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

1 files changed, 261 insertions, 0 deletions

diff --git a/src/pathops/SkDCubicLineIntersection.cpp b/src/pathops/SkDCubicLineIntersection.cpp
new file mode 100644
index 0000000000..5df3acacfd
--- /dev/null
+++ b/src/pathops/SkDCubicLineIntersection.cpp
@@ -0,0 +1,261 @@
+/*
+ * 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 "SkIntersections.h"
+#include "SkPathOpsCubic.h"
+#include "SkPathOpsLine.h"
+
+/*
+Find the interection of a line and cubic by solving for valid t values.
+
+Analogous to line-quadratic intersection, solve line-cubic intersection by
+representing the cubic as:
+  x = a(1-t)^3 + 2b(1-t)^2t + c(1-t)t^2 + dt^3
+  y = e(1-t)^3 + 2f(1-t)^2t + g(1-t)t^2 + ht^3
+and the line as:
+  y = i*x + j  (if the line is more horizontal)
+or:
+  x = i*y + j  (if the line is more vertical)
+
+Then using Mathematica, solve for the values of t where the cubic intersects the
+line:
+
+  (in) Resultant[
+        a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - x,
+        e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - i*x - j, x]
+  (out) -e     +   j     +
+       3 e t   - 3 f t   -
+       3 e t^2 + 6 f t^2 - 3 g t^2 +
+         e t^3 - 3 f t^3 + 3 g t^3 - h t^3 +
+     i ( a     -
+       3 a t + 3 b t +
+       3 a t^2 - 6 b t^2 + 3 c t^2 -
+         a t^3 + 3 b t^3 - 3 c t^3 + d t^3 )
+
+if i goes to infinity, we can rewrite the line in terms of x. Mathematica:
+
+  (in) Resultant[
+        a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - i*y - j,
+        e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - y,       y]
+  (out)  a     -   j     -
+       3 a t   + 3 b t   +
+       3 a t^2 - 6 b t^2 + 3 c t^2 -
+         a t^3 + 3 b t^3 - 3 c t^3 + d t^3 -
+     i ( e     -
+       3 e t   + 3 f t   +
+       3 e t^2 - 6 f t^2 + 3 g t^2 -
+         e t^3 + 3 f t^3 - 3 g t^3 + h t^3 )
+
+Solving this with Mathematica produces an expression with hundreds of terms;
+instead, use Numeric Solutions recipe to solve the cubic.
+
+The near-horizontal case, in terms of:  Ax^3 + Bx^2 + Cx + D == 0
+    A =   (-(-e + 3*f - 3*g + h) + i*(-a + 3*b - 3*c + d)     )
+    B = 3*(-( e - 2*f +   g    ) + i*( a - 2*b +   c    )     )
+    C = 3*(-(-e +   f          ) + i*(-a +   b          )     )
+    D =   (-( e                ) + i*( a                ) + j )
+
+The near-vertical case, in terms of:  Ax^3 + Bx^2 + Cx + D == 0
+    A =   ( (-a + 3*b - 3*c + d) - i*(-e + 3*f - 3*g + h)     )
+    B = 3*( ( a - 2*b +   c    ) - i*( e - 2*f +   g    )     )
+    C = 3*( (-a +   b          ) - i*(-e +   f          )     )
+    D =   ( ( a                ) - i*( e                ) - j )
+
+For horizontal lines:
+(in) Resultant[
+      a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - j,
+      e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - y, y]
+(out)  e     -   j     -
+     3 e t   + 3 f t   +
+     3 e t^2 - 6 f t^2 + 3 g t^2 -
+       e t^3 + 3 f t^3 - 3 g t^3 + h t^3
+ */
+
+class LineCubicIntersections {
+public:
+
+LineCubicIntersections(const SkDCubic& c, const SkDLine& l, SkIntersections& i)
+    : cubic(c)
+    , line(l)
+    , intersections(i) {
+}
+
+// see parallel routine in line quadratic intersections
+int intersectRay(double roots[3]) {
+    double adj = line[1].fX - line[0].fX;
+    double opp = line[1].fY - line[0].fY;
+    SkDCubic r;
+    for (int n = 0; n < 4; ++n) {
+        r[n].fX = (cubic[n].fY - line[0].fY) * adj - (cubic[n].fX - line[0].fX) * opp;
+    }
+    double A, B, C, D;
+    SkDCubic::Coefficients(&r[0].fX, &A, &B, &C, &D);
+    return SkDCubic::RootsValidT(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)) {
+            SkDPoint pt = line.xyAtT(lineT);
+            intersections.insert(cubicT, lineT, pt);
+        }
+    }
+    return intersections.used();
+}
+
+int horizontalIntersect(double axisIntercept, double roots[3]) {
+    double A, B, C, D;
+    SkDCubic::Coefficients(&cubic[0].fY, &A, &B, &C, &D);
+    D -= axisIntercept;
+    return SkDCubic::RootsValidT(A, B, C, D, roots);
+}
+
+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 cubicT = rootVals[index];
+        SkDPoint pt = cubic.xyAtT(cubicT);
+        double lineT = (pt.fX - left) / (right - left);
+        if (pinTs(&cubicT, &lineT)) {
+            intersections.insert(cubicT, lineT, pt);
+        }
+    }
+    if (flipped) {
+        intersections.flip();
+    }
+    return intersections.used();
+}
+
+int verticalIntersect(double axisIntercept, double roots[3]) {
+    double A, B, C, D;
+    SkDCubic::Coefficients(&cubic[0].fX, &A, &B, &C, &D);
+    D -= axisIntercept;
+    return SkDCubic::RootsValidT(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 cubicT = rootVals[index];
+        SkDPoint pt = cubic.xyAtT(cubicT);
+        double lineT = (pt.fY - top) / (bottom - top);
+        if (pinTs(&cubicT, &lineT)) {
+            intersections.insert(cubicT, lineT, pt);
+        }
+    }
+    if (flipped) {
+        intersections.flip();
+    }
+    return intersections.used();
+}
+
+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, line[lIndex]);
+            }
+        }
+    }
+}
+
+void addHorizontalEndPoints(double left, double right, double y) {
+    for (int cIndex = 0; cIndex < 4; cIndex += 3) {
+        if (cubic[cIndex].fY != y) {
+            continue;
+        }
+        if (cubic[cIndex].fX == left) {
+            intersections.insert(cIndex >> 1, 0, cubic[cIndex]);
+        }
+        if (cubic[cIndex].fX == right) {
+            intersections.insert(cIndex >> 1, 1, cubic[cIndex]);
+        }
+    }
+}
+
+void addVerticalEndPoints(double top, double bottom, double x) {
+    for (int cIndex = 0; cIndex < 4; cIndex += 3) {
+        if (cubic[cIndex].fX != x) {
+            continue;
+        }
+        if (cubic[cIndex].fY == top) {
+            intersections.insert(cIndex >> 1, 0, cubic[cIndex]);
+        }
+        if (cubic[cIndex].fY == bottom) {
+            intersections.insert(cIndex >> 1, 1, cubic[cIndex]);
+        }
+    }
+}
+
+double findLineT(double t) {
+    SkDPoint xy = cubic.xyAtT(t);
+    double dx = line[1].fX - line[0].fX;
+    double dy = line[1].fY - line[0].fY;
+    if (fabs(dx) > fabs(dy)) {
+        return (xy.fX - line[0].fX) / dx;
+    }
+    return (xy.fY - line[0].fY) / dy;
+}
+
+static bool pinTs(double* cubicT, double* lineT) {
+    if (!approximately_one_or_less(*lineT)) {
+        return false;
+    }
+    if (!approximately_zero_or_more(*lineT)) {
+        return false;
+    }
+    if (precisely_less_than_zero(*cubicT)) {
+        *cubicT = 0;
+    } else if (precisely_greater_than_one(*cubicT)) {
+        *cubicT = 1;
+    }
+    if (precisely_less_than_zero(*lineT)) {
+        *lineT = 0;
+    } else if (precisely_greater_than_one(*lineT)) {
+        *lineT = 1;
+    }
+    return true;
+}
+
+private:
+
+const SkDCubic& cubic;
+const SkDLine& line;
+SkIntersections& intersections;
+};
+
+int SkIntersections::horizontal(const SkDCubic& cubic, double left, double right, double y,
+        bool flipped) {
+    LineCubicIntersections c(cubic, *(static_cast<SkDLine*>(0)), *this);
+    return c.horizontalIntersect(y, left, right, flipped);
+}
+
+int SkIntersections::vertical(const SkDCubic& cubic, double top, double bottom, double x,
+        bool flipped) {
+    LineCubicIntersections c(cubic, *(static_cast<SkDLine*>(0)), *this);
+    return c.verticalIntersect(x, top, bottom, flipped);
+}
+
+int SkIntersections::intersect(const SkDCubic& cubic, const SkDLine& line) {
+    LineCubicIntersections c(cubic, line, *this);
+    return c.intersect();
+}
+
+int SkIntersections::intersectRay(const SkDCubic& cubic, const SkDLine& line) {
+    LineCubicIntersections c(cubic, line, *this);
+    return c.intersectRay(fT[0]);
+}