Circles, chords and epicycloids

Abstract

In a recent session of the Royal Institution Master Classes in Cambridge, the enthusiastic participants explored a computer graphics program, designed and written by Derek and Alan Ball. The program first draws a circle, and, given an integer N, marks N equally spaced points on it, these being cyclically labelled 1,2,…,N. The operator then inputs some rule, say n ↦ 5n + 8, and the program draws, for each n, the chord from the point marked n to the point marked 5n + 8 (modulo N). Before long, the children took N to be large and flower-like envelopes of the chords started to appear (as illustrated below). Here, we show how the equations of these envelopes can be obtained in a simple way with some general parameter k, and that for positive k they are all epicycloids, that is, curves obtained by rolling a circle around a fixed circle, the simplest one being a cardioid. A similar analysis shows that for negative k they are hypocycloids (see for example E. H. Lockwood’s Book of curves, C.U.P. 1961) except when k = 1 when parallel chords occur.