ABSTRACT
More generally, this problem was studied more than a century ago. Hum bert was interested by the algebraic curves with an algebraic arc length [13]. In the complex field, Raffy characterized the curves of In by considering them as envelope of their tangent lines [19], but the difficulty is to pull out the real parametric curves. Whereas it is easy to characterize polynomial curves with rational offsets (Kubota [15]), this is no longer the case if one considers rational curves. In this paper we give two answers to this problem, on the one hand giving their explicit parametric form as well as the one of their offsets (Section 2), on the other hand characterizing them as involute of the caustics of the rational parametric curves (Section 3). In the same section, we show that the set of the rational parametric curves with rational arc length is identical to the set of the caustics of the rational parametric
curves. These geometrical characterizations were already presented at the French 24th Congress of Numerical Analysis in Vittel [10].
