Optimize QuadF's PointIsInTriangle function

The original version required 34 float ops (16mul + 11sub + 5add + 2div).
This version removes some of the intermediate dot products, reducing the
max number of operations to 19 (6mul + 2div + 11sub).  It also avoids using
doubles, warranted by the removal of several intermediate operations that might
otherwise incur significance loss.

In local (release build) benchmarks, this reduced the typical CPU cost of a call to
QuadF::Contains(...) by 2/3, both with GCC and Clang on an ARM v7 device.

Review URL: https://codereview.chromium.org/291493003

git-svn-id: svn://svn.chromium.org/chrome/trunk/src@271088 0039d316-1c4b-4281-b951-d872f2087c98

ui/gfx/geometry/quad_f.cc

@@ -62,28 +62,24 @@ static inline bool PointIsInTriangle(const PointF& point,
                                      const PointF& r1,
                                      const PointF& r2,
                                      const PointF& r3) {
-  // Compute the barycentric coordinates of |point| relative to the triangle
-  // (r1, r2, r3). This algorithm comes from Christer Ericson's Real-Time
-  // Collision Detection.
-  Vector2dF v0 = r2 - r1;
-  Vector2dF v1 = r3 - r1;
-  Vector2dF v2 = point - r1;
+  // Compute the barycentric coordinates (u, v, w) of |point| relative to the
+  // triangle (r1, r2, r3) by the solving the system of equations:
+  //   1) point = u * r1 + v * r2 + w * r3
+  //   2) u + v + w = 1
+  // This algorithm comes from Christer Ericson's Real-Time Collision Detection.
 
-  double dot00 = DotProduct(v0, v0);
-  double dot01 = DotProduct(v0, v1);
-  double dot11 = DotProduct(v1, v1);
-  double dot20 = DotProduct(v2, v0);
-  double dot21 = DotProduct(v2, v1);
+  Vector2dF r31 = r1 - r3;
+  Vector2dF r32 = r2 - r3;
+  Vector2dF r3p = point - r3;
 
-  double denom = dot00 * dot11 - dot01 * dot01;
-
-  double v = (dot11 * dot20 - dot01 * dot21) / denom;
-  double w = (dot00 * dot21 - dot01 * dot20) / denom;
-  double u = 1 - v - w;
+  float denom = r32.y() * r31.x() - r32.x() * r31.y();
+  float u = (r32.y() * r3p.x() - r32.x() * r3p.y()) / denom;
+  float v = (r31.x() * r3p.y() - r31.y() * r3p.x()) / denom;
+  float w = 1.f - u - v;
 
   // Use the barycentric coordinates to test if |point| is inside the
   // triangle (r1, r2, r2).
-  return (v >= 0) && (w >= 0) && (u >= 0);
+  return (u >= 0) && (v >= 0) && (w >= 0);
 }
 
 bool QuadF::Contains(const PointF& point) const {

ui/gfx/geometry/quad_f.h

@@ -52,7 +52,7 @@ class GFX_EXPORT QuadF {
   bool IsCounterClockwise() const;
 
   // Returns true if the |point| is contained within the quad, or lies on on
-  // edge of the quad.
+  // edge of the quad. This assumes that the quad is convex.
   bool Contains(const gfx::PointF& point) const;
 
   // Returns a rectangle that bounds the four points of the quad. The points of