Constructing quadratic birational maps via their complex rational representation

Abstract

We present a method to construct quadratic birational planar maps, which is based on the observation that any rational planar map with complex rational representation possesses two special syzygies. After establishing the relation between degree one complex rational Bézier curves and quadratic rational Bézier curves, we derive conditions to determine when a quadratic rational planar map has a complex rational representation. Hence, we can make a quadratic planar map be birational by suitably choosing the middle Bézier control points and their corresponding weights. In addition, this paper explores an interesting geometric property of the isoparametric curves of quadratic birational planar maps with complex rational representation, i.e., all isoparametric curves meet at a common point.