Do not snap curve points to t = 0 / 1 with epsilon

Closes paperjs#936. Also: - Reduce CLIPPING_EPSILON to 1e-9 to fix errors that were masked by the snapping - Implement unit tests for edge cases in paperjs#936
bmacnaughton · Feb 6, 2016 · 0371f66 · 0371f66
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diff --git a/src/path/Curve.js b/src/path/Curve.js
@@ -1306,10 +1306,7 @@ new function() { // Scope for methods that require private functions
             c1x = v[2], c1y = v[3],
             c2x = v[4], c2y = v[5],
             p2x = v[6], p2y = v[7],
-            tMin = /*#=*/Numerical.CURVETIME_EPSILON,
-            tMax = 1 - tMin,
-            isZero = Numerical.isZero,
-            x, y;
+            isZero = Numerical.isZero;
         // If the curve handles are almost zero, reset the control points to the
         // anchors.
         if (isZero(c1x - p1x) && isZero(c1y - p1y)) {
@@ -1320,70 +1317,69 @@ new function() { // Scope for methods that require private functions
             c2x = p2x;
             c2y = p2y;
         }
-        // Handle special case at beginning / end of curve
-        if (type === 0 && (t < tMin || t > tMax)) {
-            var zero = t < tMin;
-            x = zero ? p1x : p2x;
-            y = zero ? p1y : p2y;
+        // Calculate the polynomial coefficients.
+        var cx = 3 * (c1x - p1x),
+            bx = 3 * (c2x - c1x) - cx,
+            ax = p2x - p1x - cx - bx,
+            cy = 3 * (c1y - p1y),
+            by = 3 * (c2y - c1y) - cy,
+            ay = p2y - p1y - cy - by,
+            x, y;
+        if (type === 0) {
+            // type === 0: getPoint()
+            // Calculate the curve point at parameter value t
+            x = ((ax * t + bx) * t + cx) * t + p1x;
+            y = ((ay * t + by) * t + cy) * t + p1y;
         } else {
-            // Calculate the polynomial coefficients.
-            var cx = 3 * (c1x - p1x),
-                bx = 3 * (c2x - c1x) - cx,
-                ax = p2x - p1x - cx - bx,
-
-                cy = 3 * (c1y - p1y),
-                by = 3 * (c2y - c1y) - cy,
-                ay = p2y - p1y - cy - by;
-            if (type === 0) {
-                // Calculate the curve point at parameter value t
-                x = ((ax * t + bx) * t + cx) * t + p1x;
-                y = ((ay * t + by) * t + cy) * t + p1y;
+            // type === 1: getTangent()
+            // type === 2: getNormal()
+            // type === 3: getCurvature()
+            var tMin = /*#=*/Numerical.CURVETIME_EPSILON,
+                tMax = 1 - tMin;
+            // 1: tangent, 1st derivative
+            // 2: normal, 1st derivative
+            // 3: curvature, 1st derivative & 2nd derivative
+            // Simply use the derivation of the bezier function for both
+            // the x and y coordinates:
+            // Prevent tangents and normals of length 0:
+            // http://stackoverflow.com/questions/10506868/
+            if (t < tMin) {
+                x = cx;
+                y = cy;
+            } else if (t > tMax) {
+                x = 3 * (p2x - c2x);
+                y = 3 * (p2y - c2y);
             } else {
-                // 1: tangent, 1st derivative
-                // 2: normal, 1st derivative
-                // 3: curvature, 1st derivative & 2nd derivative
-                // Simply use the derivation of the bezier function for both
-                // the x and y coordinates:
-                // Prevent tangents and normals of length 0:
-                // http://stackoverflow.com/questions/10506868/
-                if (t < tMin) {
-                    x = cx;
-                    y = cy;
-                } else if (t > tMax) {
-                    x = 3 * (p2x - c2x);
-                    y = 3 * (p2y - c2y);
-                } else {
-                    x = (3 * ax * t + 2 * bx) * t + cx;
-                    y = (3 * ay * t + 2 * by) * t + cy;
-                }
-                if (normalized) {
-                    // When the tangent at t is zero and we're at the beginning
-                    // or the end, we can use the vector between the handles,
-                    // but only when normalizing as its weighted length is 0.
-                    if (x === 0 && y === 0 && (t < tMin || t > tMax)) {
-                        x = c2x - c1x;
-                        y = c2y - c1y;
-                    }
-                    // Now normalize x & y
-                    var len = Math.sqrt(x * x + y * y);
-                    if (len) {
-                        x /= len;
-                        y /= len;
-                    }
+                x = (3 * ax * t + 2 * bx) * t + cx;
+                y = (3 * ay * t + 2 * by) * t + cy;
+            }
+            if (normalized) {
+                // When the tangent at t is zero and we're at the beginning
+                // or the end, we can use the vector between the handles,
+                // but only when normalizing as its weighted length is 0.
+                if (x === 0 && y === 0 && (t < tMin || t > tMax)) {
+                    x = c2x - c1x;
+                    y = c2y - c1y;
                 }
-                if (type === 3) {
-                    // Calculate 2nd derivative, and curvature from there:
-                    // http://cagd.cs.byu.edu/~557/text/ch2.pdf page#31
-                    // k = |dx * d2y - dy * d2x| / (( dx^2 + dy^2 )^(3/2))
-                    var x2 = 6 * ax * t + 2 * bx,
-                        y2 = 6 * ay * t + 2 * by,
-                        d = Math.pow(x * x + y * y, 3 / 2);
-                    // For JS optimizations we always return a Point, although
-                    // curvature is just a numeric value, stored in x:
-                    x = d !== 0 ? (x * y2 - y * x2) / d : 0;
-                    y = 0;
+                // Now normalize x & y
+                var len = Math.sqrt(x * x + y * y);
+                if (len) {
+                    x /= len;
+                    y /= len;
                 }
             }
+            if (type === 3) {
+                // Calculate 2nd derivative, and curvature from there:
+                // http://cagd.cs.byu.edu/~557/text/ch2.pdf page#31
+                // k = |dx * d2y - dy * d2x| / (( dx^2 + dy^2 )^(3/2))
+                var x2 = 6 * ax * t + 2 * bx,
+                    y2 = 6 * ay * t + 2 * by,
+                    d = Math.pow(x * x + y * y, 3 / 2);
+                // For JS optimizations we always return a Point, although
+                // curvature is just a numeric value, stored in x:
+                x = d !== 0 ? (x * y2 - y * x2) / d : 0;
+                y = 0;
+            }
         }
         // The normal is simply the rotated tangent:
         return type === 2 ? new Point(y, -x) : new Point(x, y);

diff --git a/src/util/Numerical.js b/src/util/Numerical.js
@@ -114,9 +114,10 @@ var Numerical = new function() {
          */
         TRIGONOMETRIC_EPSILON: 1e-7,
         /**
-         * The epsilon to be used in the fat-line clipping code.
+         * The epsilon to be used when comparing curve-time parameters in the
+         * fat-line clipping code.
          */
-        CLIPPING_EPSILON: 1e-7,
+        CLIPPING_EPSILON: 1e-9,
         /**
          * Kappa is the value which which to scale the curve handles when
          * drawing a circle with bezier curves.

diff --git a/test/tests/Path_Boolean.js b/test/tests/Path_Boolean.js
@@ -394,6 +394,40 @@ test('#899', function() {
         'M349.31715,176.97192l-41.08297,217.42545c-0.3852,2.03862 -25.43665,23.05386 -34.63134,21.31651c0,0 45.72613,-193.57408 61.79225,-248.85827c-4.42287,-6.99195 -6.99542,-14.22758 -1.98246,-13.27272c1.9865,0.37838 3.84798,1.03847 5.55095,1.93102c0.85247,-2.33202 1.44527,-3.46596 1.73311,-3.20238c5.69403,5.21442 8.66306,11.93238 9.02453,18.54465c0.10579,1.93513 -0.00703,3.88198 -0.35068,5.82386c-0.01706,0.09726 -0.03486,0.19455 -0.0534,0.29188z');
 });
 
+test('#936', function() {
+    var p1 = new Path({
+        segments:[
+            [437.02857142857147, 121.64285714285714],
+            [390.3571428571429, 74.97142857142856],
+            [390.3571428571429, 437.02857142857147],
+            [437.02857142857147, 390.3571428571429],
+            [74.97142857142862, 390.3571428571429],
+            [121.6428571428572, 437.02857142857147],
+            [121.64285714285714, 74.97142857142858],
+            [74.97142857142859, 121.64285714285714]
+        ],
+        closed:true
+    });
+    compareBoolean(function() { return p1.unite(); },
+        'M390.35714,121.64286l0,268.71429l-268.71429,0l0,-268.71429z M121.64286,437.02857l-46.67143,-46.67143l46.67143,0z M390.35714,121.64286l0,-46.67143l46.67143,46.67143z M437.02857,390.35714l-46.67143,46.67143l0,-46.67143z M74.97143,121.64286l46.67143,-46.67143l0,46.67143z');
+
+var p2 = new Path({
+        segments:[
+            [384.0884708724184, 194.49096034834812, -66.07592496222571, -39.741588860234515],
+            [364.2895784370851, 164.00333956484477, 0, 0, 9.430248075502885, 76.52772155228502],
+            [317.5090396516519, 384.0884708724184, 39.74158886023454, -66.07592496222564],
+            [347.9966604351552, 364.2895784370851, 0, 0, -76.52772155228439, 9.430248075502845],
+            [127.91152912758163, 317.50903965165196, 66.0759249622259, 39.74158886023463],
+            [147.71042156291495, 347.9966604351553, 0, 0, -9.43024807550288, -76.52772155228509],
+            [194.49096034834812, 127.91152912758156, -39.74158886023451, 66.07592496222576],
+            [164.00333956484474, 147.7104215629149, 0, 0, 76.52772155228499, -9.430248075502845]
+        ],
+        closed:true
+    });
+    compareBoolean(function() { return p2.unite(); },
+        'M366.1932,184.4397c3.98163,62.97187 -9.38253,125.8453 -38.6329,181.75349c-62.97187,3.98163 -125.8453,-9.38253 -181.75349,-38.6329c-3.98163,-62.97187 9.38253,-125.8453 38.6329,-181.75349c62.97187,-3.98163 125.8453,9.38253 181.75349,38.6329z M164.00334,147.71042l30.48762,-19.79889c-3.53519,5.87774 -6.88622,11.84573 -10.05126,17.89528c-6.81387,0.43083 -13.62889,1.06475 -20.43636,1.90362z M147.71042,347.99666l-19.79889,-30.48762c5.87774,3.53519 11.84573,6.88622 17.89528,10.05126c0.43083,6.81387 1.06475,13.62889 1.90362,20.43636z M366.1932,184.4397c-0.43083,-6.81387 -1.06475,-13.62889 -1.90362,-20.43636l19.79889,30.48762c-5.87774,-3.53519 -11.84573,-6.88622 -17.89528,-10.05126z M347.99666,364.28958l-30.48762,19.79889c3.53519,-5.87774 6.88622,-11.84573 10.05126,-17.89528c6.81387,-0.43083 13.62889,-1.06475 20.43636,-1.90362z');
+});
+
 test('frame.intersect(rect);', function() {
     var frame = new CompoundPath();
     frame.addChild(new Path.Rectangle(new Point(140, 10), [100, 300]));