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$e^{2\pi \mathit{ki}/m}=\cos (2k\pi /m)+i\sin (2k\pi /m)\leftrightarrow (\cos (2k\pi /m),\sin (2k\pi /m)).$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.

我们研究了贝塞尔曲线的 de Casteljau 细分算法和整数mod m上均匀 B 样条曲线的 Lane-Riesenfeld 算法，其中$米>2$是一个奇整数。我们将整数mod m均匀地放置在单位圆周围，使得整数k mod m位于单位圆上的位置$e^{2\pi \mathit{气}/米}=\mathrm{因斯}（2k\pi /米）+我罪（2k\pi /米）\leftrightarrow （\mathrm{因斯}（2k\pi /米）,罪（2k\pi /米））。$给定一个整数序列$（s_0,\dots \dots ,s_{米}）$ mod m，我们连接连续的值$s_j{s}_{j+1}$在单位圆上用直线段构成控制多边形。我们证明，如果我们按照以下顺序开始这些细分过程$（0,1,\dots \dots ,米）$ mod m，然后由这些递归细分算法生成的序列生成由规则m边多边形和规则m尖星组成的控制多边形，这些多边形以等于最小整数k 的周期重复，使得$2^k=\pm 1米哦d米$。而且，这些控制多边形表示特征值对应的相关细分矩阵的特征向量$2^{-1}米哦d米。$我们继续研究这些细分过程对更一般的初始控制多边形的影响，并且特别表明某些控制多边形，包括正m边多边形的轨道和m边多边形的完全图，是不动点这些细分程序。