Implement Conics in SkCurveMeasure

src/utils/SkCurveMeasure.cpp

Index: src/utils/SkCurveMeasure.cpp
diff --git a/src/utils/SkCurveMeasure.cpp b/src/utils/SkCurveMeasure.cpp
index a38a4fd1f0b99b79fe5544ac11905dbfb12232ed..425b6f35efb18cf7dc27306926083cf3fa6d2d1e 100644
--- a/src/utils/SkCurveMeasure.cpp
+++ b/src/utils/SkCurveMeasure.cpp
@@ -69,6 +69,7 @@ static inline SkVector evaluateDerivative(const SkPoint pts[4],
 static inline Sk8f evaluateDerivativeLength(const Sk8f& ts,
                                             const Sk8f (&xCoeff)[3],
                                             const Sk8f (&yCoeff)[3],
+                                            const SkScalar w,
                                             const SkSegType segType) {
     Sk8f x;
     Sk8f y;
@@ -81,8 +82,16 @@ static inline Sk8f evaluateDerivativeLength(const Sk8f& ts,
             x = (xCoeff[0]*ts + xCoeff[1])*ts + xCoeff[2];
             y = (yCoeff[0]*ts + yCoeff[1])*ts + yCoeff[2];
             break;
-        case kConic_SegType:
-            UNIMPLEMENTED;
+        case kConic_SegType: {
+            Sk8f xnum = (xCoeff[0]*ts + xCoeff[1])*ts + xCoeff[2];
+            Sk8f ynum = (yCoeff[0]*ts + yCoeff[1])*ts + yCoeff[2];
+
+            Sk8f denom =  ts*(-4 + 4*w + ts*(8 - 12*w + 4*w*w + ts*(-8 + 16*w -
+                            8*w*w + ts*(4 - 8*w + 4*w*w)))) + 1;

[caryclark, 2016/08/15 19:13:04]

Where did this come from? That is, please document your derivation.

C++ compilers will not do a good job of refactoring this. I see trivial common
terms:
-8 + 16*w - 8*w*w
 4 -  8*w + 4*w*w

Have you looked at the compiled output? Are you getting the SIMD you expect?

[Harry Stern, 2016/08/15 20:55:16]

I took out those terms. I honestly didn't notice them when I looked because I
was too fixated on the 12.

I am getting SIMD from this function in the output. `objdump -S -w -d
out/Release/obj/src/utils/utils.SkCurveMeasure.o` is showing me that the
function was inlined and is using e.g. mulps, addps, sqrtps

[Harry Stern, 2016/08/17 02:48:46]

Also, I have added the mathematica commands used to generate the coefficients.
Is there something better I could do?

+
+            x = xnum / denom;
+            y = ynum / denom;

[caryclark, 2016/08/15 19:13:04]

Is there a unit test for this?

[Harry Stern, 2016/08/15 20:55:16]

It's on my todo by tomorrow.

+        }
             break;
         default:
             UNIMPLEMENTED;
@@ -134,8 +143,27 @@ ArcLengthIntegrator::ArcLengthIntegrator(const SkPoint* pts, SkSegType segType)
             yCoeff[2] = Sk8f(3.0f*(-Ay + By));
         }
             break;
-        case kConic_SegType:
-            UNIMPLEMENTED;
+        case kConic_SegType: {
+            float Ax = pts[0].x();
+            float Bx = pts[1].x();
+            float Cx = pts[2].x();
+            float Ay = pts[0].y();
+            float By = pts[1].y();
+            float Cy = pts[2].y();
+
+            fConicW = pts[3].x();

[Harry Stern, 2016/08/15 18:01:21]

fConicW gets set here

+
+            SkScalar w = fConicW;
+
+            // precompute coefficients for derivative
+            xCoeff[0] = Sk8f(2*(Ax - Cx - Ax*w + Cx*w));
+            xCoeff[1] = Sk8f(-2*(Ax + Cx + 2*(Ax*w - Bx*w)));
+            xCoeff[2] = Sk8f(-2*(Ax*w + Bx*w));
+
+            yCoeff[0] = Sk8f(2*(Ay - Cy - Ay*w + Cy*w));
+            yCoeff[1] = Sk8f(-2*(Ay + Cy + 2*(Ay*w - By*w)));
+            yCoeff[2] = Sk8f(-2*(Ay*w + By*w));
+        }
             break;
         default:
             UNIMPLEMENTED;
@@ -148,7 +176,7 @@ ArcLengthIntegrator::ArcLengthIntegrator(const SkPoint* pts, SkSegType segType)
 SkScalar ArcLengthIntegrator::computeLength(SkScalar t) {
     SkScalar length = 0.0f;
 
-    Sk8f lengths = evaluateDerivativeLength(absc*t, xCoeff, yCoeff, fSegType);
+    Sk8f lengths = evaluateDerivativeLength(absc*t, xCoeff, yCoeff, fConicW, fSegType);
     lengths = weights*lengths;
     // is it faster or more accurate to sum and then multiply or vice versa?
     lengths = lengths*(t*0.5f);