Equidistants for families of surfaces

Giblin, Peter and Reeve, Graham (2020) Equidistants for families of surfaces. Journal of Singularities, 21. pp. 117-139. ISSN 1949-2006 (Accepted for Publication)

[img]
Preview
Text
Giblin-Reeve-2surfaces.pdf - Accepted Version

Download (6MB) | Preview

Abstract

For a smooth surface in R3 this article investigates certain affine equidistants, that is loci of points at a fixed ratio between points of contact of parallel tangent planes (but excluding ratios 0 and 1 where the equidistant contains one or other point of contact). The
situation studied occurs generically in a 1-parameter family, where two parabolic points of the surface have parallel tangent planes at which the unique asymptotic directions are also parallel. The singularities are classified by regarding the equidistants as critical values of a 2-parameter unfolding of maps from R4 to R3. In particular, the singularities that occur near the so-called ‘supercaustic chord’, joining the two special parabolic points, are classified. For a given ratio along this chord either one or three special points are identified at which singularities of the equidistant become more special. Many of the resulting singularities have occurred before in the literature in abstract classifications, so the article also provides a natural
geometric setting for these singularities, relating back to the geometry of the surfaces from which they are derived.

Item Type: Article
Additional Information and Comments: This article has been accepted for publication in the Journal of Singularities. When published, the final version will be available from https://www.journalofsing.org/index.html
Keywords: Affine equidistant, surface family in 3-space, critical set, map germ 4-space to 3-space.
Faculty / Department: Faculty of Science > Mathematics and Computer Science
Depositing User: Graham Reeve
Date Deposited: 25 Feb 2020 15:46
Last Modified: 25 Feb 2020 15:46
URI: https://hira.hope.ac.uk/id/eprint/3006

Actions (login required)

View Item View Item