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From Hypercircles to Units

T. Recio, J. Sendra, C. Villarino


This paper deals with a remarkable class of curves (in general $r$-space) that the two first authors have named ``hypercircles" (see \cite{ARS-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 hypercircles \cite{ARS-2}. Here we present an algorithm that, for a given parametrization of a hypercircle $\cal U$, over an algebraic extension, namely, $\varphi(t) \in {\mathbb K}(\alpha)(t)$, computes the linear fraction over ${\mathbb K}(\alpha)(t)$ that generates this hypercircle (and, in particular, a parametrization of the curve over ${\mathbb K}$).

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