The normal map as a vector field

TL;DR

This paper studies the properties of normal maps of closed plane curves as vector fields, analyzing their critical points, indices, and extensions to surfaces, with new geometric interpretations and counting theorems.

Contribution

It introduces a geometric interpretation of normal map critical points, studies their indices, and extends the analysis to focal sets of surfaces, providing new counting theorems.

Findings

Critical points analyzed as a vector field on the cylinder

Counting theorems for winding and rotation indices

Extension to focal sets of surfaces

Abstract

In this paper we consider the normal map of a closed plane curve as a vector field on the cylinder. We interpret the critical points geometrically and study their Poincar\'{e} index, including the points at infinity. After projecting the vector field to the sphere we prove some counting theorems regarding the winding and rotation index of the curve and its evolute. We finish with a description of the extension to focal sets of surfaces.