Merge pull request #765 from l8l/patch-1

Update Mandelbrot.pde

Author: shiffman

content/examples/Topics/Fractals and L-Systems/Mandelbrot/Mandelbrot.pde

@@ -1,7 +1,8 @@
 /**
  * The Mandelbrot Set
  * by Daniel Shiffman.  
- * 
+ * (slight modification by l8l)
+ *
  * Simple rendering of the Mandelbrot set.
  */
 
@@ -43,21 +44,30 @@ for (int j = 0; j < height; j++) {
   float x = xmin;
   for (int i = 0; i < width; i++) {
 
-    // Now we test, as we iterate z = z^2 + cm does z tend towards infinity?
+    // Now we test, as we iterate z = z^2 + c does z tend towards infinity?
     float a = x;
     float b = y;
     int n = 0;
+    float max = 4.0;  // Infinity in our finite world is simple, let's just consider it 4
+    float absOld = 0.0;
+    float convergeNumber = maxiterations; // this will change if the while loop breaks due to non-convergence
     while (n < maxiterations) {
+      // We suppose z = a+ib
       float aa = a * a;
       float bb = b * b;
-      float twoab = 2.0 * a * b;
-      a = aa - bb + x;
-      b = twoab + y;
-      // Infinty in our finite world is simple, let's just consider it 16
-      if (dist(aa, bb, 0, 0) > 4.0) {
+      float abs = sqrt(aa + bb);
+      if (abs > max) { // |z| = sqrt(a^2+b^2)
+        // Now measure how much we exceeded the maximum: 
+        float diffToLast = (float) (abs - absOld);
+        float diffToMax  = (float) (max - absOld);
+        convergeNumber = n + diffToMax/diffToLast;
         break;  // Bail
       }
+      float twoab = 2.0 * a * b;
+      a = aa - bb + x; // this operation corresponds to z -> z^2+c where z=a+ib c=(x,y)
+      b = twoab + y;
       n++;
+      absOld = abs;
     }
 
     // We color each pixel based on how long it takes to get to infinity
@@ -66,11 +76,11 @@ for (int j = 0; j < height; j++) {
       pixels[i+j*width] = color(0);
     } else {
       // Gosh, we could make fancy colors here if we wanted
-      float norm = map(n, 0, maxiterations, 0, 1);
+      float norm = map(convergeNumber, 0, maxiterations, 0, 1);
       pixels[i+j*width] = color(map(sqrt(norm), 0, 1, 0, 255));
     }
     x += dx;
   }
   y += dy;
 }
-updatePixels();
+updatePixels();