Tohoku Mathematical Journal
2023
June
SECOND SERIES VOL. 75, NO. 2

Tohoku Math. J.
75 (2023), 233-249

Title SINGULARITIES OF PARALLELS TO TANGENT DEVELOPABLE SURFACES

Dedicated to Professor Takashi Nishimura on the occasion of his 60th birthday
Author Goo Ishikawa

(Received July 19, 2021, revised November 15, 2021)
Abstract. 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 edge of regression. The class of tangent developable surfaces is invariant under the parallel deformations. In this paper the notions of tangent developable surfaces and their parallels are naturally generalised for frontal curves in general in Euclidean spaces of arbitrary dimensions. The singularities appearing on parallels to tangent developable surfaces of frontal curves are studied and the classification of generic singularities on them for frontal curves in 3 or 4 dimensional Euclidean spaces are given.

Mathematics Subject Classification. Primary 58K05; Secondly 53A04, 53A05, 53C05, 58K40, 53D10.
Key words and phrases. Legendre singularity, normal connection, normally flat frontal, tangent surface, open swallowtail, unfurled swallowtail.

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