adaptive division for bezierCubicToQuad fontello#25

lib/math.js

@@ -1,5 +1,8 @@
 'use strict';
 
+var maxDefect = Math.sqrt(3) / 18;
+var precision = 1.0;
+
 function Point(x, y) {
   this.x = x;
   this.y = y;
@@ -30,12 +33,15 @@ Point.prototype.sqr = function () {
 };
 
 
+
 // converts cubic bezier to quad
 function toQuad(p1, c1, c2, p2) {
   // Quad control point is (3*c2 - p2 + 3*c1 - p1)/4
   return [p1, c2.mul(3).sub(p2).add(c1.mul(3)).sub(p1).div(4), p2];
 }
 
+
+
 /*
  * Aproximates cubic curve to 2 quad curves. Returns array of quad curves
  *
@@ -47,17 +53,7 @@ function toQuad(p1, c1, c2, p2) {
  *
  * (!) We could use more slices, but FONT SIZE DOES MATTER !!!
  */
-function bezierCubicToQuad(p1, c1, c2, p2) {
-
-  // check first, if we can aproximate with quad directly
-  // |p2 - 3*c2 + 3*c1 - p1|/2 should be small (zero in ideal)
-  // http://www.caffeineowl.com/graphics/2d/vectorial/cubic2quad01.html
-  var cpDistance = p2.sub(c2.mul(3)).add(c1.mul(3)).sub(p1).dist()/2;
-
-  if (cpDistance <= 3) {
-    return [ toQuad(p1, c1, c2, p2) ];
-  }
-
+function midpointSplit(p1, c1, c2, p2){
   // Split to 2 qubic beziers
   // https://www.atalasoft.com/blogs/stevehawley/may-2013/how-to-split-a-cubic-bezier-curve
   // http://www.timotheegroleau.com/Flash/articles/cubic_bezier/bezier_lib.as
@@ -71,9 +67,50 @@ function bezierCubicToQuad(p1, c1, c2, p2) {
 
   // now replace each half with quad curve
   return [ q1, q2 ];
+}
+
 
+
+/*
+ * Adaptive division:
+ * consider a division point at the maximum value of t for which the first cubic segment
+ * can still be approximated by a quadratic within the required precision;
+ * repeat the previous step for the remaining of the cubic.
+ */
+function bezierCubicToQuad(p1, c1, c2, p2){
+  // check first, if we can aproximate with quad directly
+  // |p2 - 3*c2 + 3*c1 - p1|/2 should be small (zero in ideal)
+  // http://www.caffeineowl.com/graphics/2d/vectorial/cubic2quad01.html
+  var cpDistance = p2.sub(c2.mul(3)).add(c1.mul(3)).sub(p1).dist()/2;
+
+  if (cpDistance <= 3) {
+    return [ toQuad(p1, c1, c2, p2) ];
+  }
+
+  //compute maximum t for which the first cubic segment can still be approximated by a quadratic
+  // within the required precision
+  var t = Math.pow(precision / (cpDistance * maxDefect), 1/3);
+
+  if (t < 1) {
+    // detaching segment of cubic: p1 s11 s12 s3 - first, s3 s21 s22 p2 - second
+    var ct = c2.sub(c1).mul(t).add(c1);
+    var s11 = c1.sub(p1).mul(t).add(p1);
+    var s12 = ct.sub(s11).mul(t).add(s11);
+    var s22 = p2.sub(c2).mul(t).add(c2);
+    var s21 = s22.sub(ct).mul(t).add(ct);
+    var s3 = s21.sub(s12).mul(t).add(s12);
+
+    //midpointSplit for first segment and recursive adaptiveDivision for second
+    return midpointSplit(p1, s11, s12, s3)
+      .concat(bezierCubicToQuad(s3, s21, s22, p2));
+  }
+
+  // if t is greater than 1.0, the cubic Bezier can be directly approximated by the
+  // mid-point approximation within the specified tolerance
+  return midpointSplit(p1, c1, c2, p2);
 }
 
+
 /*
  * Check if 3 points are in line, and second in the midle.
  * Used to replace quad curves with lines or join lines