Mannheim-d'Ocagne-Koenderink type formulas for asymptotic directions

TL;DR

This paper extends classical formulas relating Gaussian curvature, normal curvature, and contour line curvature on surfaces in 3D, specifically addressing cases where the tangent vector points in an asymptotic direction, incorporating cusp singularities.

Contribution

It introduces new formulas for asymptotic directions on surfaces, including invariants related to cusp singularities, expanding the applicability of classical curvature formulas.

Findings

Formulas for asymptotic directions incorporating cusp invariants

Extension of Mannheim-d'Ocagne-Koenderink formulas to singular cases

Enhanced understanding of curvature relations at asymptotic directions

Abstract

We consider a surface embedded in the Euclidean 3-space and fix a tangential vector $v$ at a given point $p$ on the surface. In this paper, we first review a history of the formula obtained by Mannheim, d'Ocagne and Koenderink, which asserts that the Gaussian curvature of the surface at $p$ can be obtained if one knows "the normal curvature at $p$ with respect to $v$" and "the curvature of the contour line $L$ of the surface at $p$" with respect to the orthogonal projection induced by $v$. Unfortunately, this formula does not work when $v$ points in an asymptotic direction. When $v$ is just the case, we give anlogues of the formula, which include an invariant of cusp singular points of $L$.