Evolutes of hyperbolic plane curves
Let \(\mathbb{R}^3_1\) be the Minkowski 3-space, or pseudo-Euclidean space, i.e. the 3-dimensional vector-space \(\mathbb{R}^3\) 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 \(H^2_+\): \(\langle x,x\rangle=-1\), \(x_1\geq 1\) 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).
scientific article; zbMATH DE number 2104913