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 ate2πki/m=cos(2kπ/m)+isin(2kπ/m)↔(cos(2kπ/m),sin(2kπ/m)).Given a sequence of integers (s0,…,sm)mod m, we connect consecutive values sjsj+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 (0,1,…,m)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 2k=±1modm. Moreover, these control polygons represent the eigenvectors of the associated subdivision matrices corresponding to the eigenvalue 2−1modm. 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.