Contact cylindrical surfaces and a projection of a surface around a parabolic point

TL;DR

This paper explores the differential geometry of surfaces at parabolic points, introducing contact cylindrical surfaces and linking their properties to surface singularities in orthogonal projections.

Contribution

It introduces contact cylindrical surfaces at parabolic points and characterizes surface singularities via these contact properties.

Findings

Contact cylindrical surfaces have degenerate contact with the original surface at parabolic points.

Such contact properties characterize the $
$-singularity of orthogonal surface projections.

The work provides a geometric criterion for surface singularities based on contact geometry.

Abstract

We investigate differential geometric properties of a parabolic point of a surface in the Euclidean three space. We introduce the contact cylindrical surface which is a cylindrical surface having a degenerate contact type with the original surface at a parabolic point. Furthermore, we show that such a contact property gives a characterization to the $\mathcal{A}$-singularity of the orthogonal projection of a surface from the asymptotic direction.