Singularities of parallels to tangent developable surfaces

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.