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 $γ$ a smooth, closed, convex curve in $M_c$. We explicitly parametrize the \textit{$α$-evolutoid} of $γ$, i.e.\ the closed curve $γ_α$ describing the envelope of all geodesics $σ_s=σ_s(t)$ such that $σ_s(0)=γ(s)$ and $\sphericalangle(σ_s'(0),γ'(s))=α$, with $α\in[0,π/2]$ fixed and determine its lenght. Also, we deduce that for each $s$ the points $γ(s),γ_α(s),γ_{π/2}(s)$ belong to a distinct geodesic circle. A constraint for the smoothness of $γ_α$ is calculated and, using tools from singularity theory, we prove that its singularities present cuspidal features, which mimics the classical evolute ($α=π/2$) in the plane case. Also, we define the \textit{$α$-involutoids} of a given curve $η$ in $M_c$ to be any curve $γ$ in $M_c$ such that $γ_α=η$ 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 $α$-involutoid. Finally, we show that the evolutoids can be seen as singular sets of wavefronts.