Contact with circles and Euclidean invariants of smooth surfaces in R^3

TL;DR

This paper studies the vertex curve of smooth surfaces in R^3, exploring its geometric properties, relationships with other curvature-related curves, and behavior under surface deformations, with implications for differential geometry and image analysis.

Contribution

It introduces the concept of the vertex curve in relation to Euclidean invariants and analyzes its connections with other fundamental curves on surfaces.

Findings

Characterization of the vertex curve in hyperbolic regions.

Relationships between the vertex curve, parabolic, and flecnodal curves.

Behavior of the vertex curve in generic surface deformations.

Abstract

We investigate the vertex curve, that is the set of points in the hyperbolic region of a smooth surface in real 3-space at which there is a circle in the tangent plane having at least 5-point contact with the surface. The vertex curve is related to the differential geometry of planar sections of the surface parallel to and close to the tangent planes, and to the symmetry sets of isophote curves, that is level sets of intensity in a 2-dimensional image. We investigate also the relationship of the vertex curve with the parabolic and flecnodal curves, and the evolution of the vertex curve in a generic 1-parameter family of smooth surfaces.