Add base types for path ops
Paths contain lines, quads, and cubics, which are
collectively curves.
To work with path intersections, intermediary curves
are constructed. For now, those intermediates use
doubles to guarantee sufficient precision.
The DVector, DPoint, DLine, DQuad, and DCubic
structs encapsulate these intermediate curves.
The DRect and DTriangle structs are created to
describe intersectable areas of interest.
The Bounds struct inherits from SkRect to create
a SkScalar-based rectangle that intersects shared
edges.
This also includes common math equalities and
debugging that the remainder of path ops builds on,
as well as a temporary top-level interface in
include/pathops/SkPathOps.h.
Review URL: https://codereview.chromium.org/12827020
git-svn-id: http://skia.googlecode.com/svn/trunk@8551 2bbb7eff-a529-9590-31e7-b0007b416f81
diff --git a/src/pathops/SkReduceOrder.cpp b/src/pathops/SkReduceOrder.cpp
new file mode 100644
index 0000000..14da4cc
--- /dev/null
+++ b/src/pathops/SkReduceOrder.cpp
@@ -0,0 +1,450 @@
+/*
+ * 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);
+}