Butterfly Curve

Attributed to Temple Fay
See also: Chrysanthemum curve

The so called "butterfly" curve is given by the equation

x = cos(u) (e^cos(u) - 2 cos(4 u) - sin(u / 12)^5.0)
y = sin(u) (e^cos(u) - 2 cos(4 u) - sin(u / 12)^5.0)

#include "stdlib.h"
#include "stdio.h"
#include "math.h"

#define N 10000

typedef struct {
   double x,y,z;
} XYZ;

int main(int argc,char **argv)
{
   int i;
   double u;
   XYZ p,plast;

   for (i=0;i<N;i++) {
      u = i * 24.0 * PI / N;
      p.x = cos(u) * (exp(cos(u)) - 2 * cos(4 * u) - pow(sin(u / 12),5.0));
      p.y = sin(u) * (exp(cos(u)) - 2 * cos(4 * u) - pow(sin(u / 12),5.0));
      p.z = fabs(p.y) / 2;
      colour = GetColour(u,0.0,24*PI,4);
      if (i > 0) {
         Do something with the line from plast to p
      }
      plast = p;
   }
}

Anaglyph contribution by Jean Tousset.

[Click for higher resolution version]

References

Temple H. Fay, "The Butterfly Curve" American Mathematical Monthly 96, No. 5, May 1989.

Temple H. Fay, "A Study in Step Size" Mathematics Magazine 70, No 2, April 1997