Singularities of parallels to tangent developable surfaces

TL;DR

This paper generalizes the concept of tangent developable surfaces and their parallels to higher dimensions, classifies their singularities, and explores invariance properties under parallel transformations.

Contribution

It introduces a generalization of tangent developable surfaces for frontal curves in arbitrary dimensions and classifies their generic singularities in 3D and 4D spaces.

Findings

Singularities on parallels to tangent developable surfaces are classified.

Invariance of developable surface classes under parallel transformations is confirmed.

Generalization to higher dimensions extends classical surface theory.

Abstract

It is known that the class of developable surfaces which have zero Gaussian curvature in three dimensional Euclidean space is preserved by the parallel transformations. A tangent developable surface is defined as a ruled developable surface by tangent lines to a space curve and it has singularities at least along the space curve, called the directrix or the the edge of regression. Also the class of tangent developable surfaces are invariant under the parallel deformations. In this paper the notions of tangent developable surfaces and their parallels are naturally generalized for frontal curves in general in Euclidean spaces of arbitrary dimensions. We study singularities appearing on parallels to tangent developable surfaces of frontal curves and give the classification of generic singularities on them for frontal curves in 3 or 4 dimensional Euclidean spaces.