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Evolutes of hyperbolic plane curves. (English) Zbl 1065.53022
Let ${ℝ}_{1}^{3}$ be the Minkowski 3-space, or pseudo-Euclidean space, i.e. the 3-dimensional vector-space ${ℝ}^{3}$ equipped with the scalar product $〈x,y〉=-{x}_{1}{y}_{1}+{x}_{2}{y}_{2}+{x}_{3}{y}_{3}$. The Lorentzian sphere: $〈x,x〉=-1$, a two-sheet-hyperboloid, contains as one of its sheets the surface ${H}_{+}^{2}$: $〈x,x〉=-1$, ${x}_{1}\ge 1$ which is adopted as the model of the hyperbolic plane and (wrongly) called “hyperbola”. The authors develop the Frenet-Serret-type formula for hyperbolic plane curves and study in particular the following properties of these curves: hyperbolic invariants, osculating pseudo-circles, the hyperbolic evolute and its singularities. The case when a point of the hyperbolic evolute is an ordinary cusp is characterized (theorem 5.3). The last section describes how one can draw the picture of the hyperbolic evolute of a curve on the Poincaré disk: fig. 1 shows a family of Euclidean ellipses and their hyperbolic evolutes. The authors mention – what can be seen clearly in fig. 1 – that the four cusps theorem does not hold in ${H}_{+}^{2}$ (although the four vertex theorem does).
##### MSC:
 53B30 Lorentz metrics, indefinite metrics 57R70 Critical points and critical submanifolds 53A04 Curves in Euclidean space
##### Keywords:
singularity; Lorentz group; Poincaré disk
##### References:
 [1] Bruce, J. W., Giblin, P. J.: Curves and singularities (second edition), Cambridge University Press, Glasgow, 1992 [2] Torii, E.: On curves on the hyperboloid or the pseudo-sphere in Minkowski 3-space, Master thesis of Hokkaido University, 1999 (in Japanese) [3] O’Neill, B.: Semi-Riemannian Geometry, Academic Press, New York, 1983