Rationality of the offsets to algebraic curves and surfaces
The parallel curves and surfaces of algebraic curves and surfaces are algebraic as envelopes of algebraic systems of circles and spheres. However, the parallel curves and surfaces of rational curves and surfaces are not in general rational (i.e., of genus 0). The condition of rationality is clearly the existence of a rational parametrization of the system of circles or spheres, i.e., of the unit normal. In addition to the literature quoted by the author one may add the papers of \textit{E. Laguerre} [Oeuvres, Vol. 2: Géométrie. pp. 592 ff.]. Rational parallel curves have by definition a rational representation in Laguerre geometry [cf. also \textit{H. Wieleitner}, ``Spezielle ebene Kurven'' (Leipzig 1908); \S16]. Using essentially Laguerre's criterion, the author proves that the parallel surfaces to paraboloids, spheres, and hyperboloids of one sheet are rational.
scientific article; zbMATH DE number 713529