Lines of Principal Curvature near Singular End Points of Surfaces in R3

TL;DR

This paper analyzes the behavior of principal curvature lines near the ends of surfaces in three-dimensional space, classifying stable patterns and extending previous work on algebraic surfaces to more general cases.

Contribution

It extends the study of principal curvature nets to include end points where surfaces tend to infinity, providing a classification of stable patterns at these ends.

Findings

Classified stable curvature line patterns at surface ends.

Extended previous algebraic surface results to more general surfaces.

Identified generic behaviors in one-parameter surface families.

Abstract

In this paper are studied the nets of principal curvature lines on surfaces embedded in Euclidean $3-$space near their end points, at which the surfaces tend to infinity. This is a natural complement and extension to smooth surfaces of the work of Garcia and Sotomayor (1996), devoted to the study of principal curvature nets which are structurally stable -- do not change topologically-- under small perturbations on the coefficients of the equations defining algebraic surfaces. This paper goes one step further and classifies the patterns of the most common and stable behaviors at the ends, present also in generic families of surfaces depending on one-parameter.