Some results on evolutoids of convex curves in $2$-dimensional space forms

Abstract

Let $M_c$ be a $2$-dimensional space form of constant curvature $c=-1,0,1$ and $\gamma$ a smooth, closed, convex curve in $M_c$. We explicitly parametrize the \textit{$\alpha$-evolutoid} of $\gamma$, i.e.\ the closed curve $\gamma_\alpha$ describing the envelope of all geodesics $\sigma_s=\sigma_s(t)$ such that $\sigma_s(0)=\gamma(s)$ and $\sphericalangle(\sigma_s'(0),\gamma'(s))=\alpha$, with $\alpha\in[0,\pi/2]$ fixed and determine its lenght. Also, we deduce that for each $s$ the points $\gamma(s),\gamma_\alpha(s),\gamma_{\pi/2}(s)$ belong to a distinct geodesic circle. A constraint for the smoothness of $\gamma_\alpha$ is calculated and, using tools from singularity theory, we prove that its singularities present cuspidal features, which mimics the classical evolute ($\alpha=\pi/2$) in the plane case. Also, we define the \textit{$\alpha$-involutoids} of a given curve $\eta$ in $M_c$ to be any curve $\gamma$ in $M_c$ such that $\gamma_\alpha=\eta$ and study some of its properties. In particular, we prove that any convex, closed curve in $M_{-1,0}$ has associated to itself exactly one closed $\alpha$-involutoid. Finally, we show that the evolutoids can be seen as singular sets of wavefronts.