The envelope of chords in a circle, the cardioid, the nephroid, and more line graphs

The Envelope of Chords

Assuming there are N dots evenly distributed on a circle, named as a_1, a_2, a_3, \cdots, a_{N}. We can draw chords between a_{n} and a_{kn} with n\in [1,N], k=1,\ 2, \ 3, \cdots.

Cardioid (k=2)

With N=2M, M is natural number, and k=2, the envelope of these chords is a cardioid. Check this link for proof.

Nephroid (k=3)

With N=3M and k=3, the envelope of these chords is a nephroid. Check this link for proof.

Graphs for N=60, k from 1 to 61

Graphs for N=128, k from 1 to 129