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

TL;DR

The paper extends the classical plane evolute to α-evolutoids of smooth convex curves in all 2D space forms $M_c$ with $c\in\{-1,0,1\}$. It provides explicit parametrizations, a regularity criterion, and relationships to wavefronts, alongside area/length comparisons and a study of α-involutoids via a differential equation framework, using a unified Frenet-Serret and exponential-map formalism. The results show how evolutoids behave across hyperbolic, Euclidean, and spherical geometries, including singularities (cusps) and the existence/uniqueness of closed involutoids in certain space forms. The work connects geometric envelopes to singularity theory and wavefronts, offering non-Euclidean generalizations of known plane results and revealing new questions about curvature-dependent bounds and the role of geometry in evolutoid/involutoid structures.

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.

Figures (7)

• Figure 1: Examples of $\alpha$-evolutoids of closed curves. The original curve is shown in blue and the $\alpha$-evolutoid in red.
• Figure 2: $\alpha$-evolutoid of a curve $\gamma$.
• Figure 3: Geometric representation of $\alpha_0$.
• Figure 4: The behaviour of $A_{c}(\gamma_\alpha^R)/A_{c}(\gamma^R)$ with $\gamma^R$ a geodesic circle of radius $R>0$, for $c=-1$ (red), $c=0$ (blue) and $c=1$ (green).
• Figure 5: $\eta$ is an $\alpha$-involutoid of the curve $\gamma$.

Theorems & Definitions (40)

• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Definition 3.1
• Proposition 3.2
• proof
• Remark 3.3