Let

${\mathbb{R}}_{1}^{3}$ be the Minkowski 3-space, or pseudo-Euclidean space, i.e. the 3-dimensional vector-space

${\mathbb{R}}^{3}$ equipped with the scalar product

$\langle x,y\rangle =-{x}_{1}{y}_{1}+{x}_{2}{y}_{2}+{x}_{3}{y}_{3}$. 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).