2025/10/08 by Alessandro Gaio Chimenton, Junior, Ady Cambraia, Chimenton, Alessandro Gaio +4
Computer Science · Mathematics · #53A04 #53A35 and 58K05 #Differential Geometry (math.DG) #FOS: Mathematics #Mathematical Dynamics and Fractals #Point processes and geometric inequalities #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2510.07274
openalex publication_date 2025/10/08 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
Let Mc be a 2-dimensional space form of constant curvature c=-1,0,1 and γ a smooth, closed, convex curve in Mc. We explicitly parametrize the α-evolutoid of γ, i.e. the closed curve γα describing the envelope of all geodesics σs=σs(t) such that σs(0)=γ(s) and \sphericalangle(σs'(0),γ'(s))=α, with α∈[0,π/2] fixed and determine its lenght. Also, we deduce that for each s the points γ(s),γα(s),γπ/2(s) belong to a distinct geodesic circle. A constraint for the smoothness of γα 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 α-involutoids of a given curve η in Mc to be any curve γ in Mc such that γα=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.