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This paper deals with a family of spatial rational curves that wereintroduced by Andradas, Recio and Sendra, under the name of hypercircles, as analgorithmic cornerstone tool in the context of improving the rationalparametrization (simplifying the coefficients of the rational functions, whenpossible) of algebraic varieties. A real circle can be defined as the image ofthe real axis under a Moebius transformation in the complex field. Likewise,and roughly speaking, a hypercircle can be defined as the image of a line ("the${\mathbb{K}}$-axis") in a $n$-degree finite algebraic extension$\mathbb{K}(\alpha)\thickapprox\mathbb{K}^n$ under the transformation$\frac{at+b}{ct+d}:\mathbb{K}(\alpha)\to\mathbb{K}(\alpha)$.  The aim of this article is to extend, to the case of hypercircles, some ofthe specific properties of circles. We show that hypercircles are precisely,via $\mathbb{K}$-projective transformations, the rational normal curve of asuitable degree. We also obtain a complete description of the points atinfinity of these curves (generalizing the cyclic structure at infinity ofcircles). We characterize hypercircles as those curves of degree equal to thedimension of the ambient affine space and with infinitely many${\mathbb{K}}$-rational points, passing through these points at infinity.Moreover, we give explicit formulae for the parametrization and implicitationof hypercircles. Besides the intrinsic interest of this very special family ofcurves, the understanding of its properties has a direct application to thesimplification of parametrizations problem, as shown in the last section.

Generalizing circles over algebraic extensions