Some results on evolutoids of convex curves in $2$-dimensional space forms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918156720668672 |
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| author | Junior, Ady Cambraia Chimenton, Alessandro Gaio Fernandes, Marco Antônio do Couto Salarinoghabi, Mostafa |
| author_facet | Junior, Ady Cambraia Chimenton, Alessandro Gaio Fernandes, Marco Antônio do Couto Salarinoghabi, Mostafa |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_07274 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some results on evolutoids of convex curves in $2$-dimensional space forms Junior, Ady Cambraia Chimenton, Alessandro Gaio Fernandes, Marco Antônio do Couto Salarinoghabi, Mostafa Differential Geometry 53A04, 53A35 and 58K05 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. |
| title | Some results on evolutoids of convex curves in $2$-dimensional space forms |
| topic | Differential Geometry 53A04, 53A35 and 58K05 |
| url | https://arxiv.org/abs/2510.07274 |