This paper deals with a remarkable class of curves (in general
$r$space) that the two first authors have named ``hypercircles"
(see \cite{ARS2}). 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{ARS2}.
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 ${\mathbb K}$).
