what you're seeing
Put m points round a circle, numbered 0…m−1. From each point i, draw one chord to
point (k·i) mod m — the k times-table on the ring ℤ/mℤ. The chords
are never the curve; they're tangent to it, and a cardioid (k=2),
a nephroid (k=3), or a k−1-cusp epicycloid EMERGES as their envelope.
The same ring is a
cipher wheel:
(k·P) mod m is the multiplicative
half of an affine cipher — the half the additive
Volvelle never had.