Explanation
I was inspired to make this application by the work of Simon Plouffe. This construction is known as the Cremona generation (after Luigi Cremona). The recipe is: divide p points equally over the circumference of a circle and number them from n = 0 to n = p-1. Then join points n and (n × t) mod p with a straight line segment (a chord). Variable t is a constant factor. We iterate over the times table of t and each time we take the result modulo p. In figure 1:
- join n=0 and 0 × 2 mod 10 = 0
- join n=1 and 1 × 2 mod 10 = 2
- join n=2 and 2 × 2 mod 10 = 4
- join n=3 and 3 × 2 mod 10 = 6
- join n=4 and 4 × 2 mod 10 = 8
- join n=5 and 5 × 2 mod 10 = 0
- join n=6 and 6 × 2 mod 10 = 2 (exists already)
- etc.
Envelopes of some of these pencils of lines form Epicycloids. If t = 2, the envelope forms a cardioid. If t = 3, the envelope forms a nephroid. If t = 4, the envelope forms a three-cusped epicycloid (see first 3 figures below).
Examples
