bezier: geek out and write a general case

Is this a bad idea? I think it is. But the high school math competition go-er in me demands it.

src/generator/bezier.js

@@ -7,6 +7,19 @@ const Color = require('../Color');
 require('../io/lab');
 const scale = require('./scale');
 
+// nth row of the pascal triangle
+const binom_row = function(n) {
+    let row = [1, 1];
+    for (let i = 1; i < n; i++) {
+        let newrow = [1];
+        for (let j = 1; j <= row.length; j++) {
+            newrow[j] = (row[j] || 0) + row[j - 1];
+        }
+        row = newrow;
+    }
+    return row;
+}
+
 const bezier = function(colors) {
     let I, lab0, lab1, lab2;
     colors = colors.map(c => new Color(c));
@@ -32,18 +45,19 @@ const bezier = function(colors) {
             const lab = ([0, 1, 2].map((i) => ((1-t)*(1-t)*(1-t) * lab0[i]) + (3 * (1-t) * (1-t) * t * lab1[i]) + (3 * (1-t) * t * t * lab2[i]) + (t*t*t * lab3[i])));
             return new Color(lab, 'lab');
         };
-    } else if (colors.length === 5) {
-        const I0 = bezier(colors.slice(0, 3));
-        const I1 = bezier(colors.slice(2, 5));
-        I = function(t) {
-            if (t < 0.5) {
-                return I0(t*2);
-            } else {
-                return I1((t-0.5)*2);
-            }
+    } else if (colors.length >= 5) {
+        // general case (degree n bezier)
+        let labs, row, n;
+        labs = colors.map(c => c.lab());
+        n = colors.length - 1
+        row = binom_row(n);
+        I = function (t) {
+            const u = 1 - t;
+            const lab = ([0, 1, 2].map((i) => labs.reduce((sum, el, j) => (sum + row[j] * u ** (n - 1) * t ** n * el[i]), 0)))
+            return new Color(lab, 'lab');
         };
     } else {
-        throw new RangeError("Bezier can only accept between 2 to 5 colors; got `${colors.length}`.")
+        throw new RangeError("No point in running bezier with only one color.")
     }
     return I;
 };