Subdivision algorithms with modular arithmetic

Abstract

We study the de Casteljau subdivision algorithm for Bezier curves and the Lane-Riesenfeld algorithm for uniform B-spline curves over the integers mod m, where $m>2$ is an odd integer. We place the integers mod m evenly spaced around a unit circle so that the integer k mod m is located at the position on the unit circle at$e^{2\pi \mathrm{ki}/m}=\cos (2k\pi /m)+i\sin (2k\pi /m)\leftrightarrow (\cos (2k\pi /m),\sin (2k\pi /m)).$Given a sequence of integers $\left(s_0,\dots ,s_m\right)$ mod m, we connect consecutive values $s_j{s}_{j+1}$ on the unit circle with straight line segments to form a control polygon. We show that if we start these subdivision procedures with the sequence $\left(0,1,\dots ,m\right)$ mod m, then the sequences generated by these recursive subdivision algorithms spawn control polygons consisting of the regular m-sided polygon and regular m-pointed stars that repeat with a period equal to the minimal integer k such that $2^k=\pm 1modm$. Moreover, these control polygons represent the eigenvectors of the associated subdivision matrices corresponding to the eigenvalue $2^{-1}modm.$ We go on to study the effects of these subdivision procedures on more general initial control polygons, and we show in particular that certain control polygons, including the orbits of regular m-sided polygons and the complete graphs of m-sided polygons, are fixed points of these subdivision procedures.