From Hypercircles to Units

摘要

This paper deals with a remarkable class of curves (in general r-space) that the two first authors have named "hy-percircles" (see [2]). As shown there, such curves appear in the CAD context, when aiming towards finding a parametric representation with simpler coefficients (i.e. without algebraic numbers) for a given parametric curve. In fact, it turns out that the crucial point to solve the simplification problem in general is to solve this same problem for hyper-circles. Here we present an algorithm that, for a given parametrization of a hypercircle u, over an algebraic extension, namely, φ(t) ∈K(α)(t)~r, computes the linear fraction over K(α)(t) that generates this hypercircle (and, in particular, a parametrization of the curve over K).