diff --git a/src/path/Curve.js b/src/path/Curve.js
index 56360668..a7cad223 100644
--- a/src/path/Curve.js
+++ b/src/path/Curve.js
@@ -667,23 +667,35 @@ statics: {
     },
 
     getParameterOf: function(v, point) {
+        // Before solving cubics, compare the beginning and end of the curve
+        // with zero epsilon:
+        var p1 = new Point(v[0], v[1]),
+            p2 = new Point(v[6], v[7]),
+            epsilon = /*#=*/Numerical.EPSILON,
+            t = point.isClose(p1, epsilon) ? 0
+              : point.isClose(p2, epsilon) ? 1
+              : null;
+        if (t !== null)
+            return t;
+        // Solve the cubic for both x- and y-coordinates and consider all found
+        // solutions, testing with the larger / looser geometric epsilon.
         var coords = [point.x, point.y],
             roots = [],
-            epsilon = /*#=*/Numerical.GEOMETRIC_EPSILON;
+            geomEpsilon = /*#=*/Numerical.GEOMETRIC_EPSILON;
         for (var c = 0; c < 2; c++) {
             var count = Curve.solveCubic(v, c, coords[c], roots, 0, 1);
             for (var i = 0; i < count; i++) {
-                var t = roots[i];
-                if (point.isClose(Curve.getPoint(v, t), epsilon))
+                t = roots[i];
+                if (point.isClose(Curve.getPoint(v, t), geomEpsilon))
                     return t;
             }
         }
         // For very short curves (length ~ 1e-13), the above code will not
         // necessarily produce any valid roots. As a fall-back, just check the
         // beginnings and ends at the end so we can still return a valid result.
-        return point.isClose(new Point(v[0], v[1]), epsilon) ? 0
-            : point.isClose(new Point(v[6], v[7]), epsilon) ? 1
-            : null;
+        return point.isClose(p1, geomEpsilon) ? 0
+             : point.isClose(p2, geomEpsilon) ? 1
+             : null;
     },
 
     // TODO: Find better name