Web Animations: Allow timing inputs outside the range [0, 1]

Source/platform/animation/UnitBezier.h

 /*
  * Copyright (C) 2008 Apple Inc. All Rights Reserved.
  *
  * Redistribution and use in source and binary forms, with or without
  * modification, are permitted provided that the following conditions
  * are met:
  * 1. Redistributions of source code must retain the above copyright
  *    notice, this list of conditions and the following disclaimer.
  * 2. Redistributions in binary form must reproduce the above copyright
  *    notice, this list of conditions and the following disclaimer in the
@@ skipping 25 matching lines @@
     UnitBezier(double p1x, double p1y, double p2x, double p2y)
     {
         // Calculate the polynomial coefficients, implicit first and last control points are (0,0) and (1,1).
         cx = 3.0 * p1x;
         bx = 3.0 * (p2x - p1x) - cx;
         ax = 1.0 - cx -bx;
 
         cy = 3.0 * p1y;
         by = 3.0 * (p2y - p1y) - cy;
         ay = 1.0 - cy - by;
+
+        // End-point gradients are used to calculate timing function results
+        // outside the range [0, 1].
+        //
+        // There are three possibilities for the gradient at each end:
+        // (1) the closest control point is not horizontally coincident with regard to

[alancutter (OOO until 2018), 2014/04/16 04:50:42]

Change to "adjacent control point"?
Closest sounds like you're talking terms of proximity in space when you mean
proximity in the sequence.
This should say "with the end point", this is what the code appears to do.

[shans, 2014/04/17 06:04:18]

I don't think these edits make this prose any better.

+        //     (0, 0) or (1, 1). In this case the line between the end point and
+        //     the control point is tangent to the bezier at the end point.
+        // (2) the closest control point is coincident with the end point. In

[alancutter (OOO until 2018), 2014/04/16 04:50:42]

Add "and the far control point is not horizontally coincident with the end
point", the code appears to also check this condition as well.

I suppose this would technically add another condition to cover where the
adjacent control point is coincident with the end point and the far control
point is horizontally coincident but vertically distinct that should fall under
case 3.

[shans, 2014/04/17 06:04:18]

Likewise.

+        //     this case the line between the end point and the far control
+        //     point is tangent to the bezier at the end point.
+        // (3) the closest control point is horizontally coincident with the end
+        //     point, but vertically distinct. In this case the gradient at the
+        //     end point is Infinite. However, this causes issues when
+        //     interpolating. As a result, we break down to a simple case of
+        //     0 gradient under these conditions.
+
+        if (p1x > 0)
+            m_startGradient = p1y / p1x;
+        else if (!p1y && p2x > 0)
+            m_startGradient = p2y / p2x;
+        else
+            m_startGradient = 0;
+
+        if (p2x < 1)
+            m_endGradient = (p2y - 1) / (p2x - 1);
+        else if (p2x == 1 && p1x < 1)
+            m_endGradient = (p1y - 1) / (p1x - 1);
+        else
+            m_endGradient = 0;
     }
 
     double sampleCurveX(double t)
     {
         // `ax t^3 + bx t^2 + cx t' expanded using Horner's rule.
         return ((ax * t + bx) * t + cx) * t;
     }
 
     double sampleCurveY(double t)
     {
@@ skipping 47 matching lines @@
 
         // Failure.
         return t2;
     }
 
     // Evaluates y at the given x. The epsilon parameter provides a hint as to the required
     // accuracy and is not guaranteed.
     double solve(double x, double epsilon)
     {
         if (x < 0.0)
-            return 0.0;
+            return 0.0 + m_startGradient * x;
         if (x > 1.0)
-            return 1.0;
+            return 1.0 + m_endGradient * (x - 1.0);
         return sampleCurveY(solveCurveX(x, epsilon));
     }
 
 private:
     double ax;
     double bx;
     double cx;
 
     double ay;
     double by;
     double cy;
+
+    double m_startGradient;
+    double m_endGradient;
 };
 
 } // namespace WebCore
 
 #endif // UnitBezier_h