cubic, double precision, double start, double end,
+        SkTDArray<double>& ts) {
+    double tDiv = calcTDiv(cubic, precision, 0);
+    double parts = ceil(1.0 / tDiv);
+    for (double index = 0; index < parts; ++index) {
+        double newT = start + (index / parts) * (end - start);
+        if (newT > 0 && newT < 1) {
+            *ts.append() = newT;
+        }
+    }
+}
+
+// flavor that returns T values only, deferring computing the quads until they are needed
+void cubic_to_quadratics(const Cubic& cubic, double precision, SkTDArray<double>& ts) {
     Cubic reduced;
     int order = reduceOrder(cubic, reduced, kReduceOrder_QuadraticsAllowed);
     if (order < 3) {
-        memcpy(quadratics[0], reduced, order * sizeof(_Point));
-        return order;
+        return;
     }
-    double tDiv = calcTDiv(cubic, precision, 0);
-    if (approximately_greater_than_one(tDiv)) {
-        demote(cubic, quadratics[0]);
-        return 3;
+    double inflectT[2];
+    int inflections = find_cubic_inflections(cubic, inflectT);
+    SkASSERT(inflections <= 2);
+    if (inflections == 0 && addSimpleTs(cubic, precision, ts)) {
+        return;
     }
-    if (tDiv >= 0.5) {
+    if (inflections == 1) {
         CubicPair pair;
-        chop_at(cubic, pair, 0.5);
-        quadratics.setCount(2);
-        demote(pair.first(), quadratics[0]);
-        demote(pair.second(), quadratics[1]);
-        return 3;
+        chop_at(cubic, pair, inflectT[0]);
+        addTs(pair.first(), precision, 0, inflectT[0], ts);
+        addTs(pair.second(), precision, inflectT[0], 1, ts);
+        return;
     }
-    double start = 0;
-    int index = -1;
-    // if we iterate but make little progress, check to see if the curve is flat
-    // or if the control points are tangent to each other
-    do {
-        Cubic part;
-        sub_divide(cubic, start, tDiv, part);
-        quadratics.append();
-        demote(part, quadratics[++index]);
-        if (tDiv == 1) {
-            break;
+    if (inflections == 2) {
+        if (inflectT[0] > inflectT[1]) {
+            SkTSwap(inflectT[0], inflectT[1]);
         }
-        start = tDiv;
-        tDiv = calcTDiv(cubic, precision, start);
-        if (tDiv > 1) {
-            tDiv = 1;
-        }
-    } while (true);
-    return 3;
+        Cubic part;
+        sub_divide(cubic, 0, inflectT[0], part);
+        addTs(part, precision, 0, inflectT[0], ts);
+        sub_divide(cubic, inflectT[0], inflectT[1], part);
+        addTs(part, precision, inflectT[0], inflectT[1], ts);
+        sub_divide(cubic, inflectT[1], 1, part);
+        addTs(part, precision, inflectT[1], 1, ts);
+        return;
+    }
+    addTs(cubic, precision, 0, 1, ts);
 }
diff --git a/experimental/Intersection/CubicToQuadratics_Test.cpp b/experimental/Intersection/CubicToQuadratics_Test.cpp
index 24c0ffe03a..843833f92f 100644
--- a/experimental/Intersection/CubicToQuadratics_Test.cpp
+++ b/experimental/Intersection/CubicToQuadratics_Test.cpp
@@ -3,21 +3,14 @@
 #include "Intersection_Tests.h"
 #include "QuadraticIntersection_TestData.h"
 #include "TestUtilities.h"
+#include "SkGeometry.h"
 
-static double calcPrecision(const Cubic& cubic) {
-    _Rect dRect;
-    dRect.setBounds(cubic);
-    double width = dRect.right - dRect.left;
-    double height = dRect.bottom - dRect.top;
-    return (width > height ? width : height) / 256;
-}
-
-static void test(const Cubic(& cubics)[], const char* name, int firstTest, size_t testCount) {
+static void test(const Cubic* cubics, const char* name, int firstTest, size_t testCount) {
     SkTDArray<Quadratic> quads;
     for (size_t index = firstTest; index < testCount; ++index) {
         const Cubic& cubic = cubics[index];
         double precision = calcPrecision(cubic);
-        cubic_to_quadratics(cubic, precision, quads);
+        (void) cubic_to_quadratics(cubic, precision, quads);
         if (quads.count() != 1) {
             printf("%s [%d] cubic to quadratics failed count=%d\n", name, (int) index,
                     quads.count());
@@ -25,14 +18,14 @@ static void test(const Cubic(& cubics)[], const char* name, int firstTest, size_
     }
 }
 
-static void test(const Quadratic(& quadTests)[], const char* name, int firstTest, size_t testCount) {
+static void test(const Quadratic* quadTests, const char* name, int firstTest, size_t testCount) {
     SkTDArray<Quadratic> quads;
     for (size_t index = firstTest; index < testCount; ++index) {
         const Quadratic& quad = quadTests[index];
         Cubic cubic;
         quad_to_cubic(quad, cubic);
         double precision = calcPrecision(cubic);
-        cubic_to_quadratics(cubic, precision, quads);
+        (void) cubic_to_quadratics(cubic, precision, quads);
         if (quads.count() != 1) {
             printf("%s [%d] cubic to quadratics failed count=%d\n", name, (int) index,
                     quads.count());
@@ -40,7 +33,7 @@ static void test(const Quadratic(& quadTests)[], const char* name, int firstTest
     }
 }
 
-static void testC(const Cubic(& cubics)[], const char* name, int firstTest, size_t testCount) {
+static void testC(const Cubic* cubics, const char* name, int firstTest, size_t testCount) {
     SkTDArray<Quadratic> quads;
     // test if computed line end points are valid
     for (size_t index = firstTest; index < testCount; ++index) {
@@ -63,15 +56,15 @@ static void testC(const Cubic(& cubics)[], const char* name, int firstTest, size
     }
 }
 
-static void testC(const Cubic(& cubics)[][2], const char* name, int firstTest, size_t testCount) {
+static void testC(const Cubic(* cubics)[2], const char* name, int firstTest, size_t testCount) {
     SkTDArray<Quadratic> quads;
     for (size_t index = firstTest; index < testCount; ++index) {
         for (int idx2 = 0; idx2 < 2; ++idx2) {
             const Cubic& cubic = cubics[index][idx2];
             double precision = calcPrecision(cubic);
             int order = cubic_to_quadratics(cubic, precision, quads);
-            assert(order != 4);
-            if (order < 3) {
+        assert(order != 4);
+        if (order < 3) {
                 continue;
             }
             if (!AlmostEqualUlps(cubic[0].x, quads[0][0].x)
@@ -136,6 +129,8 @@ void CubicToQuadratics_Test() {
 }
 
 static Cubic locals[] = {
+ {{14.5975863, 41.632436}, {16.3518929, 26.2639684}, {18.5165519, 7.68775139}, {8.03767257, 89.1628526}},
+ {{69.7292014, 38.6877352}, {24.7648688, 23.1501713}, {84.9283191, 90.2588441}, {80.392774, 61.3533852}},
  {{
     60.776536520932126,
     71.249307306133829
@@ -148,39 +143,113 @@ static Cubic locals[] = {
   }, {
     45.261946574441133,
     17.536076632112298
-  }}
+  }},
 };
 
 static size_t localsCount = sizeof(locals) / sizeof(locals[0]);
 
+#define DEBUG_CRASH 0
+#define TEST_AVERAGE_END_POINTS 0 // must take const off to test
+extern const bool AVERAGE_END_POINTS;
+
 void CubicsToQuadratics_RandTest() {
     for (size_t x = 0; x < localsCount; ++x) {
         const Cubic& cubic = locals[x];
+        const SkPoint skcubic[4] = {{(float) cubic[0].x, (float) cubic[0].y},
+                {(float) cubic[1].x, (float) cubic[1].y}, {(float) cubic[2].x, (float) cubic[2].y},
+                {(float) cubic[3].x, (float) cubic[3].y}};
+        SkScalar skinflect[2];
+        int skin = SkFindCubicInflections(skcubic, skinflect);
+        SkDebugf("%s %d %1.9g\n", __FUNCTION__, skin, skinflect[0]);
         SkTDArray<Quadratic> quads;
         double precision = calcPrecision(cubic);
-        cubic_to_quadratics(cubic, precision, quads);
+        (void) cubic_to_quadratics(cubic, precision, quads);
+        SkDebugf("%s quads=%d\n", __FUNCTION__, quads.count());
     }
     srand(0);
-    const int arrayMax = 1000;
-    const int tests = 1000000;
+    const int arrayMax = 8;
+    const int sampleMax = 10;
+    const int tests = 1000000; // 10000000;
     int quadDist[arrayMax];
     bzero(quadDist, sizeof(quadDist));
+    Cubic samples[arrayMax][sampleMax];
+    int sampleCount[arrayMax];
+    bzero(sampleCount, sizeof(sampleCount));
     for (int x = 0; x < tests; ++x) {
         Cubic cubic;
         for (int i = 0; i < 4; ++i) {
             cubic[i].x = (double) rand() / RAND_MAX * 100;
             cubic[i].y = (double) rand() / RAND_MAX * 100;
         }
+    #if DEBUG_CRASH
+        char str[1024];
+        sprintf(str, "{{%1.9g, %1.9g}, {%1.9g, %1.9g}, {%1.9g, %1.9g}, {%1.9g, %1.9g}},\n",
+                cubic[0].x, cubic[0].y,  cubic[1].x, cubic[1].y, cubic[2].x, cubic[2].y,
+                cubic[3].x, cubic[3].y);
+    #endif
         SkTDArray<Quadratic> quads;
         double precision = calcPrecision(cubic);
-        cubic_to_quadratics(cubic, precision, quads);
-        assert(quads.count() < arrayMax);
-        quadDist[quads.count()]++;
+        (void) cubic_to_quadratics(cubic, precision, quads);
+        int count = quads.count();
+        assert(count > 0);
+        assert(--count < arrayMax);
+        quadDist[count]++;
+        int sCount = sampleCount[count];
+        if (sCount < sampleMax) {
+            memcpy(samples[count][sCount], cubic, sizeof(Cubic));
+            sampleCount[count]++;
+        }
     }
     for (int x = 0; x < arrayMax; ++x) {
         if (!quadDist[x]) {
             continue;
         }
-        printf("%d 1.9%g%%\n", x, (double) quadDist[x] / tests * 100);
+        SkDebugf("%d %1.9g%%\n", x + 1, (double) quadDist[x] / tests * 100);
+    }
+    SkDebugf("\n");
+    for (int x = 0; x < arrayMax; ++x) {
+        for (int y = 0; y < sampleCount[x]; ++y) {
+#if TEST_AVERAGE_END_POINTS
+            for (int w = 0; w < 2; ++w) {
+                AVERAGE_END_POINTS = w;
+#else
+                int w = 0;
+#endif
+                SkDebugf("<div id=\"cubic%dx%d%s\">\n", x + 1, y, w ? "x" : "");
+                const Cubic& cubic = samples[x][y];
+                SkDebugf("{{%1.9g, %1.9g}, {%1.9g, %1.9g}, {%1.9g, %1.9g}, {%1.9g, %1.9g}},\n",
+                    cubic[0].x, cubic[0].y,  cubic[1].x, cubic[1].y, cubic[2].x, cubic[2].y,
+                    cubic[3].x, cubic[3].y);
+                SkTDArray<Quadratic> quads;
+                double precision = calcPrecision(cubic);
+                (void) cubic_to_quadratics(cubic, precision, quads);
+                for (int z = 0; z < quads.count(); ++z) {
+                    const Quadratic& quad = quads[z];
+                    SkDebugf("{{%1.9g, %1.9g}, {%1.9g, %1.9g}, {%1.9g, %1.9g}},\n",
+                        quad[0].x, quad[0].y,  quad[1].x, quad[1].y, quad[2].x, quad[2].y);
+                }
+                SkDebugf("</div>\n\n");
+#if TEST_AVERAGE_END_POINTS
+            }
+#endif
+        }
+    }
+    SkDebugf("</div>\n\n");
+    SkDebugf("<script type=\"text/javascript\">\n\n");
+    SkDebugf("var testDivs = [\n");
+    for (int x = 0; x < arrayMax; ++x) {
+        for (int y = 0; y < sampleCount[x]; ++y) {
+#if TEST_AVERAGE_END_POINTS
+            for (int w = 0; w < 2; ++w) {
+#else
+            int w = 0;
+#endif
+                SkDebugf("    cubic%dx%d%s,\n", x + 1, y, w ? "x" : "");
+#if TEST_AVERAGE_END_POINTS
+            }
+#endif
+        }
     }
+    SkDebugf("\n\n\n");
+    SkDebugf("%s end\n", __FUNCTION__);
 }
diff --git a/experimental/Intersection/CubicUtilities.cpp b/experimental/Intersection/CubicUtilities.cpp
index 958cabb316..36aa9e8fde 100644
--- a/experimental/Intersection/CubicUtilities.cpp
+++ b/experimental/Intersection/CubicUtilities.cpp
@@ -7,6 +7,14 @@
 #include "CubicUtilities.h"
 #include "QuadraticUtilities.h"
 
+double calcPrecision(const Cubic& cubic) {
+    _Rect dRect;
+    dRect.setBounds(cubic);
+    double width = dRect.right - dRect.left;
+    double height = dRect.bottom - dRect.top;
+    return (width > height ? width : height) / 256;
+}
+
 void coefficients(const double* cubic, double& A, double& B, double& C, double& D) {
     A = cubic[6]; // d
     B = cubic[4] * 3; // 3*c
@@ -93,15 +101,6 @@ double derivativeAtT(const double* cubic, double t) {
     return (b - a) * one_t * one_t + 2 * (c - b) * t * one_t + (d - c) * t * t;
 }
 
-// same as derivativeAtT
-// which is more accurate? which is faster?
-double derivativeAtT_2(const double* cubic, double t) {
-    double a = cubic[2] - cubic[0];
-    double b = cubic[4] - 2 * cubic[2] + cubic[0];
-    double c = cubic[6] + 3 * (cubic[2] - cubic[4]) - cubic[0];
-    return c * c * t * t + 2 * b * t + a;
-}
-
 void dxdy_at_t(const Cubic& cubic, double t, double& dx, double& dy) {
     if (&dx) {
         dx = derivativeAtT(&cubic[0].x, t);
@@ -111,6 +110,17 @@ void dxdy_at_t(const Cubic& cubic, double t, double& dx, double& dy) {
     }
 }
 
+int find_cubic_inflections(const Cubic& src, double tValues[])
+{
+    double Ax = src[1].x - src[0].x;
+    double Ay = src[1].y - src[0].y;
+    double Bx = src[2].x - 2 * src[1].x + src[0].x;
+    double By = src[2].y - 2 * src[1].y + src[0].y;
+    double Cx = src[3].x + 3 * (src[1].x - src[2].x) - src[0].x;
+    double Cy = src[3].y + 3 * (src[1].y - src[2].y) - src[0].y;
+    return quadraticRoots(Bx * Cy - By * Cx, (Ax * Cy - Ay * Cx) / 2, Ax * By - Ay * Bx, tValues);
+}
+
 bool rotate(const Cubic& cubic, int zero, int index, Cubic& rotPath) {
     double dy = cubic[index].y - cubic[zero].y;
     double dx = cubic[index].x - cubic[zero].x;
diff --git a/experimental/Intersection/CubicUtilities.h b/experimental/Intersection/CubicUtilities.h
index 890ff00732..6159472f7e 100644
--- a/experimental/Intersection/CubicUtilities.h
+++ b/experimental/Intersection/CubicUtilities.h
@@ -10,15 +10,18 @@
 #include "DataTypes.h"
 #include "SkTDArray.h"
 
+double calcPrecision(const Cubic& cubic);
 double cube_root(double x);
 int cubic_to_quadratics(const Cubic& cubic, double precision,
         SkTDArray<Quadratic>& quadratics);
+void cubic_to_quadratics(const Cubic& cubic, double precision, SkTDArray<double>& ts);
 void coefficients(const double* cubic, double& A, double& B, double& C, double& D);
 int cubicRoots(double A, double B, double C, double D, double t[3]);
+int cubicRootsX(double A, double B, double C, double D, double s[3]);
+void demote_cubic_to_quad(const Cubic& cubic, Quadratic& quad);
 double derivativeAtT(const double* cubic, double t);
-// competing version that should produce same results
-double derivativeAtT_2(const double* cubic, double t);
 void dxdy_at_t(const Cubic& , double t, double& x, double& y);
+int find_cubic_inflections(const Cubic& src, double tValues[]);
 bool rotate(const Cubic& cubic, int zero, int index, Cubic& rotPath);
 double secondDerivativeAtT(const double* cubic, double t);
 void xy_at_t(const Cubic& , double t, double& x, double& y);
diff --git a/experimental/Intersection/CurveIntersection.h b/experimental/Intersection/CurveIntersection.h
index 664145b104..4b34b31adb 100644
--- a/experimental/Intersection/CurveIntersection.h
+++ b/experimental/Intersection/CurveIntersection.h
@@ -50,7 +50,11 @@ int horizontalIntersect(const Quadratic& quad, double left, double right,
 int horizontalIntersect(const Quadratic& quad, double left, double right,
         double y, bool flipped, Intersections& );
 bool intersect(const Cubic& cubic1, const Cubic& cubic2, Intersections& );
-int intersect(const Cubic& cubic, const _Line& line, double cRange[3], double lRange[3]);
+// the following flavor uses quadratic approximation instead of convex hulls
+bool intersect2(const Cubic& cubic1, const Cubic& cubic2, Intersections& );
+bool intersect(const Cubic& cubic, Intersections& i); // return true if cubic self-intersects
+int intersect(const Cubic& cubic, const Quadratic& quad, Intersections& );
+int intersect(const Cubic& cubic, const _Line& line, Intersections& );
 bool intersect(const Quadratic& q1, const Quadratic& q2, Intersections& );
 int intersect(const Quadratic& quad, const _Line& line, Intersections& );
 // the following flavor uses the implicit form instead of convex hulls
diff --git a/experimental/Intersection/DataTypes.cpp b/experimental/Intersection/DataTypes.cpp
index 2832ab2c85..2be8938e07 100644
--- a/experimental/Intersection/DataTypes.cpp
+++ b/experimental/Intersection/DataTypes.cpp
@@ -74,11 +74,4 @@ int UlpsDiff(float A, float B)
 
     return abs(uA.i - uB.i);
 }
-
-int FloatAsInt(float A)
-{
-    Float_t uA(A);
-    return uA.i;
-}
 #endif
-
diff --git a/experimental/Intersection/DataTypes.h b/experimental/Intersection/DataTypes.h
index 265e3b769c..56931d49bd 100644
--- a/experimental/Intersection/DataTypes.h
+++ b/experimental/Intersection/DataTypes.h
@@ -23,13 +23,9 @@ inline bool AlmostEqualUlps(double A, double B) { return AlmostEqualUlps((float)
 
 // FIXME: delete
 int UlpsDiff(float A, float B);
-int FloatAsInt(float A);
 
-#if defined(IN_TEST)
-// FIXME: move to test-only header
-const double PointEpsilon = 0.000001;
-const double SquaredEpsilon = PointEpsilon * PointEpsilon;
-#endif
+const double FLT_EPSILON_SQUARED = FLT_EPSILON * FLT_EPSILON;
+const double FLT_EPSILON_SQRT = sqrt(FLT_EPSILON);
 
 inline bool approximately_zero(double x) {
 
@@ -47,7 +43,11 @@ inline bool approximately_zero(float x) {
 }
 
 inline bool approximately_zero_squared(double x) {
-    return fabs(x) < FLT_EPSILON * FLT_EPSILON;
+    return fabs(x) < FLT_EPSILON_SQUARED;
+}
+
+inline bool approximately_zero_sqrt(double x) {
+    return fabs(x) < FLT_EPSILON_SQRT;
 }
 
 inline bool approximately_equal(double x, double y) {
@@ -107,7 +107,7 @@ inline bool approximately_positive(double x) {
 }
 
 inline bool approximately_positive_squared(double x) {
-    return x > -(FLT_EPSILON * FLT_EPSILON);
+    return x > -(FLT_EPSILON_SQUARED);
 }
 
 inline bool approximately_zero_or_more(double x) {
@@ -130,6 +130,11 @@ struct _Point {
     double x;
     double y;
 
+    friend _Point operator-(const _Point& a, const _Point& b) {
+        _Point v = {a.x - b.x, a.y - b.y};
+        return v;
+    }
+
     void operator-=(const _Point& v) {
         x -= v.x;
         y -= v.y;
@@ -150,7 +155,10 @@ struct _Point {
         return AlmostEqualUlps((float) x, (float) a.x)
                 && AlmostEqualUlps((float) y, (float) a.y);
     }
-
+    
+    double dot(const _Point& a) {
+        return x * a.x + y * a.y;
+    }
 };
 
 typedef _Point _Line[2];
diff --git a/experimental/Intersection/DataTypes_Test.h b/experimental/Intersection/DataTypes_Test.h
new file mode 100644
index 0000000000..a8f14fa112
--- /dev/null
+++ b/experimental/Intersection/DataTypes_Test.h
@@ -0,0 +1,7 @@
+#ifndef __DataTypes_Test_h__
+#define __DataTypes_Test_h__
+
+const double PointEpsilon = 0.000001;
+const double SquaredEpsilon = PointEpsilon * PointEpsilon;
+
+#endif
diff --git a/experimental/Intersection/EdgeWalker.cpp b/experimental/Intersection/EdgeWalker.cpp
index 12fe30da79..79334b41ed 100644
--- a/experimental/Intersection/EdgeWalker.cpp
+++ b/experimental/Intersection/EdgeWalker.cpp
@@ -80,7 +80,7 @@ static int CubicLineIntersect(const SkPoint a[2], const SkPoint b[3],
     const Cubic aCubic = {{a[0].fX, a[0].fY}, {a[1].fX, a[1].fY}, {a[2].fX, a[2].fY},
             {a[3].fX, a[3].fY}};
     const _Line bLine = {{b[0].fX, b[0].fY}, {b[1].fX, b[1].fY}};
-    return intersect(aCubic, bLine, intersections.fT[0], intersections.fT[1]);
+    return intersect(aCubic, bLine, intersections);
 }
 
 static int QuadIntersect(const SkPoint a[3], const SkPoint b[3],
diff --git a/experimental/Intersection/Intersection_Tests.cpp b/experimental/Intersection/Intersection_Tests.cpp
index 6bad957f31..8e6e63e0ec 100644
--- a/experimental/Intersection/Intersection_Tests.cpp
+++ b/experimental/Intersection/Intersection_Tests.cpp
@@ -15,9 +15,12 @@ void cubecode_test(int test);
 void Intersection_Tests() {
     int testsRun = 0;
 
+    QuadraticIntersection_Test();
+    CubicIntersection_OneOffTest();
+    CubicIntersection_RandTest();
+    SimplifyNew_Test();
     CubicsToQuadratics_RandTest();
     CubicToQuadratics_Test();
-    SimplifyNew_Test();
     Simplify4x4RectsThreaded_Test(testsRun);
     Simplify4x4QuadraticsThreaded_Test(testsRun);
     QuadLineIntersectThreaded_Test(testsRun);
@@ -26,7 +29,6 @@ void Intersection_Tests() {
     Simplify4x4QuadralateralsThreaded_Test(testsRun);
     ShapeOps4x4RectsThreaded_Test(testsRun);
     SkDebugf("%s total testsRun=%d\n", __FUNCTION__, testsRun);
-    QuadraticIntersection_Test();
     LineQuadraticIntersection_Test();
     MiniSimplify_Test();
     SimplifyAngle_Test();
diff --git a/experimental/Intersection/Intersection_Tests.h b/experimental/Intersection/Intersection_Tests.h
index 7da4e097f5..2ca1e056c8 100644
--- a/experimental/Intersection/Intersection_Tests.h
+++ b/experimental/Intersection/Intersection_Tests.h
@@ -4,16 +4,16 @@
  * Use of this source code is governed by a BSD-style license that can be
  * found in the LICENSE file.
  */
-#if !defined(IN_TEST)
-    #define IN_TEST 1
-#endif
+#include "DataTypes_Test.h"
 
 void ActiveEdge_Test();
 void ConvexHull_Test();
 void ConvexHull_X_Test();
 void CubicBezierClip_Test();
 void CubicCoincidence_Test();
+void CubicIntersection_OneOffTest();
 void CubicIntersection_Test();
+void CubicIntersection_RandTest();
 void CubicParameterization_Test();
 void CubicReduceOrder_Test();
 void CubicsToQuadratics_RandTest();
diff --git a/experimental/Intersection/Intersections.cpp b/experimental/Intersection/Intersections.cpp
index 3f4e8cf32b..d1e0f9bdb2 100644
--- a/experimental/Intersection/Intersections.cpp
+++ b/experimental/Intersection/Intersections.cpp
@@ -8,9 +8,108 @@
 #include "DataTypes.h"
 #include "Intersections.h"
 
+void Intersections::addCoincident(double s1, double e1, double s2, double e2) {
+    assert((fCoincidentUsed & 1) != 1);
+    for (int index = 0; index < fCoincidentUsed; index += 2) {
+        double cs1 = fCoincidentT[fSwap][index];
+        double ce1 = fCoincidentT[fSwap][index + 1];
+        bool s1in = approximately_between(cs1, s1, ce1);
+        bool e1in = approximately_between(cs1, e1, ce1);
+        double cs2 = fCoincidentT[fSwap ^ 1][index];
+        double ce2 = fCoincidentT[fSwap ^ 1][index + 1];
+        bool s2in = approximately_between(cs2, s2, ce2);
+        bool e2in = approximately_between(cs2, e2, ce2);
+        if ((s1in | e1in) & (s2in | e2in)) {
+            double lesser1 = std::min(cs1, ce1);
+            index += cs1 > ce1;
+            if (s1in < lesser1) {
+                fCoincidentT[fSwap][index] = s1in;
+            } else if (e1in < lesser1) {
+                fCoincidentT[fSwap][index] = e1in;
+            }
+            index ^= 1;
+            double greater1 = fCoincidentT[fSwap][index];
+            if (s1in > greater1) {
+                fCoincidentT[fSwap][index] = s1in;
+            } else if (e1in > greater1) {
+                fCoincidentT[fSwap][index] = e1in;
+            }
+            index &= ~1;
+            double lesser2 = std::min(cs2, ce2);
+            index += cs2 > ce2;
+            if (s2in < lesser2) {
+                fCoincidentT[fSwap ^ 1][index] = s2in;
+            } else if (e2in < lesser2) {
+                fCoincidentT[fSwap ^ 1][index] = e2in;
+            }
+            index ^= 1;
+            double greater2 = fCoincidentT[fSwap ^ 1][index];
+            if (s2in > greater2) {
+                fCoincidentT[fSwap ^ 1][index] = s2in;
+            } else if (e2in > greater2) {
+                fCoincidentT[fSwap ^ 1][index] = e2in;
+            }
+            return;
+        }
+    }
+    assert(fCoincidentUsed < 9);
+    fCoincidentT[fSwap][fCoincidentUsed] = s1;
+    fCoincidentT[fSwap ^ 1][fCoincidentUsed] = s2;
+    ++fCoincidentUsed;
+    fCoincidentT[fSwap][fCoincidentUsed] = e1;
+    fCoincidentT[fSwap ^ 1][fCoincidentUsed] = e2;
+    ++fCoincidentUsed;
+}
+
 void Intersections::cleanUp() {
     assert(fCoincidentUsed);
     assert(fUsed);
     // find any entries in fT that could be part of the coincident range
 
 }
+
+void Intersections::insert(double one, double two) {
+    assert(fUsed <= 1 || fT[0][0] < fT[0][1]);
+    int index;
+    for (index = 0; index < fUsed; ++index) {
+        if (approximately_equal(fT[0][index], one)
+                && approximately_equal(fT[1][index], two)) {
+            return;
+        }
+        if (fT[0][index] > one) {
+            break;
+        }
+    }
+    assert(fUsed < 9);
+    int remaining = fUsed - index;
+    if (remaining > 0) {
+        memmove(&fT[0][index + 1], &fT[0][index], sizeof(fT[0][0]) * remaining);
+        memmove(&fT[1][index + 1], &fT[1][index], sizeof(fT[1][0]) * remaining);
+    }
+    fT[0][index] = one;
+    fT[1][index] = two;
+    ++fUsed;
+}
+
+// FIXME: all callers should be moved to regular insert. Failures are likely
+// if two separate callers differ on whether ts are equal or not
+void Intersections::insertOne(double t, int side) {
+    int used = side ? fUsed2 : fUsed;
+    assert(used <= 1 || fT[side][0] < fT[side][1]);
+    int index;
+    for (index = 0; index < used; ++index) {
+        if (approximately_equal(fT[side][index], t)) {
+            return;
+        }
+        if (fT[side][index] > t) {
+            break;
+        }
+    }
+    assert(used < 9);
+    int remaining = used - index;
+    if (remaining > 0) {
+        memmove(&fT[side][index + 1], &fT[side][index], sizeof(fT[side][0]) * remaining);
+    }
+    fT[side][index] = t;
+    side ? ++fUsed2 : ++fUsed;
+}
diff --git a/experimental/Intersection/Intersections.h b/experimental/Intersection/Intersections.h
index fe12b25902..618c2c1b56 100644
--- a/experimental/Intersection/Intersections.h
+++ b/experimental/Intersection/Intersections.h
@@ -7,7 +7,8 @@
 #ifndef Intersections_DEFINE
 #define Intersections_DEFINE
 
-#include <algorithm> // for std::min
+#include <algorithm> // for std::min -- Skia doesn't have a SkMinDouble
+#include "SkTypes.h"
 
 class Intersections {
 public:
@@ -15,8 +16,9 @@ public:
         : fUsed(0)
         , fUsed2(0)
         , fCoincidentUsed(0)
+        , fFlip(false)
+        , fUnsortable(false)
         , fSwap(0)
-        , fFlip(0)
     {
         // OPTIMIZE: don't need to be initialized in release
         bzero(fT, sizeof(fT));
@@ -50,64 +52,13 @@ public:
         ++fCoincidentUsed;
     }
 
-    void addCoincident(double s1, double e1, double s2, double e2) {
-        assert((fCoincidentUsed & 1) != 1);
-        for (int index = 0; index < fCoincidentUsed; index += 2) {
-            double cs1 = fCoincidentT[fSwap][index];
-            double ce1 = fCoincidentT[fSwap][index + 1];
-            bool s1in = approximately_between(cs1, s1, ce1);
-            bool e1in = approximately_between(cs1, e1, ce1);
-            double cs2 = fCoincidentT[fSwap ^ 1][index];
-            double ce2 = fCoincidentT[fSwap ^ 1][index + 1];
-            bool s2in = approximately_between(cs2, s2, ce2);
-            bool e2in = approximately_between(cs2, e2, ce2);
-            if ((s1in | e1in) & (s2in | e2in)) {
-                double lesser1 = std::min(cs1, ce1);
-                index += cs1 > ce1;
-                if (s1in < lesser1) {
-                    fCoincidentT[fSwap][index] = s1in;
-                } else if (e1in < lesser1) {
-                    fCoincidentT[fSwap][index] = e1in;
-                }
-                index ^= 1;
-                double greater1 = fCoincidentT[fSwap][index];
-                if (s1in > greater1) {
-                    fCoincidentT[fSwap][index] = s1in;
-                } else if (e1in > greater1) {
-                    fCoincidentT[fSwap][index] = e1in;
-                }
-                index &= ~1;
-                double lesser2 = std::min(cs2, ce2);
-                index += cs2 > ce2;
-                if (s2in < lesser2) {
-                    fCoincidentT[fSwap ^ 1][index] = s2in;
-                } else if (e2in < lesser2) {
-                    fCoincidentT[fSwap ^ 1][index] = e2in;
-                }
-                index ^= 1;
-                double greater2 = fCoincidentT[fSwap ^ 1][index];
-                if (s2in > greater2) {
-                    fCoincidentT[fSwap ^ 1][index] = s2in;
-                } else if (e2in > greater2) {
-                    fCoincidentT[fSwap ^ 1][index] = e2in;
-                }
-                return;
-            }
-        }
-        assert(fCoincidentUsed < 9);
-        fCoincidentT[fSwap][fCoincidentUsed] = s1;
-        fCoincidentT[fSwap ^ 1][fCoincidentUsed] = s2;
-        ++fCoincidentUsed;
-        fCoincidentT[fSwap][fCoincidentUsed] = e1;
-        fCoincidentT[fSwap ^ 1][fCoincidentUsed] = e2;
-        ++fCoincidentUsed;
-    }
+    void addCoincident(double s1, double e1, double s2, double e2);
 
     // FIXME: this is necessary because curve/curve intersections are noisy
     // remove once curve/curve intersections are improved
     void cleanUp();
 
-    int coincidentUsed() {
+    int coincidentUsed() const{
         return fCoincidentUsed;
     }
 
@@ -120,51 +71,8 @@ public:
         }
     }
 
-    void insert(double one, double two) {
-        assert(fUsed <= 1 || fT[0][0] < fT[0][1]);
-        int index;
-        for (index = 0; index < fUsed; ++index) {
-            if (approximately_equal(fT[0][index], one)
-                    && approximately_equal(fT[1][index], two)) {
-                return;
-            }
-            if (fT[0][index] > one) {
-                break;
-            }
-        }
-        assert(fUsed < 9);
-        int remaining = fUsed - index;
-        if (remaining > 0) {
-            memmove(&fT[0][index + 1], &fT[0][index], sizeof(fT[0][0]) * remaining);
-            memmove(&fT[1][index + 1], &fT[1][index], sizeof(fT[1][0]) * remaining);
-        }
-        fT[0][index] = one;
-        fT[1][index] = two;
-        ++fUsed;
-    }
-
-    // FIXME: all callers should be moved to regular insert. Failures are likely
-    // if two separate callers differ on whether ts are equal or not
-    void insertOne(double t, int side) {
-        int used = side ? fUsed2 : fUsed;
-        assert(used <= 1 || fT[side][0] < fT[side][1]);
-        int index;
-        for (index = 0; index < used; ++index) {
-            if (approximately_equal(fT[side][index], t)) {
-                return;
-            }
-            if (fT[side][index] > t) {
-                break;
-            }
-        }
-        assert(used < 9);
-        int remaining = used - index;
-        if (remaining > 0) {
-            memmove(&fT[side][index + 1], &fT[side][index], sizeof(fT[side][0]) * remaining);
-        }
-        fT[side][index] = t;
-        side ? ++fUsed2 : ++fUsed;
-    }
+    void insert(double one, double two);
+    void insertOne(double t, int side);
 
     bool intersected() const {
         return fUsed > 0;
@@ -175,14 +83,25 @@ public:
     }
 
     void swap() {
-        fSwap ^= 1;
+        fSwap ^= true;
+    }
+    
+    void swapPts() {
+        int index;
+        for (index = 0; index < fUsed; ++index) {
+            SkTSwap(fT[0][index], fT[1][index]);
+        }
     }
 
-    bool swapped() {
+    bool swapped() const {
         return fSwap;
     }
 
-    int used() {
+    bool unsortable() const {
+        return fUnsortable;
+    }
+
+    int used() const {
         return fUsed;
     }
 
@@ -191,10 +110,10 @@ public:
     int fUsed;
     int fUsed2;
     int fCoincidentUsed;
-    int fFlip;
+    bool fFlip;
+    bool fUnsortable;
 private:
     int fSwap;
 };
 
 #endif
-
diff --git a/experimental/Intersection/LineCubicIntersection.cpp b/experimental/Intersection/LineCubicIntersection.cpp
index 91a0b7df92..333e30f0e3 100644
--- a/experimental/Intersection/LineCubicIntersection.cpp
+++ b/experimental/Intersection/LineCubicIntersection.cpp
@@ -77,160 +77,213 @@ So the cubic coefficients are:
 
  */
 
-class LineCubicIntersections : public Intersections {
+class LineCubicIntersections {
 public:
 
-LineCubicIntersections(const Cubic& c, const _Line& l, double r[3])
+LineCubicIntersections(const Cubic& c, const _Line& l, Intersections& i)
     : cubic(c)
     , line(l)
-    , range(r) {
+    , intersections(i) {
 }
 
-int intersect() {
-    double slope;
-    double axisIntercept;
-    moreHorizontal = implicitLine(line, slope, axisIntercept);
+// see parallel routine in line quadratic intersections
+int intersectRay(double roots[3]) {
+    double adj = line[1].x - line[0].x;
+    double opp = line[1].y - line[0].y;
+    Cubic r;
+    for (int n = 0; n < 4; ++n) {
+        r[n].x = (cubic[n].y - line[0].y) * adj - (cubic[n].x - line[0].x) * opp;
+    }
     double A, B, C, D;
-    coefficients(&cubic[0].x, A, B, C, D);
-    double E, F, G, H;
-    coefficients(&cubic[0].y, E, F, G, H);
-    if (moreHorizontal) {
-        A = A * slope - E;
-        B = B * slope - F;
-        C = C * slope - G;
-        D = D * slope - H + axisIntercept;
-    } else {
-        A = A - E * slope;
-        B = B - F * slope;
-        C = C - G * slope;
-        D = D - H * slope - axisIntercept;
-    }
-    return cubicRoots(A, B, C, D, range);
-}
-
-int horizontalIntersect(double axisIntercept) {
+    coefficients(&r[0].x, A, B, C, D);
+    return cubicRoots(A, B, C, D, roots);
+}
+
+int intersect() {
+    addEndPoints();
+    double rootVals[3];
+    int roots = intersectRay(rootVals);
+    for (int index = 0; index < roots; ++index) {
+        double cubicT = rootVals[index];
+        double lineT = findLineT(cubicT);
+        if (pinTs(cubicT, lineT)) {
+            intersections.insert(cubicT, lineT);
+        }
+    }
+    return intersections.fUsed;
+}
+
+int horizontalIntersect(double axisIntercept, double roots[3]) {
     double A, B, C, D;
     coefficients(&cubic[0].y, A, B, C, D);
     D -= axisIntercept;
-    return cubicRoots(A, B, C, D, range);
+    return cubicRoots(A, B, C, D, roots);
 }
 
-int verticalIntersect(double axisIntercept) {
+int horizontalIntersect(double axisIntercept, double left, double right, bool flipped) {
+    addHorizontalEndPoints(left, right, axisIntercept);
+    double rootVals[3];
+    int roots = horizontalIntersect(axisIntercept, rootVals);
+    for (int index = 0; index < roots; ++index) {
+        double x;
+        double cubicT = rootVals[index];
+        xy_at_t(cubic, cubicT, x, *(double*) NULL);
+        double lineT = (x - left) / (right - left);
+        if (pinTs(cubicT, lineT)) {
+            intersections.insert(cubicT, lineT);
+        }
+    }
+    if (flipped) {
+        flip();
+    }
+    return intersections.fUsed;
+}
+
+int verticalIntersect(double axisIntercept, double roots[3]) {
     double A, B, C, D;
     coefficients(&cubic[0].x, A, B, C, D);
     D -= axisIntercept;
-    return cubicRoots(A, B, C, D, range);
+    return cubicRoots(A, B, C, D, roots);
+}
+
+int verticalIntersect(double axisIntercept, double top, double bottom, bool flipped) {
+    addVerticalEndPoints(top, bottom, axisIntercept);
+    double rootVals[3];
+    int roots = verticalIntersect(axisIntercept, rootVals);
+    for (int index = 0; index < roots; ++index) {
+        double y;
+        double cubicT = rootVals[index];
+        xy_at_t(cubic, cubicT, *(double*) NULL, y);
+        double lineT = (y - top) / (bottom - top);
+        if (pinTs(cubicT, lineT)) {
+            intersections.insert(cubicT, lineT);
+        }
+    }
+    if (flipped) {
+        flip();
+    }
+    return intersections.fUsed;
+}
+
+protected:
+
+void addEndPoints()
+{
+    for (int cIndex = 0; cIndex < 4; cIndex += 3) {
+        for (int lIndex = 0; lIndex < 2; lIndex++) {
+            if (cubic[cIndex] == line[lIndex]) {
+                intersections.insert(cIndex >> 1, lIndex);
+            }
+        }
+    }
+}
+
+void addHorizontalEndPoints(double left, double right, double y)
+{
+    for (int cIndex = 0; cIndex < 4; cIndex += 3) {
+        if (cubic[cIndex].y != y) {
+            continue;
+        }
+        if (cubic[cIndex].x == left) {
+            intersections.insert(cIndex >> 1, 0);
+        }
+        if (cubic[cIndex].x == right) {
+            intersections.insert(cIndex >> 1, 1);
+        }
+    }
+}
+
+void addVerticalEndPoints(double top, double bottom, double x)
+{
+    for (int cIndex = 0; cIndex < 4; cIndex += 3) {
+        if (cubic[cIndex].x != x) {
+            continue;
+        }
+        if (cubic[cIndex].y == top) {
+            intersections.insert(cIndex >> 1, 0);
+        }
+        if (cubic[cIndex].y == bottom) {
+            intersections.insert(cIndex >> 1, 1);
+        }
+    }
 }
 
 double findLineT(double t) {
-    const double* cPtr;
-    const double* lPtr;
-    if (moreHorizontal) {
-        cPtr = &cubic[0].x;
-        lPtr = &line[0].x;
-    } else {
-        cPtr = &cubic[0].y;
-        lPtr = &line[0].y;
-    }
-    // FIXME: should fold the following in with TestUtilities.cpp xy_at_t()
-    double s = 1 - t;
-    double cubicVal = cPtr[0] * s * s * s + 3 * cPtr[2] * s * s * t
-                + 3 * cPtr[4] * s * t * t + cPtr[6] * t * t * t;
-    return (cubicVal - lPtr[0]) / (lPtr[2] - lPtr[0]);
+    double x, y;
+    xy_at_t(cubic, t, x, y);
+    double dx = line[1].x - line[0].x;
+    double dy = line[1].y - line[0].y;
+    if (fabs(dx) > fabs(dy)) {
+        return (x - line[0].x) / dx;
+    }
+    return (y - line[0].y) / dy;
+}
+
+void flip() {
+    // OPTIMIZATION: instead of swapping, pass original line, use [1].y - [0].y
+    int roots = intersections.fUsed;
+    for (int index = 0; index < roots; ++index) {
+        intersections.fT[1][index] = 1 - intersections.fT[1][index];
+    }
+}
+
+bool pinTs(double& cubicT, double& lineT) {
+    if (!approximately_one_or_less(lineT)) {
+        return false;
+    }
+    if (!approximately_zero_or_more(lineT)) {
+        return false;
+    }
+    if (cubicT < 0) {
+        cubicT = 0;
+    } else if (cubicT > 1) {
+        cubicT = 1;
+    }
+    if (lineT < 0) {
+        lineT = 0;
+    } else if (lineT > 1) {
+        lineT = 1;
+    }
+    return true;
 }
 
 private:
 
 const Cubic& cubic;
 const _Line& line;
-double* range;
-bool moreHorizontal;
-
+Intersections& intersections;
 };
 
-int horizontalIntersect(const Cubic& cubic, double y, double tRange[3]) {
-    LineCubicIntersections c(cubic, *((_Line*) 0), tRange);
-    return c.horizontalIntersect(y);
-}
-
 int horizontalIntersect(const Cubic& cubic, double left, double right, double y,
         double tRange[3]) {
-    LineCubicIntersections c(cubic, *((_Line*) 0), tRange);
-    int result = c.horizontalIntersect(y);
-    for (int index = 0; index < result; ) {
+    LineCubicIntersections c(cubic, *((_Line*) 0), *((Intersections*) 0));
+    double rootVals[3];
+    int result = c.horizontalIntersect(y, rootVals);
+    int tCount = 0;
+    for (int index = 0; index < result; ++index) {
         double x, y;
-        xy_at_t(cubic, tRange[index], x, y);
+        xy_at_t(cubic, rootVals[index], x, y);
         if (x < left || x > right) {
-            if (--result > index) {
-                tRange[index] = tRange[result];
-            }
             continue;
         }
-        ++index;
+        tRange[tCount++] = rootVals[index];
     }
     return result;
 }
 
 int horizontalIntersect(const Cubic& cubic, double left, double right, double y,
         bool flipped, Intersections& intersections) {
-    LineCubicIntersections c(cubic, *((_Line*) 0), intersections.fT[0]);
-    int result = c.horizontalIntersect(y);
-    for (int index = 0; index < result; ) {
-        double x, y;
-        xy_at_t(cubic, intersections.fT[0][index], x, y);
-        if (x < left || x > right) {
-            if (--result > index) {
-                intersections.fT[0][index] = intersections.fT[0][result];
-            }
-            continue;
-        }
-        intersections.fT[1][index] = (x - left) / (right - left);
-        ++index;
-    }
-    if (flipped) {
-        // OPTIMIZATION: instead of swapping, pass original line, use [1].x - [0].x
-        for (int index = 0; index < result; ++index) {
-            intersections.fT[1][index] = 1 - intersections.fT[1][index];
-        }
-    }
-    return result;
+    LineCubicIntersections c(cubic, *((_Line*) 0), intersections);
+    return c.horizontalIntersect(y, left, right, flipped);
 }
 
 int verticalIntersect(const Cubic& cubic, double top, double bottom, double x,
         bool flipped, Intersections& intersections) {
-    LineCubicIntersections c(cubic, *((_Line*) 0), intersections.fT[0]);
-    int result = c.verticalIntersect(x);
-    for (int index = 0; index < result; ) {
-        double x, y;
-        xy_at_t(cubic, intersections.fT[0][index], x, y);
-        if (y < top || y > bottom) {
-            if (--result > index) {
-                intersections.fT[1][index] = intersections.fT[0][result];
-            }
-            continue;
-        }
-        intersections.fT[0][index] = (y - top) / (bottom - top);
-        ++index;
-    }
-    if (flipped) {
-        // OPTIMIZATION: instead of swapping, pass original line, use [1].x - [0].x
-        for (int index = 0; index < result; ++index) {
-            intersections.fT[1][index] = 1 - intersections.fT[1][index];
-        }
-    }
-    return result;
+    LineCubicIntersections c(cubic, *((_Line*) 0), intersections);
+    return c.verticalIntersect(x, top, bottom, flipped);
 }
 
-int intersect(const Cubic& cubic, const _Line& line, double cRange[3], double lRange[3]) {
-    LineCubicIntersections c(cubic, line, cRange);
-    int roots;
-    if (AlmostEqualUlps(line[0].y, line[1].y)) {
-        roots = c.horizontalIntersect(line[0].y);
-    } else {
-        roots = c.intersect();
-    }
-    for (int index = 0; index < roots; ++index) {
-        lRange[index] = c.findLineT(cRange[index]);
-    }
-    return roots;
+int intersect(const Cubic& cubic, const _Line& line, Intersections& i) {
+    LineCubicIntersections c(cubic, line, i);
+    return c.intersect();
 }
diff --git a/experimental/Intersection/LineCubicIntersection_Test.cpp b/experimental/Intersection/LineCubicIntersection_Test.cpp
index 48754ddc1d..b81a87e421 100644
--- a/experimental/Intersection/LineCubicIntersection_Test.cpp
+++ b/experimental/Intersection/LineCubicIntersection_Test.cpp
@@ -36,8 +36,10 @@ void LineCubicIntersection_Test() {
             printf("[%d] line order=%d\n", (int) index, order2);
         }
         if (order1 == 4 && order2 == 2) {
-            double range1[2], range2[2];
-            int roots = intersect(reduce1, reduce2, range1, range2);
+            Intersections i;
+            double* range1 = i.fT[0];
+            double* range2 = i.fT[1];
+            int roots = intersect(reduce1, reduce2, i);
             for (int pt = 0; pt < roots; ++pt) {
                 double tt1 = range1[pt];
                 double tx1, ty1;
diff --git a/experimental/Intersection/LineIntersection.cpp b/experimental/Intersection/LineIntersection.cpp
index ab23c442ce..5308a26474 100644
--- a/experimental/Intersection/LineIntersection.cpp
+++ b/experimental/Intersection/LineIntersection.cpp
@@ -9,12 +9,28 @@
 #include "LineIntersection.h"
 #include <algorithm> // used for std::swap
 
+/* Determine the intersection point of two lines. This assumes the lines are not parallel,
+   and that that the lines are infinite.
+   From http://en.wikipedia.org/wiki/Line-line_intersection
+ */
+void lineIntersect(const _Line& a, const _Line& b, _Point& p) {
+    double axLen = a[1].x - a[0].x;
+    double ayLen = a[1].y - a[0].y;
+    double bxLen = b[1].x - b[0].x;
+    double byLen = b[1].y - b[0].y;
+    double denom = byLen * axLen - ayLen * bxLen;
+    assert(denom);
+    double term1 = a[1].x * a[0].y - a[1].y * a[0].x;
+    double term2 = b[1].x * b[0].y - b[1].y * b[0].x;
+    p.x = (term1 * bxLen - axLen * term2) / denom;
+    p.y = (term1 * byLen - ayLen * term2) / denom;
+}
 
 /*
    Determine the intersection point of two line segments
    Return FALSE if the lines don't intersect
    from: http://paulbourke.net/geometry/lineline2d/
-*/
+ */
 
 int intersect(const _Line& a, const _Line& b, double aRange[2], double bRange[2]) {
     double axLen = a[1].x - a[0].x;
@@ -27,7 +43,7 @@ int intersect(const _Line& a, const _Line& b, double aRange[2], double bRange[2]
              byLen  * axLen         ==  ayLen          * bxLen
              byLen  * axLen         -   ayLen          * bxLen == 0 ( == denom )
      */
-    double denom  = byLen * axLen - ayLen * bxLen;
+    double denom = byLen * axLen - ayLen * bxLen;
     if (approximately_zero(denom)) {
        /* See if the axis intercepts match:
                   ay - ax * ayLen / axLen  ==          by - bx * ayLen / axLen
diff --git a/experimental/Intersection/LineIntersection.h b/experimental/Intersection/LineIntersection.h
index 33076b68e9..adb578971a 100644
--- a/experimental/Intersection/LineIntersection.h
+++ b/experimental/Intersection/LineIntersection.h
@@ -12,9 +12,10 @@
 int horizontalIntersect(const _Line& line, double y, double tRange[2]);
 int horizontalLineIntersect(const _Line& line, double left, double right,
         double y, double tRange[2]);
-int verticalLineIntersect(const _Line& line, double top, double bottom,
-        double x, double tRange[2]);
+void lineIntersect(const _Line& a, const _Line& b, _Point& p);
 int intersect(const _Line& a, const _Line& b, double aRange[2], double bRange[2]);
 bool testIntersect(const _Line& a, const _Line& b);
+int verticalLineIntersect(const _Line& line, double top, double bottom,
+        double x, double tRange[2]);
 
 #endif
diff --git a/experimental/Intersection/LineParameters.h b/experimental/Intersection/LineParameters.h
index fc1bcc873a..8155ad8b4e 100644
--- a/experimental/Intersection/LineParameters.h
+++ b/experimental/Intersection/LineParameters.h
@@ -80,15 +80,9 @@ public:
         }
     }
 
-    void controlPtDistance(const Cubic& pts, double distance[2]) const {
-        for (int index = 0; index < 2; ++index) {
-            distance[index] = a * pts[index + 1].x + b * pts[index + 1].y + c;
-        }
-    }
-
-    void controlPtDistance(const Cubic& pts, int i, int j, double distance[2]) const {
-        distance[0] = a * pts[i].x + b * pts[i].y + c;
-        distance[1] = a * pts[j].x + b * pts[j].y + c;
+    double controlPtDistance(const Cubic& pts, int index) const {
+        assert(index == 1 || index == 2);
+        return a * pts[index].x + b * pts[index].y + c;
     }
 
     double controlPtDistance(const Quadratic& pts) const {
diff --git a/experimental/Intersection/LineParameteters_Test.cpp b/experimental/Intersection/LineParameteters_Test.cpp
index 6d853255ac..326477e792 100644
--- a/experimental/Intersection/LineParameteters_Test.cpp
+++ b/experimental/Intersection/LineParameteters_Test.cpp
@@ -45,7 +45,8 @@ void LineParameter_Test() {
         const Cubic& cubic = tests[index];
         lineParameters.cubicEndPoints(cubic);
         double denormalizedDistance[2];
-        lineParameters.controlPtDistance(cubic, denormalizedDistance);
+        denormalizedDistance[0] = lineParameters.controlPtDistance(cubic, 1);
+        denormalizedDistance[1] = lineParameters.controlPtDistance(cubic, 2);
         double normalSquared = lineParameters.normalSquared();
         size_t inner;
         for (inner = 0; inner < 2; ++inner) {
@@ -64,7 +65,8 @@ void LineParameter_Test() {
         }
         lineParameters.normalize();
         double normalizedDistance[2];
-        lineParameters.controlPtDistance(cubic, normalizedDistance);
+        normalizedDistance[0] = lineParameters.controlPtDistance(cubic, 1);
+        normalizedDistance[1] = lineParameters.controlPtDistance(cubic, 2);
         for (inner = 0; inner < 2; ++inner) {
             if (AlmostEqualUlps(fabs(normalizedDistance[inner]), answers[index][inner])) {
                 continue;
diff --git a/experimental/Intersection/QuadraticImplicit.cpp b/experimental/Intersection/QuadraticImplicit.cpp
index d892ae97e0..72a4186203 100644
--- a/experimental/Intersection/QuadraticImplicit.cpp
+++ b/experimental/Intersection/QuadraticImplicit.cpp
@@ -5,11 +5,15 @@
 // http://planetmath.org/encyclopedia/GaloisTheoreticDerivationOfTheQuarticFormula.html#step
 
 
+#include "CubicUtilities.h"
 #include "CurveIntersection.h"
 #include "Intersections.h"
 #include "QuadraticParameterization.h"
 #include "QuarticRoot.h"
 #include "QuadraticUtilities.h"
+#include "TSearch.h"
+
+#include <algorithm> // for std::min, max
 
 /* given the implicit form 0 = Ax^2 + Bxy + Cy^2 + Dx + Ey + F
  * and given x = at^2 + bt + c  (the parameterized form)
@@ -17,8 +21,15 @@
  * then
  * 0 = A(at^2+bt+c)(at^2+bt+c)+B(at^2+bt+c)(dt^2+et+f)+C(dt^2+et+f)(dt^2+et+f)+D(at^2+bt+c)+E(dt^2+et+f)+F
  */
+ 
+#if SK_DEBUG
+#define QUARTIC_DEBUG 1
+#else
+#define QUARTIC_DEBUG 0
+#endif
 
-static int findRoots(const QuadImplicitForm& i, const Quadratic& q2, double roots[4]) {
+static int findRoots(const QuadImplicitForm& i, const Quadratic& q2, double roots[4],
+        bool useCubic, bool& disregardCount) {
     double a, b, c;
     set_abc(&q2[0].x, a, b, c);
     double d, e, f;
@@ -45,6 +56,42 @@ static int findRoots(const QuadImplicitForm& i, const Quadratic& q2, double root
                     +     i.x()  *  c
                     +     i.y()  *  f
                     +     i.c();
+#if QUARTIC_DEBUG
+    // create a string mathematica understands
+    char str[1024];
+    bzero(str, sizeof(str));
+    sprintf(str, "Solve[%1.19g x^4 + %1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]",
+        t4, t3, t2, t1, t0);
+#endif
+    if (approximately_zero(t4)) {
+        disregardCount = true;
+        if (approximately_zero(t3)) {
+            return quadraticRootsX(t2, t1, t0, roots);
+        }
+        return cubicRootsX(t3, t2, t1, t0, roots);
+    }
+    if (approximately_zero(t0)) { // 0 is one root
+        disregardCount = true;
+        int num = cubicRootsX(t4, t3, t2, t1, roots);
+        for (int i = 0; i < num; ++i) {
+            if (approximately_zero(roots[i])) {
+                return num;
+            }
+        }
+        roots[num++] = 0;
+        return num;
+    }
+    if (useCubic) {
+        assert(approximately_zero(t4 + t3 + t2 + t1 + t0)); // 1 is one root
+        int num = cubicRootsX(t4, t4 + t3, -(t1 + t0), -t0, roots); // note that -C==A+B+D+E
+        for (int i = 0; i < num; ++i) {
+            if (approximately_equal(roots[i], 1)) {
+                return num;
+            }
+        }
+        roots[num++] = 1;
+        return num;
+    }
     return quarticRoots(t4, t3, t2, t1, t0, roots);
 }
 
@@ -83,7 +130,9 @@ static bool onlyEndPtsInCommon(const Quadratic& q1, const Quadratic& q2, Interse
         double adj = endPt[1]->x - origX;
         double opp = endPt[1]->y - origY;
         double sign = (q1[oddMan].y - origY) * adj - (q1[oddMan].x - origX) * opp;
-        assert(!approximately_zero(sign));
+        if (approximately_zero(sign)) {
+            goto tryNextHalfPlane;
+        }
         for (int n = 0; n < 3; ++n) {
             double test = (q2[n].y - origY) * adj - (q2[n].x - origX) * opp;
             if (test * sign > 0) {
@@ -105,12 +154,227 @@ tryNextHalfPlane:
     return false;
 }
 
+// http://www.blackpawn.com/texts/pointinpoly/default.html
+static bool pointInTriangle(const _Point& pt, const _Line* testLines[]) {
+    const _Point& A = (*testLines[0])[0];
+    const _Point& B = (*testLines[1])[0];
+    const _Point& C = (*testLines[2])[0];
+
+// Compute vectors        
+    _Point v0 = C - A;
+    _Point v1 = B - A;
+    _Point v2 = pt - A;
+
+// Compute dot products
+    double dot00 = v0.dot(v0);
+    double dot01 = v0.dot(v1);
+    double dot02 = v0.dot(v2);
+    double dot11 = v1.dot(v1);
+    double dot12 = v1.dot(v2);
+
+// Compute barycentric coordinates
+    double invDenom = 1 / (dot00 * dot11 - dot01 * dot01);
+    double u = (dot11 * dot02 - dot01 * dot12) * invDenom;
+    double v = (dot00 * dot12 - dot01 * dot02) * invDenom;
+
+// Check if point is in triangle
+    return (u >= 0) && (v >= 0) && (u + v < 1);
+}
+
+static bool addIntercept(const Quadratic& q1, const Quadratic& q2, double tMin, double tMax,
+        Intersections& i) {
+    double tMid = (tMin + tMax) / 2;
+    _Point mid;
+    xy_at_t(q2, tMid, mid.x, mid.y);
+    _Line line;
+    line[0] = line[1] = mid;
+    double dx, dy;
+    dxdy_at_t(q2, tMid, dx, dy);
+    line[0].x -= dx;
+    line[0].y -= dy;
+    line[1].x += dx;
+    line[1].y += dy;
+    Intersections rootTs;
+    int roots = intersect(q1, line, rootTs);
+    assert(roots == 1);
+    _Point pt2;
+    xy_at_t(q1, rootTs.fT[0][0], pt2.x, pt2.y);
+    if (!pt2.approximatelyEqual(mid)) {
+        return false;
+    }
+    i.add(rootTs.fT[0][0], tMid);
+    return true;
+}
+
+static bool isLinearInner(const Quadratic& q1, double t1s, double t1e, const Quadratic& q2,
+        double t2s, double t2e, Intersections& i) {
+    Quadratic hull;
+    sub_divide(q1, t1s, t1e, hull);
+    _Line line = {hull[2], hull[0]};
+    const _Line* testLines[] = { &line, (const _Line*) &hull[0], (const _Line*) &hull[1] };
+    size_t testCount = sizeof(testLines) / sizeof(testLines[0]);
+    SkTDArray<double> tsFound;
+    for (size_t index = 0; index < testCount; ++index) {
+        Intersections rootTs;
+        int roots = intersect(q2, *testLines[index], rootTs);
+        for (int idx2 = 0; idx2 < roots; ++idx2) {
+            double t = rootTs.fT[0][idx2];
+            if (approximately_negative(t - t2s) || approximately_positive(t - t2e)) {
+                continue;
+            }
+            *tsFound.append() = rootTs.fT[0][idx2];
+        }
+    }
+    int tCount = tsFound.count();
+    if (!tCount) {
+        return true;
+    }
+    double tMin, tMax;
+    _Point dxy1, dxy2;
+    if (tCount == 1) {
+        tMin = tMax = tsFound[0];
+    } else if (tCount > 1) {
+        QSort<double>(tsFound.begin(), tsFound.end() - 1);
+        tMin = tsFound[0];
+        tMax = tsFound[1];
+    }
+    _Point end;
+    xy_at_t(q2, t2s, end.x, end.y);
+    bool startInTriangle = pointInTriangle(end, testLines);
+    if (startInTriangle) {
+        tMin = t2s;
+    } 
+    xy_at_t(q2, t2e, end.x, end.y);
+    bool endInTriangle = pointInTriangle(end, testLines);
+    if (endInTriangle) {
+        tMax = t2e;
+    }
+    int split = 0;
+    if (tMin != tMax || tCount > 2) {
+        dxdy_at_t(q2, tMin, dxy2.x, dxy2.y);
+        for (int index = 1; index < tCount; ++index) {
+            dxy1 = dxy2;
+            dxdy_at_t(q2, tsFound[index], dxy2.x, dxy2.y);
+            double dot = dxy1.dot(dxy2);
+            if (dot < 0) {
+                split = index - 1;
+                break;
+            }
+        }
+        
+    }
+    if (split == 0) { // there's one point
+        if (addIntercept(q1, q2, tMin, tMax, i)) {
+            return true;
+        }
+        i.swap();
+        return isLinearInner(q2, tMin, tMax, q1, t1s, t1e, i);
+    }
+    // At this point, we have two ranges of t values -- treat each separately at the split
+    bool result;
+    if (addIntercept(q1, q2, tMin, tsFound[split - 1], i)) {
+        result = true;
+    } else {
+        i.swap();
+        result = isLinearInner(q2, tMin, tsFound[split - 1], q1, t1s, t1e, i);
+    }
+    if (addIntercept(q1, q2, tsFound[split], tMax, i)) {
+        result = true;
+    } else {
+        i.swap();
+        result |= isLinearInner(q2, tsFound[split], tMax, q1, t1s, t1e, i);
+    }
+    return result;
+}
+
+static double flatMeasure(const Quadratic& q) {
+    _Point mid;
+    xy_at_t(q, 0.5, mid.x, mid.y);
+    double dx = q[2].x - q[0].x;
+    double dy = q[2].y - q[0].y;
+    double length = sqrt(dx * dx + dy * dy); // OPTIMIZE: get rid of sqrt
+    return ((mid.x - q[0].x) * dy - (mid.y - q[0].y) * dx) / length;
+}
+
+// FIXME ? should this measure both and then use the quad that is the flattest as the line?
+static bool isLinear(const Quadratic& q1, const Quadratic& q2, Intersections& i) {
+    double measure = flatMeasure(q1);
+    // OPTIMIZE: (get rid of sqrt) use approximately_zero
+    if (!approximately_zero_sqrt(measure)) {
+        return false;
+    }
+    return isLinearInner(q1, 0, 1, q2, 0, 1, i);
+}
+
+static bool relaxedIsLinear(const Quadratic& q1, const Quadratic& q2, Intersections& i) {
+    double m1 = flatMeasure(q1);
+    double m2 = flatMeasure(q2);
+    if (fabs(m1) < fabs(m2)) {
+        isLinearInner(q1, 0, 1, q2, 0, 1, i);
+        return false;
+    } else {
+        isLinearInner(q2, 0, 1, q1, 0, 1, i);
+        return true;
+    }
+}
+
+#if 0
+static void unsortableExpanse(const Quadratic& q1, const Quadratic& q2, Intersections& i) {
+    const Quadratic* qs[2] = { &q1, &q2 };
+    // need t values for start and end of unsortable expanse on both curves
+    // try projecting lines parallel to the end points
+    i.fT[0][0] = 0;
+    i.fT[0][1] = 1;
+    int flip = -1; // undecided
+    for (int qIdx = 0; qIdx < 2; qIdx++) {
+        for (int t = 0; t < 2; t++) {
+            _Point dxdy;
+            dxdy_at_t(*qs[qIdx], t, dxdy.x, dxdy.y);
+            _Line perp;
+            perp[0] = perp[1] = (*qs[qIdx])[t == 0 ? 0 : 2];
+            perp[0].x += dxdy.y;
+            perp[0].y -= dxdy.x;
+            perp[1].x -= dxdy.y;
+            perp[1].y += dxdy.x;
+            Intersections hitData;
+            int hits = intersectRay(*qs[qIdx ^ 1], perp, hitData);
+            assert(hits <= 1);
+            if (hits) {
+                if (flip < 0) {
+                    _Point dxdy2;
+                    dxdy_at_t(*qs[qIdx ^ 1], hitData.fT[0][0], dxdy2.x, dxdy2.y);
+                    double dot = dxdy.dot(dxdy2);
+                    flip = dot < 0;
+                    i.fT[1][0] = flip;
+                    i.fT[1][1] = !flip;
+                }
+                i.fT[qIdx ^ 1][t ^ flip] = hitData.fT[0][0];
+            }
+        }
+    }
+    i.fUnsortable = true; // failed, probably coincident or near-coincident
+    i.fUsed = 2;
+}
+#endif
+
 bool intersect2(const Quadratic& q1, const Quadratic& q2, Intersections& i) {
     // if the quads share an end point, check to see if they overlap
 
     if (onlyEndPtsInCommon(q1, q2, i)) {
         return i.intersected();
     }
+    if (onlyEndPtsInCommon(q2, q1, i)) {
+        i.swapPts();
+        return i.intersected();
+    }
+    // see if either quad is really a line
+    if (isLinear(q1, q2, i)) {
+        return i.intersected();
+    }
+    if (isLinear(q2, q1, i)) {
+        i.swapPts();
+        return i.intersected();
+    }
     QuadImplicitForm i1(q1);
     QuadImplicitForm i2(q2);
     if (i1.implicit_match(i2)) {
@@ -135,15 +399,20 @@ bool intersect2(const Quadratic& q1, const Quadratic& q2, Intersections& i) {
         return i.fCoincidentUsed > 0;
     }
     double roots1[4], roots2[4];
-    int rootCount = findRoots(i2, q1, roots1);
+    bool disregardCount1 = false;
+    bool disregardCount2 = false;
+    bool useCubic = q1[0] == q2[0] || q1[0] == q2[2] || q1[2] == q2[0];
+    int rootCount = findRoots(i2, q1, roots1, useCubic, disregardCount1);
     // OPTIMIZATION: could short circuit here if all roots are < 0 or > 1
-#ifndef NDEBUG
-    int rootCount2 =
-#endif
-        findRoots(i1, q2, roots2);
-    assert(rootCount == rootCount2);
+    int rootCount2 = findRoots(i1, q2, roots2, useCubic, disregardCount2);
+ #if 0
+    if (rootCount != rootCount2 && !disregardCount1 && !disregardCount2) {
+        unsortableExpanse(q1, q2, i);
+        return false;
+    }
+ #endif
     addValidRoots(roots1, rootCount, 0, i);
-    addValidRoots(roots2, rootCount, 1, i);
+    addValidRoots(roots2, rootCount2, 1, i);
     if (i.insertBalanced() && i.fUsed <= 1) {
         if (i.fUsed == 1) {
             _Point xy1, xy2;
@@ -157,16 +426,13 @@ bool intersect2(const Quadratic& q1, const Quadratic& q2, Intersections& i) {
         return i.intersected();
     }
     _Point pts[4];
-    bool matches[4];
-    int flipCheck[4];
     int closest[4];
     double dist[4];
     int index, ndex2;
-    int flipIndex = 0;
     for (ndex2 = 0; ndex2 < i.fUsed2; ++ndex2) {
         xy_at_t(q2, i.fT[1][ndex2], pts[ndex2].x, pts[ndex2].y);
-        matches[ndex2] = false;
     }
+    bool foundSomething = false;
     for (index = 0; index < i.fUsed; ++index) {
         _Point xy;
         xy_at_t(q1, i.fT[0][index], xy.x, xy.y);
@@ -193,38 +459,41 @@ bool intersect2(const Quadratic& q1, const Quadratic& q2, Intersections& i) {
             }
             dist[index] = distance;
             closest[index] = ndex2;
+            foundSomething = true;
         next:
             ;
         }
     }
-    for (index = 0; index < i.fUsed; ) {
-        for (ndex2 = 0; ndex2 < i.fUsed2; ++ndex2) {
-             if (closest[index] == ndex2) {
-                assert(flipIndex < 4);
-                flipCheck[flipIndex++] = ndex2;
-                matches[ndex2] = true;
-                goto next2;
-             }
-        }
-        if (--i.fUsed > index) {
-            memmove(&i.fT[0][index], &i.fT[0][index + 1], (i.fUsed - index) * sizeof(i.fT[0][0]));
-            memmove(&closest[index], &closest[index + 1], (i.fUsed - index) * sizeof(closest[0]));
-            continue;
+    if (i.fUsed && i.fUsed2 && !foundSomething) {
+        if (relaxedIsLinear(q1, q2, i)) {
+            i.swapPts();
         }
-    next2:
-        ++index;
+        return i.intersected();
     }
-    for (ndex2 = 0; ndex2 < i.fUsed2; ) {
-        if (!matches[ndex2]) {
-             if (--i.fUsed2 > ndex2) {
-                memmove(&i.fT[1][ndex2], &i.fT[1][ndex2 + 1], (i.fUsed2 - ndex2) * sizeof(i.fT[1][0]));
-                memmove(&matches[ndex2], &matches[ndex2 + 1], (i.fUsed2 - ndex2) * sizeof(matches[0]));
+    double roots1Copy[4], roots2Copy[4];
+    memcpy(roots1Copy, i.fT[0], i.fUsed * sizeof(double));
+    memcpy(roots2Copy, i.fT[1], i.fUsed2 * sizeof(double));
+    int used = 0;
+    do {
+        double lowest = DBL_MAX;
+        int lowestIndex = -1;
+        for (index = 0; index < i.fUsed; ++index) {
+            if (closest[index] < 0) {
                 continue;
-             }
+            }
+            if (roots1Copy[index] < lowest) {
+                lowestIndex = index;
+                lowest = roots1Copy[index];
+            }
         }
-        ++ndex2;
-    }
-    i.fFlip = i.fUsed >= 2 && flipCheck[0] > flipCheck[1];
-    assert(i.insertBalanced());
+        if (lowestIndex < 0) {
+            break;
+        }
+        i.fT[0][used] = roots1Copy[lowestIndex];
+        i.fT[1][used] = roots2Copy[closest[lowestIndex]];
+        closest[lowestIndex] = -1;
+    } while (++used < i.fUsed);
+    i.fUsed = i.fUsed2 = used;
+    i.fFlip = false;
     return i.intersected();
 }
diff --git a/experimental/Intersection/QuadraticIntersection_Test.cpp b/experimental/Intersection/QuadraticIntersection_Test.cpp
index bab6c73718..829dffebcc 100644
--- a/experimental/Intersection/QuadraticIntersection_Test.cpp
+++ b/experimental/Intersection/QuadraticIntersection_Test.cpp
@@ -59,9 +59,41 @@ static void standardTestCases() {
 }
 
 static const Quadratic testSet[] = {
+{{53.774852327053594, 53.318060789841951}, {45.787877803416805, 51.393492026284981}, {46.703936967162392, 53.06860709822206}},
+{{46.703936967162392, 53.06860709822206}, {47.619996130907957, 54.74372217015916}, {53.020051653535361, 48.633140968832024}},
+
+{{67.4265481,37.9937726}, {51.1295132,57.5422812}, {44.5947482,65.6442674}},
+{{61.3365082,82.6931328}, {54.8250789,71.6639328}, {47.7274442,61.4049645}},
+
+{{50.934805397717923, 51.52391952648901}, {56.803308902971423, 44.246234610627596}, {69.776888596721406, 40.166645096692555}},
+{{50.230212796400401, 38.386469101526998}, {49.855620812184917, 38.818990392153609}, {56.356567496227363, 47.229909093319407}},
+
+{{36.148792695174222, 70.336952793070424}, {36.141613037691357, 70.711654739870085}, {36.154708826402597, 71.088492662905836}},
+{{35.216235592661825, 70.580199617313212}, {36.244476835123969, 71.010897787304074}, {37.230244263238326, 71.423156953613102}},
+
+// this pair is nearly coincident, and causes the quartic code to produce bad
+// data. Mathematica doesn't think they touch. Graphically, I can't tell.
+// it may not be so bad to pretend that they don't touch, if I can detect that
+{{369.848602,145.680267}, {382.360413,121.298294}, {406.207703,121.298294}},
+{{369.850525,145.675964}, {382.362915,121.29287}, {406.211273,121.29287}},
+
+{{33.567436351153468, 62.336347586395924}, {35.200980274619084, 65.038561460144479}, {36.479571811084995, 67.632178905412445}},
+{{41.349524945572696, 67.886658677862641}, {39.125562529359087, 67.429772735149214}, {35.600314083992416, 66.705372160552685}},
+
+{{67.25299631583178, 21.109080184767524}, {43.617595267398613, 33.658034168577529}, {33.38371819435676, 44.214192553988745}},
+{{40.476838859398541, 39.543209911285999}, {36.701186108431131, 34.8817994016458}, {30.102144288878023, 26.739063172945315}},
+
+{{25.367434474345036, 50.4712103169743}, {17.865013304933097, 37.356741010559439}, {16.818988838905465, 37.682915484123129}},
+{{16.818988838905465, 37.682915484123129}, {15.772964372877833, 38.009089957686811}, {20.624104547604965, 41.825131596683121}},
+
+{{26.440225044088567, 79.695009812848298}, {26.085525979582247, 83.717928354134784}, {27.075079976297072, 84.820633667838905}},
+{{27.075079976297072, 84.820633667838905}, {28.276546859574015, 85.988574184029034}, {25.649263209500006, 87.166762066617025}},
+
+{{34.879150914024962, 83.862726601601125}, {35.095810134304429, 83.693473210169543}, {35.359284111931586, 83.488069234177502}},
+{{54.503204203015471, 76.094098492518242}, {51.366889541918894, 71.609856061299155}, {46.53086955445437, 69.949863036494207}},
+
 {{0, 0}, {1, 0}, {0, 3}},
 {{1, 0}, {0, 1}, {1, 1}},
-{{369.848602,145.680267}, {382.360413,121.298294}, {406.207703,121.298294}},
 {{369.961151,137.980698}, {383.970093,121.298294}, {406.213287,121.298294}},
 {{353.2948,194.351074}, {353.2948,173.767563}, {364.167572,160.819855}},
 {{360.416077,166.795715}, {370.126831,147.872162}, {388.635406,147.872162}},
@@ -71,7 +103,6 @@ static const Quadratic testSet[] = {
 {{369.8543701171875, 145.66734313964844}, {382.36788940429688, 121.28203582763672}, {406.21844482421875, 121.28203582763672}},
 {{369.96469116210938, 137.96672058105469}, {383.97555541992188, 121.28203582763672}, {406.2218017578125, 121.28203582763672}},
 
-    {{369.850525, 145.675964}, {382.362915, 121.29287}, {406.211273, 121.29287}},
     {{369.962311, 137.976044}, {383.971893, 121.29287}, {406.216125, 121.29287}},
 
     {{400.121704, 149.468719}, {391.949493, 161.037186}, {391.949493, 181.202423}},
@@ -90,25 +121,27 @@ static void oneOffTest() {
         for (size_t inner = outer + 1; inner < testSetCount; ++inner) {
             const Quadratic& quad1 = testSet[outer];
             const Quadratic& quad2 = testSet[inner];
-            double tt1, tt2;
             Intersections intersections2;
             intersect2(quad1, quad2, intersections2);
+            if (intersections2.fUnsortable) {
+                continue;
+            }
             for (int pt = 0; pt < intersections2.used(); ++pt) {
-                tt1 = intersections2.fT[0][pt];
+                double tt1 = intersections2.fT[0][pt];
                 double tx1, ty1;
                 xy_at_t(quad1, tt1, tx1, ty1);
                 int pt2 = intersections2.fFlip ? intersections2.used() - pt - 1 : pt;
-                tt2 = intersections2.fT[1][pt2];
+                double tt2 = intersections2.fT[1][pt2];
                 double tx2, ty2;
                 xy_at_t(quad2, tt2, tx2, ty2);
-                if (!approximately_equal(tx1, tx2)) {
+                if (!AlmostEqualUlps(tx1, tx2)) {
                     SkDebugf("%s [%d,%d] x!= t1=%g (%g,%g) t2=%g (%g,%g)\n",
-                        __FUNCTION__, (int)index, pt, tt1, tx1, ty1, tt2, tx2, ty2);
+                        __FUNCTION__, (int)outer, (int)inner, tt1, tx1, ty1, tt2, tx2, ty2);
                     SkASSERT(0);
                 }
-                if (!approximately_equal(ty1, ty2)) {
+                if (!AlmostEqualUlps(ty1, ty2)) {
                     SkDebugf("%s [%d,%d] y!= t1=%g (%g,%g) t2=%g (%g,%g)\n",
-                        __FUNCTION__, (int)index, pt, tt1, tx1, ty1, tt2, tx2, ty2);
+                        __FUNCTION__, (int)outer, (int)inner, tt1, tx1, ty1, tt2, tx2, ty2);
                     SkASSERT(0);
                 }
                 SkDebugf("%s [%d][%d] t1=%1.9g (%1.9g, %1.9g) t2=%1.9g\n", __FUNCTION__,
diff --git a/experimental/Intersection/QuadraticIntersection_TestData.h b/experimental/Intersection/QuadraticIntersection_TestData.h
index 4c95d5efad..2dbf34a29c 100644
--- a/experimental/Intersection/QuadraticIntersection_TestData.h
+++ b/experimental/Intersection/QuadraticIntersection_TestData.h
@@ -4,11 +4,8 @@
  * Use of this source code is governed by a BSD-style license that can be
  * found in the LICENSE file.
  */
-#if !defined(IN_TEST)
-    #define IN_TEST 1
-#endif
-
 #include "DataTypes.h"
+#include "DataTypes_Test.h"
 
 extern const Quadratic quadraticLines[];
 extern const Quadratic quadraticModEpsilonLines[];
diff --git a/experimental/Intersection/QuadraticUtilities.h b/experimental/Intersection/QuadraticUtilities.h
index 5bc15eac92..3230e9beb5 100644
--- a/experimental/Intersection/QuadraticUtilities.h
+++ b/experimental/Intersection/QuadraticUtilities.h
@@ -4,6 +4,10 @@
  * Use of this source code is governed by a BSD-style license that can be
  * found in the LICENSE file.
  */
+ 
+#if !defined QUADRATIC_UTILITIES_H
+#define QUADRATIC_UTILITIES_H
+
 #include "DataTypes.h"
 
 void dxdy_at_t(const Quadratic& , double t, double& x, double& y);
@@ -24,5 +28,7 @@ inline void set_abc(const double* quad, double& a, double& b, double& c) {
 }
 
 int quadraticRoots(double A, double B, double C, double t[2]);
-
+int quadraticRootsX(const double A, const double B, const double C, double s[2]);
 void xy_at_t(const Quadratic& , double t, double& x, double& y);
+
+#endif
diff --git a/experimental/Intersection/QuarticRoot.cpp b/experimental/Intersection/QuarticRoot.cpp
index 66ce3bf415..86ea7a63fd 100644
--- a/experimental/Intersection/QuarticRoot.cpp
+++ b/experimental/Intersection/QuarticRoot.cpp
@@ -27,13 +27,20 @@
 
 #include    <math.h>
 #include "CubicUtilities.h"
+#include "QuadraticUtilities.h"
 #include "QuarticRoot.h"
 
+#if SK_DEBUG
+#define QUARTIC_DEBUG 1
+#else
+#define QUARTIC_DEBUG 0
+#endif
+
 const double PI = 4 * atan(1);
 
 // unlike quadraticRoots in QuadraticUtilities.cpp, this does not discard
 // real roots <= 0 or >= 1
-static int quadraticRootsX(const double A, const double B, const double C,
+int quadraticRootsX(const double A, const double B, const double C,
         double s[2]) {
     if (approximately_zero(A)) {
         if (approximately_zero(B)) {
@@ -68,7 +75,7 @@ static int quadraticRootsX(const double A, const double B, const double C,
 #if USE_GEMS
 // unlike cubicRoots in CubicUtilities.cpp, this does not discard
 // real roots <= 0 or >= 1
-static int cubicRootsX(const double A, const double B, const double C,
+int cubicRootsX(const double A, const double B, const double C,
         const double D, double s[3]) {
     int num;
     /* normal form: x^3 + Ax^2 + Bx + C = 0 */
@@ -119,7 +126,13 @@ static int cubicRootsX(const double A, const double B, const double C,
 }
 #else
 
-static int cubicRootsX(double A, double B, double C, double D, double s[3]) {
+int cubicRootsX(double A, double B, double C, double D, double s[3]) {
+#if QUARTIC_DEBUG
+    // create a string mathematica understands
+    char str[1024];
+    bzero(str, sizeof(str));
+    sprintf(str, "Solve[%1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]", A, B, C, D);
+#endif
     if (approximately_zero(A)) {  // we're just a quadratic
         return quadraticRootsX(B, C, D, s);
     }
@@ -202,34 +215,6 @@ static int cubicRootsX(double A, double B, double C, double D, double s[3]) {
 
 int quarticRoots(const double A, const double B, const double C, const double D,
         const double E, double s[4]) {
-    if (approximately_zero(A)) {
-        if (approximately_zero(B)) {
-            return quadraticRootsX(C, D, E, s);
-        }
-        return cubicRootsX(B, C, D, E, s);
-    }
-    int num;
-    int i;
-    if (approximately_zero(E)) { // 0 is one root
-        num = cubicRootsX(A, B, C, D, s);
-        for (i = 0; i < num; ++i) {
-            if (approximately_zero(s[i])) {
-                return num;
-            }
-        }
-        s[num++] = 0;
-        return num;
-    }
-    if (approximately_zero_squared(A + B + C + D + E)) { // 1 is one root
-        num = cubicRootsX(A, A + B, -(D + E), -E, s); // note that -C==A+B+D+E
-        for (i = 0; i < num; ++i) {
-            if (approximately_equal(s[i], 1)) {
-                return num;
-            }
-        }
-        s[num++] = 1;
-        return num;
-    }
     double  u, v;
     /* normal form: x^4 + Ax^3 + Bx^2 + Cx + D = 0 */
     const double invA = 1 / A;
@@ -243,6 +228,7 @@ int quarticRoots(const double A, const double B, const double C, const double D,
     const double p = -3 * a2 / 8 + b;
     const double q = a2 * a / 8 - a * b / 2 + c;
     const double r = -3 * a2 * a2 / 256 + a2 * b / 16 - a * c / 4 + d;
+    int num;
     if (approximately_zero(r)) {
     /* no absolute term: y(y^3 + py + q) = 0 */
         num = cubicRootsX(1, 0, p, q, s);
@@ -250,19 +236,19 @@ int quarticRoots(const double A, const double B, const double C, const double D,
     } else {
         /* solve the resolvent cubic ... */
         (void) cubicRootsX(1, -p / 2, -r, r * p / 2 - q * q / 8, s);
-        /* ... and take the one real solution ... */
+        /* ... and take one real solution ... */
         const double z = s[0];
         /* ... to build two quadric equations */
         u = z * z - r;
         v = 2 * z - p;
-        if (approximately_zero(u)) {
+        if (approximately_zero_squared(u)) {
             u = 0;
         } else if (u > 0) {
             u = sqrt(u);
         } else {
             return 0;
         }
-        if (approximately_zero(v)) {
+        if (approximately_zero_squared(v)) {
             v = 0;
         } else if (v > 0) {
             v = sqrt(v);
@@ -273,7 +259,7 @@ int quarticRoots(const double A, const double B, const double C, const double D,
         num += quadraticRootsX(1, q < 0 ? v : -v, z + u, s + num);
     }
     // eliminate duplicates
-    for (i = 0; i < num - 1; ++i) {
+    for (int i = 0; i < num - 1; ++i) {
         for (int j = i + 1; j < num; ) {
             if (approximately_equal(s[i], s[j])) {
                 if (j < --num) {
@@ -286,10 +272,8 @@ int quarticRoots(const double A, const double B, const double C, const double D,
     }
     /* resubstitute */
     const double sub = a / 4;
-    for (i = 0; i < num; ++i) {
+    for (int i = 0; i < num; ++i) {
         s[i] -= sub;
     }
     return num;
 }
-
-
diff --git a/experimental/Intersection/QuarticRoot_Test.cpp b/experimental/Intersection/QuarticRoot_Test.cpp
index 2fe3e728c1..0f310bb08e 100644
--- a/experimental/Intersection/QuarticRoot_Test.cpp
+++ b/experimental/Intersection/QuarticRoot_Test.cpp
@@ -2,12 +2,8 @@
 #include <math.h>
 #include "CubicUtilities.h"
 #include "Intersection_Tests.h"
-
-namespace QuarticRootTest {
-
-#include "QuarticRoot.cpp"
-
-}
+#include "QuadraticUtilities.h"
+#include "QuarticRoot.h"
 
 double mulA[] = {-3, -1, 1, 3};
 size_t mulACount = sizeof(mulA) / sizeof(mulA[0]);
@@ -20,6 +16,7 @@ size_t rootDCount = sizeof(rootD) / sizeof(rootD[0]);
 double rootE[] = {-5, -1, 0, 1, 7};
 size_t rootECount = sizeof(rootE) / sizeof(rootE[0]);
 
+
 static void quadraticTest() {
     // (x - a)(x - b) == x^2 - (a + b)x + ab
     for (size_t aIndex = 0; aIndex < mulACount; ++aIndex) {
@@ -31,7 +28,7 @@ static void quadraticTest() {
                 const double b = A * (B + C);
                 const double c = A * B * C;
                 double roots[2];
-                const int rootCount = QuarticRootTest::quadraticRootsX(A, b, c, roots);
+                const int rootCount = quadraticRootsX(A, b, c, roots);
                 const int expected = 1 + (B != C);
                 assert(rootCount == expected);
                 assert(approximately_equal(roots[0], -B)
@@ -60,7 +57,7 @@ static void cubicTest() {
                     const double c = A * (B * C + C * D + B * D);
                     const double d = A * B * C * D;
                     double roots[3];
-                    const int rootCount = QuarticRootTest::cubicRootsX(A, b, c, d, roots);
+                    const int rootCount = cubicRootsX(A, b, c, d, roots);
                     const int expected = 1 + (B != C) + (B != D && C != D);
                     assert(rootCount == expected);
                     assert(approximately_equal(roots[0], -B)
@@ -103,7 +100,7 @@ static void quarticTest() {
                         const double d = A * (B * C * D + B * C * E + B * D * E + C * D * E);
                         const double e = A * B * C * D * E;
                         double roots[4];
-                        const int rootCount = QuarticRootTest::quarticRoots(A, b, c, d, e, roots);
+                        const int rootCount = quarticRoots(A, b, c, d, e, roots);
                         const int expected = 1 + (B != C) + (B != D && C != D) + (B != E && C != E && D != E);
                         assert(rootCount == expected);
                         assert(approximately_equal(roots[0], -B)
diff --git a/experimental/Intersection/Simplify.cpp b/experimental/Intersection/Simplify.cpp
index e29b0df7c0..5fd0fd5223 100644
--- a/experimental/Intersection/Simplify.cpp
+++ b/experimental/Intersection/Simplify.cpp
@@ -28,9 +28,10 @@ int gDebugMaxWindValue = SK_MaxS32;
 #define PIN_ADD_T 0
 #define TRY_ROTATE 1
 #define ONE_PASS_COINCIDENCE_CHECK 0
+#define APPROXIMATE_CUBICS 1
 
 #define DEBUG_UNUSED 0 // set to expose unused functions
-#define FORCE_RELEASE 1  // set force release to 1 for multiple thread -- no debugging
+#define FORCE_RELEASE 0  // set force release to 1 for multiple thread -- no debugging
 
 #if FORCE_RELEASE || defined SK_RELEASE
 
@@ -118,7 +119,7 @@ static int CubicLineIntersect(const SkPoint a[4], const SkPoint b[2],
         Intersections& intersections) {
     MAKE_CONST_CUBIC(aCubic, a);
     MAKE_CONST_LINE(bLine, b);
-    return intersect(aCubic, bLine, intersections.fT[0], intersections.fT[1]);
+    return intersect(aCubic, bLine, intersections);
 }
 
 static int QuadIntersect(const SkPoint a[3], const SkPoint b[3],
@@ -134,11 +135,23 @@ static int QuadIntersect(const SkPoint a[3], const SkPoint b[3],
     return intersections.fUsed ? intersections.fUsed : intersections.fCoincidentUsed;
 }
 
-static int CubicIntersect(const SkPoint a[4], const SkPoint b[4],
+#if APPROXIMATE_CUBICS
+static int CubicQuadIntersect(const SkPoint a[4], const SkPoint b[3],
         Intersections& intersections) {
     MAKE_CONST_CUBIC(aCubic, a);
+    MAKE_CONST_QUAD(bQuad, b);
+    return intersect(aCubic, bQuad, intersections);
+}
+#endif
+
+static int CubicIntersect(const SkPoint a[4], const SkPoint b[4], Intersections& intersections) {
+    MAKE_CONST_CUBIC(aCubic, a);
     MAKE_CONST_CUBIC(bCubic, b);
+#if APPROXIMATE_CUBICS
+    intersect2(aCubic, bCubic, intersections);
+#else
     intersect(aCubic, bCubic, intersections);
+#endif
     return intersections.fUsed;
 }
 
@@ -1665,6 +1678,23 @@ public:
             other.addCancelOutsides(tStart, oStart, *this, endT);
         }
     }
+    
+    int addUnsortableT(double newT, Segment* other, bool start) {
+        int result = addT(newT, other);
+        Span* span = &fTs[result];
+        if (start) {
+            if (result > 0) {
+                span[result - 1].fUnsortableEnd = true;
+            }
+            span[result].fUnsortableStart = true;
+        } else {
+            span[result].fUnsortableEnd = true;
+            if (result + 1 < fTs.count()) {
+                span[result + 1].fUnsortableStart = true;
+            }
+        }
+        return result;
+    }
 
     int bumpCoincidentThis(const Span* oTest, bool opp, int index,
             SkTDArray<double>& outsideTs) {
@@ -2722,6 +2752,8 @@ public:
         int local = spanSign(start, end);
         int oppLocal = oppSign(start, end);
         (void) markAndChaseWinding(start, end, local, oppLocal);
+        // OPTIMIZATION: the reverse mark and chase could skip the first marking
+        (void) markAndChaseWinding(end, start, local, oppLocal);
     }
 
     void initWinding(int start, int end, int winding, int oppWinding) {
@@ -4138,6 +4170,10 @@ public:
         return fSegments[segIndex].addT(newT, &other->fSegments[otherIndex]);
     }
 
+    int addUnsortableT(int segIndex, double newT, Contour* other, int otherIndex, bool start) {
+        return fSegments[segIndex].addUnsortableT(newT, &other->fSegments[otherIndex], start);
+    }
+
     const Bounds& bounds() const {
         return fBounds;
     }
@@ -4820,6 +4856,10 @@ public:
         return fContour->addT(fIndex, newT, other.fContour, other.fIndex);
     }
 
+    int addUnsortableT(double newT, const Work& other, bool start) {
+        return fContour->addUnsortableT(fIndex, newT, other.fContour, other.fIndex, start);
+    }
+
     bool advance() {
         return ++fIndex < fLast;
     }
@@ -4832,9 +4872,11 @@ public:
         return fContour->segments()[fIndex].bounds();
     }
 
+#if !APPROXIMATE_CUBICS
     const SkPoint* cubic() const {
         return fCubic;
     }
+#endif
 
     void init(Contour* contour) {
         fContour = contour;
@@ -4855,6 +4897,7 @@ public:
         return bounds().fLeft;
     }
 
+#if !APPROXIMATE_CUBICS
     void promoteToCubic() {
         fCubic[0] = pts()[0];
         fCubic[2] = pts()[1];
@@ -4864,6 +4907,7 @@ public:
         fCubic[2].fX = (fCubic[3].fX + fCubic[2].fX * 2) / 3;
         fCubic[2].fY = (fCubic[3].fY + fCubic[2].fY * 2) / 3;
     }
+#endif
 
     const SkPoint* pts() const {
         return fContour->segments()[fIndex].pts();
@@ -4923,7 +4967,9 @@ public:
 
 protected:
     Contour* fContour;
+#if !APPROXIMATE_CUBICS
     SkPoint fCubic[4];
+#endif
     int fIndex;
     int fLast;
 };
@@ -4990,6 +5036,7 @@ static void debugShowQuadLineIntersection(int pts, const Work& wt,
     SkDebugf("\n");
 }
 
+// FIXME: show more than two intersection points
 static void debugShowQuadIntersection(int pts, const Work& wt,
         const Work& wn, const double wtTs[2], const double wnTs[2]) {
     if (!pts) {
@@ -5021,6 +5068,106 @@ static void debugShowQuadIntersection(int pts, const Work& wt,
     }
     SkDebugf("\n");
 }
+
+static void debugShowCubicLineIntersection(int pts, const Work& wt,
+        const Work& wn, const double wtTs[2], const double wnTs[2]) {
+    if (!pts) {
+        SkDebugf("%s no intersect (%1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g)"
+                " (%1.9g,%1.9g %1.9g,%1.9g)\n",
+                __FUNCTION__, wt.pts()[0].fX, wt.pts()[0].fY, wt.pts()[1].fX, wt.pts()[1].fY,
+                wt.pts()[2].fX, wt.pts()[2].fY, wt.pts()[3].fX, wt.pts()[3].fY,
+                wn.pts()[0].fX, wn.pts()[0].fY, wn.pts()[1].fX, wn.pts()[1].fY);
+        return;
+    }
+    SkPoint wtOutPt, wnOutPt;
+    CubicXYAtT(wt.pts(), wtTs[0], &wtOutPt);
+    LineXYAtT(wn.pts(), wnTs[0], &wnOutPt);
+    SkDebugf("%s wtTs[0]=%1.9g (%1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g) (%1.9g,%1.9g)",
+            __FUNCTION__,
+            wtTs[0], wt.pts()[0].fX, wt.pts()[0].fY, wt.pts()[1].fX, wt.pts()[1].fY,
+            wt.pts()[2].fX, wt.pts()[2].fY, wt.pts()[3].fX, wt.pts()[3].fY,
+            wtOutPt.fX, wtOutPt.fY);
+    if (pts == 2) {
+        CubicXYAtT(wt.pts(), wtTs[1], &wtOutPt);
+        SkDebugf(" wtTs[1]=%1.9g (%1.9g,%1.9g)", wtTs[1], wtOutPt.fX, wtOutPt.fY);
+    }
+    SkDebugf(" wnTs[0]=%g (%1.9g,%1.9g %1.9g,%1.9g) (%1.9g,%1.9g)",
+            wtTs[0], wn.pts()[0].fX, wn.pts()[0].fY, wn.pts()[1].fX, wn.pts()[1].fY,
+            wnOutPt.fX, wnOutPt.fY);
+    if (pts == 2) {
+        LineXYAtT(wn.pts(), wnTs[1], &wnOutPt);
+        SkDebugf(" wnTs[1]=%1.9g (%1.9g,%1.9g)", wnTs[1], wnOutPt.fX, wnOutPt.fY);
+    }
+    SkDebugf("\n");
+}
+
+#if APPROXIMATE_CUBICS
+// FIXME: show more than two intersection points
+static void debugShowCubicQuadIntersection(int pts, const Work& wt,
+        const Work& wn, const double wtTs[2], const double wnTs[2]) {
+    if (!pts) {
+        SkDebugf("%s no intersect (%1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g)"
+                " (%1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g)\n",
+                __FUNCTION__, wt.pts()[0].fX, wt.pts()[0].fY, wt.pts()[1].fX, wt.pts()[1].fY,
+                wt.pts()[2].fX, wt.pts()[2].fY, wt.pts()[3].fX, wt.pts()[3].fY,
+                wn.pts()[0].fX, wn.pts()[0].fY, wn.pts()[1].fX, wn.pts()[1].fY,
+                wn.pts()[2].fX, wn.pts()[2].fY );
+        return;
+    }
+    SkPoint wtOutPt, wnOutPt;
+    CubicXYAtT(wt.pts(), wtTs[0], &wtOutPt);
+    CubicXYAtT(wn.pts(), wnTs[0], &wnOutPt);
+    SkDebugf("%s wtTs[0]=%1.9g (%1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g) (%1.9g,%1.9g)",
+            __FUNCTION__,
+            wtTs[0], wt.pts()[0].fX, wt.pts()[0].fY, wt.pts()[1].fX, wt.pts()[1].fY,
+            wt.pts()[2].fX, wt.pts()[2].fY, wt.pts()[3].fX, wt.pts()[3].fY,
+            wtOutPt.fX, wtOutPt.fY);
+    if (pts == 2) {
+        SkDebugf(" wtTs[1]=%1.9g", wtTs[1]);
+    }
+    SkDebugf(" wnTs[0]=%g (%1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g) (%1.9g,%1.9g)",
+            wnTs[0], wn.pts()[0].fX, wn.pts()[0].fY, wn.pts()[1].fX, wn.pts()[1].fY,
+            wn.pts()[2].fX, wn.pts()[2].fY,
+            wnOutPt.fX, wnOutPt.fY);
+    if (pts == 2) {
+        SkDebugf(" wnTs[1]=%1.9g", wnTs[1]);
+    }
+    SkDebugf("\n");
+}
+
+// FIXME: show more than two intersection points
+static void debugShowCubicIntersection(int pts, const Work& wt,
+        const Work& wn, const double wtTs[2], const double wnTs[2]) {
+    if (!pts) {
+        SkDebugf("%s no intersect (%1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g)"
+                " (%1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g)\n",
+                __FUNCTION__, wt.pts()[0].fX, wt.pts()[0].fY, wt.pts()[1].fX, wt.pts()[1].fY,
+                wt.pts()[2].fX, wt.pts()[2].fY, wt.pts()[3].fX, wt.pts()[3].fY,
+                wn.pts()[0].fX, wn.pts()[0].fY, wn.pts()[1].fX, wn.pts()[1].fY,
+                wn.pts()[2].fX, wn.pts()[2].fY, wn.pts()[3].fX, wn.pts()[3].fY );
+        return;
+    }
+    SkPoint wtOutPt, wnOutPt;
+    CubicXYAtT(wt.pts(), wtTs[0], &wtOutPt);
+    CubicXYAtT(wn.pts(), wnTs[0], &wnOutPt);
+    SkDebugf("%s wtTs[0]=%1.9g (%1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g) (%1.9g,%1.9g)",
+            __FUNCTION__,
+            wtTs[0], wt.pts()[0].fX, wt.pts()[0].fY, wt.pts()[1].fX, wt.pts()[1].fY,
+            wt.pts()[2].fX, wt.pts()[2].fY, wt.pts()[3].fX, wt.pts()[3].fY,
+            wtOutPt.fX, wtOutPt.fY);
+    if (pts == 2) {
+        SkDebugf(" wtTs[1]=%1.9g", wtTs[1]);
+    }
+    SkDebugf(" wnTs[0]=%g (%1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g) (%1.9g,%1.9g)",
+            wnTs[0], wn.pts()[0].fX, wn.pts()[0].fY, wn.pts()[1].fX, wn.pts()[1].fY,
+            wn.pts()[2].fX, wn.pts()[2].fY, wn.pts()[3].fX, wn.pts()[3].fY,
+            wnOutPt.fX, wnOutPt.fY);
+    if (pts == 2) {
+        SkDebugf(" wnTs[1]=%1.9g", wnTs[1]);
+    }
+    SkDebugf("\n");
+}
+#endif
 #else
 static void debugShowLineIntersection(int , const Work& ,
         const Work& , const double [2], const double [2]) {
@@ -5033,6 +5180,20 @@ static void debugShowQuadLineIntersection(int , const Work& ,
 static void debugShowQuadIntersection(int , const Work& ,
         const Work& , const double [2], const double [2]) {
 }
+
+static void debugShowCubicLineIntersection(int , const Work& ,
+        const Work& , const double [2], const double [2]) {
+}
+
+#if APPROXIMATE_CUBICS
+static void debugShowCubicQuadIntersection(int , const Work& ,
+        const Work& , const double [2], const double [2]) {
+}
+
+static void debugShowCubicIntersection(int , const Work& ,
+        const Work& , const double [2], const double [2]) {
+}
+#endif
 #endif
 
 static bool addIntersectTs(Contour* test, Contour* next) {
@@ -5082,6 +5243,7 @@ static bool addIntersectTs(Contour* test, Contour* next) {
                         case Work::kCubic_Segment: {
                             pts = HCubicIntersect(wn.pts(), wt.left(),
                                     wt.right(), wt.y(), wt.xFlipped(), ts);
+                            debugShowCubicLineIntersection(pts, wn, wt, ts.fT[0], ts.fT[1]);
                             break;
                         }
                         default:
@@ -5108,6 +5270,7 @@ static bool addIntersectTs(Contour* test, Contour* next) {
                         case Work::kCubic_Segment: {
                             pts = VCubicIntersect(wn.pts(), wt.top(),
                                     wt.bottom(), wt.x(), wt.yFlipped(), ts);
+                            debugShowCubicLineIntersection(pts, wn, wt, ts.fT[0], ts.fT[1]);
                             break;
                         }
                         default:
@@ -5144,6 +5307,7 @@ static bool addIntersectTs(Contour* test, Contour* next) {
                         case Work::kCubic_Segment: {
                             swap = true;
                             pts = CubicLineIntersect(wn.pts(), wt.pts(), ts);
+                            debugShowCubicLineIntersection(pts, wn, wt, ts.fT[0], ts.fT[1]);
                             break;
                         }
                         default:
@@ -5173,8 +5337,14 @@ static bool addIntersectTs(Contour* test, Contour* next) {
                             break;
                         }
                         case Work::kCubic_Segment: {
+                    #if APPROXIMATE_CUBICS
+                            swap = true;
+                            pts = CubicQuadIntersect(wn.pts(), wt.pts(), ts);
+                            debugShowCubicQuadIntersection(pts, wn, wt, ts.fT[0], ts.fT[1]);
+                    #else
                             wt.promoteToCubic();
                             pts = CubicIntersect(wt.cubic(), wn.pts(), ts);
+                    #endif
                             break;
                         }
                         default:
@@ -5190,18 +5360,30 @@ static bool addIntersectTs(Contour* test, Contour* next) {
                         case Work::kVerticalLine_Segment:
                             pts = VCubicIntersect(wt.pts(), wn.top(),
                                     wn.bottom(), wn.x(), wn.yFlipped(), ts);
+                            debugShowCubicLineIntersection(pts, wt, wn, ts.fT[0], ts.fT[1]);
                             break;
                         case Work::kLine_Segment: {
                             pts = CubicLineIntersect(wt.pts(), wn.pts(), ts);
+                            debugShowCubicLineIntersection(pts, wt, wn, ts.fT[0], ts.fT[1]);
                             break;
                         }
                         case Work::kQuad_Segment: {
+                    #if APPROXIMATE_CUBICS
+                            pts = CubicQuadIntersect(wt.pts(), wn.pts(), ts);
+                            debugShowCubicQuadIntersection(pts, wt, wn, ts.fT[0], ts.fT[1]);
+                    #else
                             wn.promoteToCubic();
                             pts = CubicIntersect(wt.pts(), wn.cubic(), ts);
+                    #endif
                             break;
                         }
                         case Work::kCubic_Segment: {
+                    #if APPROXIMATE_CUBICS
+                            pts = CubicIntersect(wt.pts(), wn.pts(), ts);
+                            debugShowCubicIntersection(pts, wt, wn, ts.fT[0], ts.fT[1]);
+                    #else
                             pts = CubicIntersect(wt.pts(), wn.pts(), ts);
+                    #endif
                             break;
                         }
                         default:
@@ -5217,6 +5399,19 @@ static bool addIntersectTs(Contour* test, Contour* next) {
                 foundCommonContour = true;
             }
             // in addition to recording T values, record matching segment
+            if (ts.unsortable()) {
+                bool start = true;
+                for (int pt = 0; pt < ts.used(); ++pt) {
+                    // FIXME: if unsortable, the other points to the original. This logic is
+                    // untested downstream.
+                    int testTAt = wt.addUnsortableT(ts.fT[swap][pt], wt, start);
+                    wt.addOtherT(testTAt, ts.fT[swap][pt], testTAt);
+                    testTAt = wn.addUnsortableT(ts.fT[!swap][pt], wn, start ^ ts.fFlip);
+                    wn.addOtherT(testTAt, ts.fT[!swap][pt], testTAt);
+                    start ^= true;
+                }
+                continue;
+            }
             if (pts == 2) {
                 if (wn.segmentType() <= Work::kLine_Segment
                         && wt.segmentType() <= Work::kLine_Segment) {
@@ -6047,4 +6242,3 @@ void simplifyx(const SkPath& path, SkPath& result) {
         result = *assembled.nativePath();
     }
 }
-
diff --git a/experimental/Intersection/SimplifyNew_Test.cpp b/experimental/Intersection/SimplifyNew_Test.cpp
index 2559e80a05..b6cc26e159 100644
--- a/experimental/Intersection/SimplifyNew_Test.cpp
+++ b/experimental/Intersection/SimplifyNew_Test.cpp
@@ -3510,12 +3510,40 @@ static void testLine85() {
     testSimplifyx(path);
 }
 
-static void (*firstTest)() = testLine85;
+static void testQuadralateral1() {
+    SkPath path;
+    path.moveTo(0, 0);
+    path.lineTo(0, 0);
+    path.lineTo(0, 0);
+    path.lineTo(3, 2);
+    path.close();
+    path.moveTo(0, 0);
+    path.lineTo(2, 1);
+    path.lineTo(2, 2);
+    path.lineTo(2, 3);
+    path.close();
+    testSimplifyx(path);
+}
+
+static void testCubic1() {
+    SkPath path;
+    path.moveTo(0, 0);
+    path.cubicTo(0, 1, 1, 1, 1, 0);
+    path.close();
+    path.moveTo(1, 0);
+    path.cubicTo(0, 0, 0, 1, 1, 1);
+    path.close();
+    testSimplifyx(path);
+}
+
+static void (*firstTest)() = 0;
 
 static struct {
     void (*fun)();
     const char* str;
 } tests[] = {
+    TEST(testCubic1),
+    TEST(testQuadralateral1),
     TEST(testLine85),
     TEST(testLine84),
     TEST(testLine84x),
@@ -3997,12 +4025,22 @@ static void testOp2u() {
     testShapeOp(path, pathB, kUnion_Op);
 }
 
+static void testOp8d() {
+    SkPath path, pathB;
+    path.addRect(0, 0, 640, 480);
+    pathB.moveTo(577330, 1971.72f);
+    pathB.cubicTo(10.7082f, -116.596f, 262.057f, 45.6468f, 294.694f, 1.96237f);
+    pathB.close();
+    testShapeOp(path, pathB, kDifference_Op);
+}    
+
 static const size_t testCount = sizeof(tests) / sizeof(tests[0]);
 
 static struct {
     void (*fun)();
     const char* str;
 } subTests[] = {
+    TEST(testOp8d),
     TEST(testDiff1),
     TEST(testIntersect1),
     TEST(testUnion1),
@@ -4024,10 +4062,10 @@ static struct {
 
 static const size_t subTestCount = sizeof(subTests) / sizeof(subTests[0]);
 
-static void (*firstBinaryTest)() = 0;
+static void (*firstBinaryTest)() = testOp8d;
 
 static bool skipAll = false;
-static bool runBinaryTestsFirst = false;
+static bool runBinaryTestsFirst = true;
 static bool runReverse = false;
 static void (*stopTest)() = 0;
 
diff --git a/experimental/Intersection/TSearch.h b/experimental/Intersection/TSearch.h
index 010e69f5e7..6952425585 100644
--- a/experimental/Intersection/TSearch.h
+++ b/experimental/Intersection/TSearch.h
@@ -12,6 +12,35 @@
 // FIXME: Move this templated version into SKTSearch.h
 
 template <typename T>
+static T* QSort_Partition(T* left, T* right, T* pivot)
+{
+    T pivotValue = *pivot;
+    SkTSwap(*pivot, *right);
+    T* newPivot = left;
+    while (left < right) {
+        if (*left < pivotValue) {
+            SkTSwap(*left, *newPivot);
+            newPivot += 1;
+        }
+        left += 1;
+    }
+    SkTSwap(*newPivot, *right);
+    return newPivot;
+}
+
+template <typename T>
+void QSort(T* left, T* right)
+{
+    if (left >= right) {
+        return;
+    }
+    T* pivot = left + (right - left >> 1);
+    pivot = QSort_Partition(left, right, pivot);
+    QSort(left, pivot - 1);
+    QSort(pivot + 1, right);
+}
+
+template <typename T>
 static T** QSort_Partition(T** left, T** right, T** pivot)
 {
     T* pivotValue = *pivot;
diff --git a/experimental/Intersection/op.htm b/experimental/Intersection/op.htm
index 50c26e90c2..8c8a9919ea 100644
--- a/experimental/Intersection/op.htm
+++ b/experimental/Intersection/op.htm
@@ -3295,11 +3295,35 @@ path.addRect(4, 13, 13, 16, SkPath::kCCW_Direction);
     path.addRect(32, 0, 36, 41, SkPath::kCCW_Direction);
 </div>
 
+<div id="testQuadralateral1">
+    path.moveTo(0, 0);
+    path.lineTo(0, 0);
+    path.lineTo(0, 0);
+    path.lineTo(3, 2);
+    path.close();
+    path.moveTo(0, 0);
+    path.lineTo(2, 1);
+    path.lineTo(2, 2);
+    path.lineTo(2, 3);
+    path.close();
+</div>
+
+<div id="testCubic1">
+    path.moveTo(0, 0);
+    path.cubicTo(0, 1, 1, 1, 1, 0);
+    path.close();
+    path.moveTo(1, 0);
+    path.cubicTo(0, 0, 0, 1, 1, 1);
+    path.close();
+</div>
+
 </div>
 
 <script type="text/javascript">
 
 var testDivs = [
+    testCubic1,
+    testQuadralateral1,
     testLine85,
     testLine84x,
     testLine83,
@@ -3763,9 +3787,8 @@ function draw(test, title, _at_x, _at_y, scale) {
         }
         ctx.closePath();
     }
-    if (hasXor) {
-        ctx.fillType=xor; // how is this done?
-    }
+  // uncomment if ever part of the standard
+  //  ctx.fillRule=hasXor ? evenodd : nonzero;
     ctx.stroke();
     ctx.fillStyle="rgba(192,192,255, 0.3)";
     ctx.fill();
diff --git a/experimental/Intersection/qc.htm b/experimental/Intersection/qc.htm
new file mode 100644
index 0000000000..29fedab866
--- /dev/null
+++ b/experimental/Intersection/qc.htm
@@ -0,0 +1,2079 @@
+<html>
+<head>
+<div style="height:0">
+
+<div id="cubic1">
+$1 = (Cubic &) @0x297c40: {{x = 60.776536520932126, y = 71.249307306133829}, {x = 87.107894191103014, y = 22.377669868235323}, {x = 1.4974754310666936, y = 68.069569937917208}, {x = 45.261946574441133, y = 17.536076632112298}}
+$3 = {{{x = 60.776536520932126, y = 71.249307306133829}, {x = 66.996745328074098, y = 59.419614231505768}, {x = 65.760655441289899, y = 53.975522936482086}}, {{x = 65.760655441289899, y = 53.975522936482086}, {x = 64.524565554505699, y = 48.531431641458411}, {x = 59.040356119613065, y = 46.936502854722001}}, {{x = 59.040356119613065, y = 46.936502854722001}, {x = 53.556146684720431, y = 45.341574067985597}, {x = 47.031996847537108, y = 45.059368518219323}}, {{x = 47.031996847537108, y = 45.059368518219323}, {x = 40.29980253329046, y = 44.781843489000011}, {x = 35.915024002796116, y = 43.168182836391942}}, {{x = 35.915024002796116, y = 43.168182836391942}, {x = 31.530245472301775, y = 41.554522183783902}, {x = 32.992157437282373, y = 35.838141687728616}}, {{x = 32.992157437282373, y = 35.838141687728616}, {x = 34.454069402262967, y = 30.121761191673329}, {x = 45.261946574441133, y = 17.536076632112298}}}
+</div>
+
+<div id="cubic2">
+$1 = {{x = 73.565270739405079, y = 11.505317181118446}, {x = 69.865863057722279, y = 35.56041113825534}, {x = 63.830000657509075, y = 90.821050755130614}, {x = 29.400041480269302, y = 26.497158886164968}}
+</div>
+
+<div id="cubic3">
+$3 = {{x = 69.729201388419241, y = 38.687735162064307}, {x = 24.764868814854356, y = 23.150171257159752}, {x = 84.928319083959011, y = 90.258844099128083}, {x = 80.39277404565027, y = 61.35338524419506}}
+</div>
+
+<div id="cubic1x0">
+{{14.5975863, 41.632436}, {16.3518929, 26.2639684}, {18.5165519, 7.68775139}, {8.03767257, 89.1628526}},
+{{14.5975863, 41.632436}, {8.03767257, 89.1628526}, {8.03767257, 89.1628526}},
+</div>
+
+<div id="cubic1x0x">
+{{14.5975863, 41.632436}, {16.3518929, 26.2639684}, {18.5165519, 7.68775139}, {8.03767257, 89.1628526}},
+{{14.5975863, 41.632436}, {8.03767257, 89.1628526}, {8.03767257, 89.1628526}},
+</div>
+
+<div id="cubic1x1">
+{{32.0437334, 59.0267425}, {62.4615541, 91.340573}, {61.0102145, 98.6747985}, {27.3387826, 82.9194526}},
+{{32.0437334, 59.0267425}, {75.9425223, 105.66184}, {27.3387826, 82.9194526}},
+</div>
+
+<div id="cubic1x1x">
+{{32.0437334, 59.0267425}, {62.4615541, 91.340573}, {61.0102145, 98.6747985}, {27.3387826, 82.9194526}},
+{{32.0437334, 59.0267425}, {77.7581975, 107.02498}, {27.3387826, 82.9194526}},
+</div>
+
+<div id="cubic1x2">
+{{57.8949944, 41.1707465}, {56.7368674, 76.5309905}, {56.356649, 86.19953}, {55.5002867, 93.318629}},
+{{57.8949944, 41.1707465}, {55.5002867, 93.318629}, {55.5002867, 93.318629}},
+</div>
+
+<div id="cubic1x2x">
+{{57.8949944, 41.1707465}, {56.7368674, 76.5309905}, {56.356649, 86.19953}, {55.5002867, 93.318629}},
+{{57.8949944, 41.1707465}, {55.5002867, 93.318629}, {55.5002867, 93.318629}},
+</div>
+
+<div id="cubic1x3">
+{{41.1844187, 86.52533}, {31.2211043, 33.1005529}, {20.992908, 27.8044979}, {10.1965708, 73.7658388}},
+{{41.1844187, 86.52533}, {26.1439273, 5.87597256}, {10.1965708, 73.7658388}},
+</div>
+
+<div id="cubic1x3x">
+{{41.1844187, 86.52533}, {31.2211043, 33.1005529}, {20.992908, 27.8044979}, {10.1965708, 73.7658388}},
+{{41.1844187, 86.52533}, {26.3152619, 5.60599593}, {10.1965708, 73.7658388}},
+</div>
+
+<div id="cubic1x4">
+{{51.1608132, 59.7881237}, {58.9955693, 38.5338731}, {38.8048957, 93.8817224}, {70.1083283, 10.6741861}},
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+    cubic6x9,
+    cubic6x9x,
+    cubic7x0,
+    cubic7x0x,
+    cubic7x1,
+    cubic7x1x,
+    cubic7x2,
+    cubic7x2x,
+    cubic7x3,
+    cubic7x3x,
+    cubic7x4,
+    cubic7x4x,
+    cubic7x5,
+    cubic7x5x,
+    cubic7x6,
+    cubic7x6x,
+    cubic7x7,
+    cubic7x7x,
+    cubic7x8,
+    cubic7x8x,
+    cubic7x9,
+    cubic7x9x,
+    cubic8x0,
+    cubic8x0x,
+    cubic8x1,
+    cubic8x1x,
+    cubic8x2,
+    cubic8x2x,
+    cubic8x3,
+    cubic8x3x,
+    cubic8x4,
+    cubic8x4x,
+    cubic8x5,
+    cubic8x5x,
+    cubic8x6,
+    cubic8x6x,
+    cubic8x7,
+    cubic8x7x,
+    cubic8x8,
+    cubic8x8x,
+    cubic8x9,
+    cubic8x9x,
+    cubic3,
+    cubic2,
+    cubic1,
+];
+
+var scale, columns, rows, xStart, yStart;
+
+var ticks = 10;
+var at_x = 13 + 0.5;
+var at_y = 23 + 0.5;
+var init_decimal_places = 1; // make this 3 to show more precision
+var decimal_places;
+var tests = [];
+var testTitles = [];
+var testIndex = 0;
+var ctx;
+var minScale = 1;
+var subscale = 1;
+var curveT = -1;
+var drawCubics = true;
+var drawQuads = true;
+var drawControlLines = true;
+
+function parse(test, title) {
+    var curveStrs = test.split("{{");
+    if (curveStrs.length == 1)
+        curveStrs = test.split("=(");
+    var pattern = /[a-z$=]?-?\d+\.*\d*/g;
+    var curves = [];
+    for (var c in curveStrs) {
+        var curveStr = curveStrs[c];
+        var points = curveStr.match(pattern);
+        var pts = [];
+        for (var wd in points) {
+            var num = parseFloat(points[wd]);
+            if (isNaN(num)) continue;
+            pts.push(num);
+        }
+        if (pts.length > 2)
+            curves.push(pts);
+    }
+    if (curves.length >= 1) {
+        tests.push(curves);
+        testTitles.push(title);
+    }
+}
+
+function init(test) {
+    var canvas = document.getElementById('canvas');
+    if (!canvas.getContext) return;
+    canvas.width = window.innerWidth - at_x;
+    canvas.height = window.innerHeight - at_y;
+    ctx = canvas.getContext('2d');
+    var xmin = Infinity;
+    var xmax = -Infinity;
+    var ymin = Infinity;
+    var ymax = -Infinity;
+    for (var curves in test) {
+        var curve = test[curves];
+        var last = curve.length;
+        for (var idx = 0; idx < last; idx += 2) {
+            xmin = Math.min(xmin, curve[idx]);
+            xmax = Math.max(xmax, curve[idx]);
+            ymin = Math.min(ymin, curve[idx + 1]);
+            ymax = Math.max(ymax, curve[idx + 1]);
+        }
+    }
+    var testW = xmax - xmin;
+    var testH = ymax - ymin;
+    subscale = 1;
+    while (testW * subscale < 0.1 && testH * subscale < 0.1) {
+        subscale *= 10;
+    }
+    while (testW * subscale > 10 && testH * subscale > 10) {
+        subscale /= 10;
+    }
+    xStart = Math.floor(xmin * subscale) / subscale;
+    yStart = Math.floor(ymin * subscale) / subscale;
+    var xEnd = Math.ceil(xmin * subscale) / subscale;
+    var yEnd = Math.ceil(ymin * subscale) / subscale;
+    var cCelsW = Math.floor(ctx.canvas.width / 10);
+    var cCelsH = Math.floor(ctx.canvas.height / 10);
+    testW = xEnd - xStart;
+    testH = yEnd - yStart; 
+    var scaleWH = 1;
+    while (cCelsW > testW * scaleWH * 10 && cCelsH > testH * scaleWH * 10) {
+        scaleWH *= 10;
+    }
+    while (cCelsW * 10 < testW * scaleWH && cCelsH * 10 < testH * scaleWH) {
+        scaleWH /= 10;
+    }
+    
+    columns = Math.ceil(xmax * subscale) - Math.floor(xmin * subscale) + 1;
+    rows = Math.ceil(ymax * subscale) - Math.floor(ymin * subscale) + 1;
+    
+    var hscale = ctx.canvas.width / columns / ticks;
+    var vscale = ctx.canvas.height / rows / ticks;
+    minScale = Math.floor(Math.min(hscale, vscale));
+    scale = minScale * subscale;
+}
+
+function drawPoint(px, py, xoffset, yoffset, unit) {
+    var label = px.toFixed(decimal_places) + ", " + py.toFixed(decimal_places);
+    var _px = px * unit + xoffset;
+    var _py = py * unit + yoffset;
+    ctx.beginPath();
+    ctx.arc(_px, _py, 3, 0, Math.PI*2, true);
+    ctx.closePath();
+    ctx.fill();
+    ctx.fillText(label, _px + 5, _py);
+}
+
+function draw(test, title, scale) {
+    ctx.fillStyle = "rgba(0,0,0, 0.1)";
+    ctx.font = "normal 50px Arial";
+    ctx.fillText(title, 50, 50);
+    ctx.font = "normal 10px Arial";
+
+    var unit = scale * ticks;
+    ctx.lineWidth = 1;
+    var i;
+    for (i = 0; i <= rows * ticks; ++i) {
+        ctx.strokeStyle = (i % ticks) != 0 ? "rgb(200,200,200)" : "black";
+        ctx.beginPath();
+        ctx.moveTo(at_x + 0, at_y + i * minScale);
+        ctx.lineTo(at_x + ticks * columns * minScale, at_y + i * minScale);
+        ctx.stroke();
+    }
+    for (i = 0; i <= columns * ticks; ++i) {
+        ctx.strokeStyle = (i % ticks) != 0 ? "rgb(200,200,200)" : "black";
+        ctx.beginPath();
+        ctx.moveTo(at_x + i * minScale, at_y + 0);
+        ctx.lineTo(at_x + i * minScale, at_y + ticks * rows * minScale);
+        ctx.stroke();
+    }
+ 
+    var xoffset = xStart * -unit + at_x;
+    var yoffset = yStart * -unit + at_y;
+
+    ctx.fillStyle = "rgb(40,80,60)"
+    for (i = 0; i <= columns; i += 1)
+    {
+        num = xStart + i / subscale; 
+        ctx.fillText(num.toFixed(decimal_places), xoffset + num * unit - 5, 10);
+    }
+    for (i = 0; i <= rows; i += 1)
+    {
+        num = yStart + i / subscale; 
+        ctx.fillText(num.toFixed(decimal_places), 0, yoffset + num * unit + 0);
+    }
+    var curves, pts;
+    for (curves in test) {
+        var curve = test[curves];
+        if (curve.length == 6 && !drawQuads) {
+            continue;
+        }
+        if (curve.length == 8 && !drawCubics) {
+            continue;
+        }
+        ctx.beginPath();
+        ctx.moveTo(xoffset + curve[0] * unit, yoffset + curve[1] * unit);
+        switch (curve.length) {
+            case 6:
+                ctx.quadraticCurveTo(
+                    xoffset + curve[2] * unit, yoffset + curve[3] * unit,
+                    xoffset + curve[4] * unit, yoffset + curve[5] * unit);
+                break;
+            case 8:
+                ctx.bezierCurveTo(
+                    xoffset + curve[2] * unit, yoffset + curve[3] * unit,
+                    xoffset + curve[4] * unit, yoffset + curve[5] * unit,
+                    xoffset + curve[6] * unit, yoffset + curve[7] * unit);
+                break;
+        }
+        if (curves == 2) ctx.strokeStyle = curves ? "red" : "blue";
+        ctx.stroke();
+        if (drawControlLines) {
+            ctx.strokeStyle = "rgba(0,0,0, 0.3)";
+            ctx.beginPath();
+            ctx.moveTo(xoffset + curve[0] * unit, yoffset + curve[1] * unit);
+            ctx.lineTo(xoffset + curve[2] * unit, yoffset + curve[3] * unit);
+            ctx.lineTo(xoffset + curve[4] * unit, yoffset + curve[5] * unit);
+            if (curve.length == 8)
+                ctx.lineTo(xoffset + curve[6] * unit, yoffset + curve[7] * unit);
+            ctx.stroke();
+        }
+        if (curveT >= 0 && curveT <= 1) {
+            var x, y;
+            var t = curveT;
+            switch (curve.length) {
+                case 6:
+                    var one_t = 1 - t;
+                    var a = one_t * one_t;
+                    var b = 2 * one_t * t;
+                    var c = t * t;
+                    x = a * curve[0] + b * curve[2] + c * curve[4];
+                    y = a * curve[1] + b * curve[3] + c * curve[5];
+                    break;
+                case 8:
+                    var one_t = 1 - t;
+                    var one_t2 = one_t * one_t;
+                    var a = one_t2 * one_t;
+                    var b = 3 * one_t2 * t;
+                    var t2 = t * t;
+                    var c = 3 * one_t * t2;
+                    var d = t2 * t;
+                    x = a * curve[0] + b * curve[2] + c * curve[4] + d * curve[6];
+                    y = a * curve[1] + b * curve[3] + c * curve[5] + d * curve[7];
+                    break;
+            }
+            drawPoint(x, y, xoffset, yoffset, unit);
+            var num = curveT.toFixed(3);
+            ctx.beginPath();
+            ctx.rect(200,10,200,10);
+            ctx.fillStyle="white";
+            ctx.fill();
+            ctx.fillStyle="black";
+            ctx.fillText(num, 230, 18);
+            if (curve.length == 8) {
+                var one_t = 1 - t;
+                var a = curve[0];
+                var b = curve[2];
+                var c = curve[4];
+                var d = curve[6];
+                var dx = (b - a) * one_t * one_t + 2 * (c - b) * t * one_t + (d - c) * t * t;
+                a = curve[1];
+                b = curve[3];
+                c = curve[5];
+                d = curve[7];
+                var dy = (b - a) * one_t * one_t + 2 * (c - b) * t * one_t + (d - c) * t * t;
+                ctx.beginPath();
+                ctx.moveTo(xoffset + (x - dx) * unit, yoffset + (y - dy) * unit);
+                ctx.lineTo(xoffset + (x + dx) * unit, yoffset + (y + dy) * unit);
+                ctx.stroke();
+            }
+        }
+    }
+}
+
+function drawTop() {
+    init(tests[testIndex]);
+    redraw();
+}
+
+function redraw() {
+    ctx.beginPath();
+    ctx.rect(0, 0, ctx.canvas.width, ctx.canvas.height);
+    ctx.fillStyle="white";
+    ctx.fill();
+    draw(tests[testIndex], testTitles[testIndex], scale);
+}
+
+function doKeyPress(evt) {
+    var char = String.fromCharCode(evt.charCode);
+    switch (char) {
+    case 'c':
+        drawCubics ^= true;
+        redraw();
+        break;
+    case 'l':
+        drawControlLines ^= true;
+        redraw();
+        break;
+    case 'N':
+        testIndex += 9;
+    case 'n':
+        if (++testIndex >= tests.length)
+            testIndex = 0;
+        mouseX = Infinity;
+        drawTop();
+        break;
+    case 'P':
+        testIndex -= 9;
+    case 'p':
+        if (--testIndex < 0)
+            testIndex = tests.length - 1;
+        mouseX = Infinity;
+        drawTop();
+        break;
+    case 'q':
+        drawQuads ^= true;
+        redraw();
+        break;
+    case 'x':
+        drawCubics ^= true;
+        drawQuads ^= true;
+        redraw();
+        break;
+    }
+}
+
+function handleMouseClick() {
+    var e = window.event;
+	var tgt = e.target || e.srcElement;
+    var min = tgt.offsetTop + Math.ceil(at_y);
+    var max = min + ticks * rows * minScale;
+    curveT = (e.clientY - min) / (max - min);
+    redraw();
+}
+
+function calcXY() {
+    var e = window.event;
+	var tgt = e.target || e.srcElement;
+    var left = tgt.offsetLeft;
+    var top = tgt.offsetTop;
+    var unit = scale * ticks;
+    mouseX = (e.clientX - left - Math.ceil(at_x) + 1) / unit + xStart;
+    mouseY = (e.clientY - top - Math.ceil(at_y)) / unit + yStart;
+}
+
+function handleMouseOver() {
+    calcXY();
+    var num = mouseX.toFixed(3) + ", " + mouseY.toFixed(3);
+    ctx.beginPath();
+    ctx.rect(30,10,200,10);
+    ctx.fillStyle="white";
+    ctx.fill();
+    ctx.fillStyle="black";
+    ctx.fillText(num, 30, 18);
+}
+
+function start() {
+    for (i = 0; i < testDivs.length; ++i) {
+        var title = testDivs[i].id.toString();
+        var str = testDivs[i].firstChild.data;
+        parse(str, title);
+    }
+    drawTop();
+    window.addEventListener('keypress', doKeyPress, true);
+    window.onresize = function() {
+        drawTop();
+    }
+}
+
+</script>
+</head>
+
+<body onLoad="start();">
+<canvas id="canvas" width="750" height="500"
+    onmousemove="handleMouseOver()"
+    onclick="handleMouseClick()"
+    ></canvas >
+</body>
+</html>