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Evolutes of hyperbolic plane curves. (English) Zbl 1065.53022
Let $\bbfR^3_1$ be the Minkowski 3-space, or pseudo-Euclidean space, i.e. the 3-dimensional vector-space $\bbfR^3$ equipped with the scalar product $\langle x,y\rangle=-x_1y_1+x_2y_2+x_3y_3$. The Lorentzian sphere: $\langle x,x\rangle=-1$, a two-sheet-hyperboloid, contains as one of its sheets the surface $H^2_+$: $\langle x,x\rangle=-1$, $x_1\ge 1$ 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).

53B30Lorentz metrics, indefinite metrics
57R70Critical points and critical submanifolds
53A04Curves in Euclidean space
Full Text: DOI
[1] Bruce, J. W., Giblin, P. J.: Curves and singularities (second edition), Cambridge University Press, Glasgow, 1992 · Zbl 0770.53002
[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