Add intersections for path ops

src/pathops/SkReduceOrder.cpp

+/*
+ * Copyright 2012 Google Inc.
+ *
+ * Use of this source code is governed by a BSD-style license that can be
+ * found in the LICENSE file.
+ */
+#include "SkReduceOrder.h"
+
+int SkReduceOrder::reduce(const SkDLine& line) {
+    fLine[0] = line[0];
+    int different = line[0] != line[1];
+    fLine[1] = line[different];
+    return 1 + different;
+}
+
+static double interp_quad_coords(double a, double b, double c, double t) {
+    double ab = SkDInterp(a, b, t);
+    double bc = SkDInterp(b, c, t);
+    return SkDInterp(ab, bc, t);
+}
+
+static int coincident_line(const SkDQuad& quad, SkDQuad& reduction) {
+    reduction[0] = reduction[1] = quad[0];
+    return 1;
+}
+
+static int reductionLineCount(const SkDQuad& reduction) {
+    return 1 + !reduction[0].approximatelyEqual(reduction[1]);
+}
+
+static int vertical_line(const SkDQuad& quad, SkReduceOrder::Style reduceStyle,
+        SkDQuad& reduction) {
+    double tValue;
+    reduction[0] = quad[0];
+    reduction[1] = quad[2];
+    if (reduceStyle == SkReduceOrder::kFill_Style) {
+        return reductionLineCount(reduction);
+    }
+    int smaller = reduction[1].fY > reduction[0].fY;
+    int larger = smaller ^ 1;
+    if (SkDQuad::FindExtrema(quad[0].fY, quad[1].fY, quad[2].fY, &tValue)) {
+        double yExtrema = interp_quad_coords(quad[0].fY, quad[1].fY, quad[2].fY, tValue);
+        if (reduction[smaller].fY > yExtrema) {
+            reduction[smaller].fY = yExtrema;
+        } else if (reduction[larger].fY < yExtrema) {
+            reduction[larger].fY = yExtrema;
+        }
+    }
+    return reductionLineCount(reduction);
+}
+
+static int horizontal_line(const SkDQuad& quad, SkReduceOrder::Style reduceStyle,
+        SkDQuad& reduction) {
+    double tValue;
+    reduction[0] = quad[0];
+    reduction[1] = quad[2];
+    if (reduceStyle == SkReduceOrder::kFill_Style) {
+        return reductionLineCount(reduction);
+    }
+    int smaller = reduction[1].fX > reduction[0].fX;
+    int larger = smaller ^ 1;
+    if (SkDQuad::FindExtrema(quad[0].fX, quad[1].fX, quad[2].fX, &tValue)) {
+        double xExtrema = interp_quad_coords(quad[0].fX, quad[1].fX, quad[2].fX, tValue);
+        if (reduction[smaller].fX > xExtrema) {
+            reduction[smaller].fX = xExtrema;
+        }  else if (reduction[larger].fX < xExtrema) {
+            reduction[larger].fX = xExtrema;
+        }
+    }
+    return reductionLineCount(reduction);
+}
+
+static int check_linear(const SkDQuad& quad, SkReduceOrder::Style reduceStyle,
+        int minX, int maxX, int minY, int maxY, SkDQuad& reduction) {
+    int startIndex = 0;
+    int endIndex = 2;
+    while (quad[startIndex].approximatelyEqual(quad[endIndex])) {
+        --endIndex;
+        if (endIndex == 0) {
+            SkDebugf("%s shouldn't get here if all four points are about equal", __FUNCTION__);
+            SkASSERT(0);
+        }
+    }
+    if (!quad.isLinear(startIndex, endIndex)) {
+        return 0;
+    }
+    // four are colinear: return line formed by outside
+    reduction[0] = quad[0];
+    reduction[1] = quad[2];
+    if (reduceStyle == SkReduceOrder::kFill_Style) {
+        return reductionLineCount(reduction);
+    }
+    int sameSide;
+    bool useX = quad[maxX].fX - quad[minX].fX >= quad[maxY].fY - quad[minY].fY;
+    if (useX) {
+        sameSide = SkDSign(quad[0].fX - quad[1].fX) + SkDSign(quad[2].fX - quad[1].fX);
+    } else {
+        sameSide = SkDSign(quad[0].fY - quad[1].fY) + SkDSign(quad[2].fY - quad[1].fY);
+    }
+    if ((sameSide & 3) != 2) {
+        return reductionLineCount(reduction);
+    }
+    double tValue;
+    int root;
+    if (useX) {
+        root = SkDQuad::FindExtrema(quad[0].fX, quad[1].fX, quad[2].fX, &tValue);
+    } else {
+        root = SkDQuad::FindExtrema(quad[0].fY, quad[1].fY, quad[2].fY, &tValue);
+    }
+    if (root) {
+        SkDPoint extrema;
+        extrema.fX = interp_quad_coords(quad[0].fX, quad[1].fX, quad[2].fX, tValue);
+        extrema.fY = interp_quad_coords(quad[0].fY, quad[1].fY, quad[2].fY, tValue);
+        // sameSide > 0 means mid is smaller than either [0] or [2], so replace smaller
+        int replace;
+        if (useX) {
+            if (extrema.fX < quad[0].fX ^ extrema.fX < quad[2].fX) {
+                return reductionLineCount(reduction);
+            }
+            replace = (extrema.fX < quad[0].fX | extrema.fX < quad[2].fX)
+                    ^ (quad[0].fX < quad[2].fX);
+        } else {
+            if (extrema.fY < quad[0].fY ^ extrema.fY < quad[2].fY) {
+                return reductionLineCount(reduction);
+            }
+            replace = (extrema.fY < quad[0].fY | extrema.fY < quad[2].fY)
+                    ^ (quad[0].fY < quad[2].fY);
+        }
+        reduction[replace] = extrema;
+    }
+    return reductionLineCount(reduction);
+}
+
+// reduce to a quadratic or smaller
+// look for identical points
+// look for all four points in a line
+    // note that three points in a line doesn't simplify a cubic
+// look for approximation with single quadratic
+    // save approximation with multiple quadratics for later
+int SkReduceOrder::reduce(const SkDQuad& quad, Style reduceStyle) {
+    int index, minX, maxX, minY, maxY;
+    int minXSet, minYSet;
+    minX = maxX = minY = maxY = 0;
+    minXSet = minYSet = 0;
+    for (index = 1; index < 3; ++index) {
+        if (quad[minX].fX > quad[index].fX) {
+            minX = index;
+        }
+        if (quad[minY].fY > quad[index].fY) {
+            minY = index;
+        }
+        if (quad[maxX].fX < quad[index].fX) {
+            maxX = index;
+        }
+        if (quad[maxY].fY < quad[index].fY) {
+            maxY = index;
+        }
+    }
+    for (index = 0; index < 3; ++index) {
+        if (AlmostEqualUlps(quad[index].fX, quad[minX].fX)) {
+            minXSet |= 1 << index;
+        }
+        if (AlmostEqualUlps(quad[index].fY, quad[minY].fY)) {
+            minYSet |= 1 << index;
+        }
+    }
+    if (minXSet == 0x7) {  // test for vertical line
+        if (minYSet == 0x7) {  // return 1 if all four are coincident
+            return coincident_line(quad, fQuad);
+        }
+        return vertical_line(quad, reduceStyle, fQuad);
+    }
+    if (minYSet == 0xF) {  // test for horizontal line
+        return horizontal_line(quad, reduceStyle, fQuad);
+    }
+    int result = check_linear(quad, reduceStyle, minX, maxX, minY, maxY, fQuad);
+    if (result) {
+        return result;
+    }
+    fQuad = quad;
+    return 3;
+}
+
+////////////////////////////////////////////////////////////////////////////////////
+
+static double interp_cubic_coords(const double* src, double t) {
+    double ab = SkDInterp(src[0], src[2], t);
+    double bc = SkDInterp(src[2], src[4], t);
+    double cd = SkDInterp(src[4], src[6], t);
+    double abc = SkDInterp(ab, bc, t);
+    double bcd = SkDInterp(bc, cd, t);
+    return SkDInterp(abc, bcd, t);
+}
+
+static int coincident_line(const SkDCubic& cubic, SkDCubic& reduction) {
+    reduction[0] = reduction[1] = cubic[0];
+    return 1;
+}
+
+static int reductionLineCount(const SkDCubic& reduction) {
+    return 1 + !reduction[0].approximatelyEqual(reduction[1]);
+}
+
+static int vertical_line(const SkDCubic& cubic, SkReduceOrder::Style reduceStyle,
+                         SkDCubic& reduction) {
+    double tValues[2];
+    reduction[0] = cubic[0];
+    reduction[1] = cubic[3];
+    if (reduceStyle == SkReduceOrder::kFill_Style) {
+        return reductionLineCount(reduction);
+    }
+    int smaller = reduction[1].fY > reduction[0].fY;
+    int larger = smaller ^ 1;
+    int roots = SkDCubic::FindExtrema(cubic[0].fY, cubic[1].fY, cubic[2].fY, cubic[3].fY, tValues);
+    for (int index = 0; index < roots; ++index) {
+        double yExtrema = interp_cubic_coords(&cubic[0].fY, tValues[index]);
+        if (reduction[smaller].fY > yExtrema) {
+            reduction[smaller].fY = yExtrema;
+            continue;
+        }
+        if (reduction[larger].fY < yExtrema) {
+            reduction[larger].fY = yExtrema;
+        }
+    }
+    return reductionLineCount(reduction);
+}
+
+static int horizontal_line(const SkDCubic& cubic, SkReduceOrder::Style reduceStyle,
+                           SkDCubic& reduction) {
+    double tValues[2];
+    reduction[0] = cubic[0];
+    reduction[1] = cubic[3];
+    if (reduceStyle == SkReduceOrder::kFill_Style) {
+        return reductionLineCount(reduction);
+    }
+    int smaller = reduction[1].fX > reduction[0].fX;
+    int larger = smaller ^ 1;
+    int roots = SkDCubic::FindExtrema(cubic[0].fX, cubic[1].fX, cubic[2].fX, cubic[3].fX, tValues);
+    for (int index = 0; index < roots; ++index) {
+        double xExtrema = interp_cubic_coords(&cubic[0].fX, tValues[index]);
+        if (reduction[smaller].fX > xExtrema) {
+            reduction[smaller].fX = xExtrema;
+            continue;
+        }
+        if (reduction[larger].fX < xExtrema) {
+            reduction[larger].fX = xExtrema;
+        }
+    }
+    return reductionLineCount(reduction);
+}
+
+// check to see if it is a quadratic or a line
+static int check_quadratic(const SkDCubic& cubic, SkDCubic& reduction) {
+    double dx10 = cubic[1].fX - cubic[0].fX;
+    double dx23 = cubic[2].fX - cubic[3].fX;
+    double midX = cubic[0].fX + dx10 * 3 / 2;
+    if (!AlmostEqualUlps(midX - cubic[3].fX, dx23 * 3 / 2)) {
+        return 0;
+    }
+    double dy10 = cubic[1].fY - cubic[0].fY;
+    double dy23 = cubic[2].fY - cubic[3].fY;
+    double midY = cubic[0].fY + dy10 * 3 / 2;
+    if (!AlmostEqualUlps(midY - cubic[3].fY, dy23 * 3 / 2)) {
+        return 0;
+    }
+    reduction[0] = cubic[0];
+    reduction[1].fX = midX;
+    reduction[1].fY = midY;
+    reduction[2] = cubic[3];
+    return 3;
+}
+
+static int check_linear(const SkDCubic& cubic, SkReduceOrder::Style reduceStyle,
+        int minX, int maxX, int minY, int maxY, SkDCubic& reduction) {
+    int startIndex = 0;
+    int endIndex = 3;
+    while (cubic[startIndex].approximatelyEqual(cubic[endIndex])) {
+        --endIndex;
+        if (endIndex == 0) {
+            SkDebugf("%s shouldn't get here if all four points are about equal\n", __FUNCTION__);
+            SkASSERT(0);
+        }
+    }
+    if (!cubic.isLinear(startIndex, endIndex)) {
+        return 0;
+    }
+    // four are colinear: return line formed by outside
+    reduction[0] = cubic[0];
+    reduction[1] = cubic[3];
+    if (reduceStyle == SkReduceOrder::kFill_Style) {
+        return reductionLineCount(reduction);
+    }
+    int sameSide1;
+    int sameSide2;
+    bool useX = cubic[maxX].fX - cubic[minX].fX >= cubic[maxY].fY - cubic[minY].fY;
+    if (useX) {
+        sameSide1 = SkDSign(cubic[0].fX - cubic[1].fX) + SkDSign(cubic[3].fX - cubic[1].fX);
+        sameSide2 = SkDSign(cubic[0].fX - cubic[2].fX) + SkDSign(cubic[3].fX - cubic[2].fX);
+    } else {
+        sameSide1 = SkDSign(cubic[0].fY - cubic[1].fY) + SkDSign(cubic[3].fY - cubic[1].fY);
+        sameSide2 = SkDSign(cubic[0].fY - cubic[2].fY) + SkDSign(cubic[3].fY - cubic[2].fY);
+    }
+    if (sameSide1 == sameSide2 && (sameSide1 & 3) != 2) {
+        return reductionLineCount(reduction);
+    }
+    double tValues[2];
+    int roots;
+    if (useX) {
+        roots = SkDCubic::FindExtrema(cubic[0].fX, cubic[1].fX, cubic[2].fX, cubic[3].fX, tValues);
+    } else {
+        roots = SkDCubic::FindExtrema(cubic[0].fY, cubic[1].fY, cubic[2].fY, cubic[3].fY, tValues);
+    }
+    for (int index = 0; index < roots; ++index) {
+        SkDPoint extrema;
+        extrema.fX = interp_cubic_coords(&cubic[0].fX, tValues[index]);
+        extrema.fY = interp_cubic_coords(&cubic[0].fY, tValues[index]);
+        // sameSide > 0 means mid is smaller than either [0] or [3], so replace smaller
+        int replace;
+        if (useX) {
+            if (extrema.fX < cubic[0].fX ^ extrema.fX < cubic[3].fX) {
+                continue;
+            }
+            replace = (extrema.fX < cubic[0].fX | extrema.fX < cubic[3].fX)
+                    ^ (cubic[0].fX < cubic[3].fX);
+        } else {
+            if (extrema.fY < cubic[0].fY ^ extrema.fY < cubic[3].fY) {
+                continue;
+            }
+            replace = (extrema.fY < cubic[0].fY | extrema.fY < cubic[3].fY)
+                    ^ (cubic[0].fY < cubic[3].fY);
+        }
+        reduction[replace] = extrema;
+    }
+    return reductionLineCount(reduction);
+}
+
+/* food for thought:
+http://objectmix.com/graphics/132906-fast-precision-driven-cubic-quadratic-piecewise-degree-reduction-algos-2-a.html
+
+Given points c1, c2, c3 and c4 of a cubic Bezier, the points of the
+corresponding quadratic Bezier are (given in convex combinations of
+points):
+
+q1 = (11/13)c1 + (3/13)c2 -(3/13)c3 + (2/13)c4
+q2 = -c1 + (3/2)c2 + (3/2)c3 - c4
+q3 = (2/13)c1 - (3/13)c2 + (3/13)c3 + (11/13)c4
+
+Of course, this curve does not interpolate the end-points, but it would
+be interesting to see the behaviour of such a curve in an applet.
+
+--
+Kalle Rutanen
+http://kaba.hilvi.org
+
+*/
+
+// reduce to a quadratic or smaller
+// look for identical points
+// look for all four points in a line
+    // note that three points in a line doesn't simplify a cubic
+// look for approximation with single quadratic
+    // save approximation with multiple quadratics for later
+int SkReduceOrder::reduce(const SkDCubic& cubic, Quadratics allowQuadratics,
+        Style reduceStyle) {
+    int index, minX, maxX, minY, maxY;
+    int minXSet, minYSet;
+    minX = maxX = minY = maxY = 0;
+    minXSet = minYSet = 0;
+    for (index = 1; index < 4; ++index) {
+        if (cubic[minX].fX > cubic[index].fX) {
+            minX = index;
+        }
+        if (cubic[minY].fY > cubic[index].fY) {
+            minY = index;
+        }
+        if (cubic[maxX].fX < cubic[index].fX) {
+            maxX = index;
+        }
+        if (cubic[maxY].fY < cubic[index].fY) {
+            maxY = index;
+        }
+    }
+    for (index = 0; index < 4; ++index) {
+        double cx = cubic[index].fX;
+        double cy = cubic[index].fY;
+        double denom = SkTMax(fabs(cx), SkTMax(fabs(cy),
+                SkTMax(fabs(cubic[minX].fX), fabs(cubic[minY].fY))));
+        if (denom == 0) {
+            minXSet |= 1 << index;
+            minYSet |= 1 << index;
+            continue;
+        }
+        double inv = 1 / denom;
+        if (approximately_equal_half(cx * inv, cubic[minX].fX * inv)) {
+            minXSet |= 1 << index;
+        }
+        if (approximately_equal_half(cy * inv, cubic[minY].fY * inv)) {
+            minYSet |= 1 << index;
+        }
+    }
+    if (minXSet == 0xF) {  // test for vertical line
+        if (minYSet == 0xF) {  // return 1 if all four are coincident
+            return coincident_line(cubic, fCubic);
+        }
+        return vertical_line(cubic, reduceStyle, fCubic);
+    }
+    if (minYSet == 0xF) {  // test for horizontal line
+        return horizontal_line(cubic, reduceStyle, fCubic);
+    }
+    int result = check_linear(cubic, reduceStyle, minX, maxX, minY, maxY, fCubic);
+    if (result) {
+        return result;
+    }
+    if (allowQuadratics == SkReduceOrder::kAllow_Quadratics
+            && (result = check_quadratic(cubic, fCubic))) {
+        return result;
+    }
+    fCubic = cubic;
+    return 4;
+}
+
+SkPath::Verb SkReduceOrder::Quad(const SkPoint a[3], SkTDArray<SkPoint>* reducePts) {
+    SkDQuad quad;
+    quad.set(a);
+    SkReduceOrder reducer;
+    int order = reducer.reduce(quad, kFill_Style);
+    if (order == 2) {  // quad became line
+        for (int index = 0; index < order; ++index) {
+            SkPoint* pt = reducePts->append();
+            pt->fX = SkDoubleToScalar(reducer.fLine[index].fX);
+            pt->fY = SkDoubleToScalar(reducer.fLine[index].fY);
+        }
+    }
+    return (SkPath::Verb) (order - 1);
+}
+
+SkPath::Verb SkReduceOrder::Cubic(const SkPoint a[4], SkTDArray<SkPoint>* reducePts) {
+    SkDCubic cubic;
+    cubic.set(a);
+    SkReduceOrder reducer;
+    int order = reducer.reduce(cubic, kAllow_Quadratics, kFill_Style);
+    if (order == 2 || order == 3) {  // cubic became line or quad
+        for (int index = 0; index < order; ++index) {
+            SkPoint* pt = reducePts->append();
+            pt->fX = SkDoubleToScalar(reducer.fQuad[index].fX);
+            pt->fY = SkDoubleToScalar(reducer.fQuad[index].fY);
+        }
+    }
+    return (SkPath::Verb) (order - 1);
+}