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

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 $\alpha \mathit{\text{-evolutoid}}$ 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 $\mathrm{\sphericalangle }({\sigma }_s^{\mathrm{\prime }}(0),{\gamma }^{\mathrm{\prime }}(s))=\alpha$, with $\alpha \in [0,\pi \mathrm{/}2]$ fixed and determine its lenght. Also, we deduce that for each $s$ the points $\gamma (s),{\gamma }_{\alpha }(s),{\gamma }_{\pi \mathrm{/}2}(s)$ 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 ($\alpha =\pi \mathrm{/}2$) in the plane case. Also, we define the $\alpha \mathit{\text{-involutoids}}$ 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.