arXiv subjects

Samuel Pacitti Gentil

Publications and source records attributed to Samuel Pacitti Gentil.

2 recordsLinked to original sources

arXiv2025

A discrete analog of Segre's theorem on spherical curves

We prove a discrete analog of a certain four-vertex theorem for space curves. The smooth case goes back to the work of Beniamino Segre and states that a closed and smooth curve whose tangent indicatrix has no self-intersections admits at least four points at which its torsion vanishes. Our approach uses the notion of discrete tangent indicatrix of a (closed) polygon. Our theorem then states that a polygon with at least four vertices and whose discrete tangent indicatrix has no self-intersections admits at least four flattenings, i.e., triples of vertices such that the preceding and following vertices are on the same side of the plane spanned by this triple.

math.DG↗

arXiv2024

New four-vertex type theorems for spherical polygons

We prove discrete analogs of four-vertex type theorems of spherical curves, which imply corresponding results for space polygons. The smooth theory goes back to the work of Beniamino Segre and, more recently, by Mohammad Ghomi, and consists of theorems that state, for a given closed spherical curve, nontrivial lower bounds on the number of spherical inflections plus the number of self and/or antipodal intersections counted with multiplicity. We study these concepts and results adapted to the case of spherical polygons and prove, using only discrete tools, the corresponding theorems.

math.DG↗