Equidistants for families of surfaces

Peter Giblin and Graham Reeve

Journal of Singularities
volume 21 (2020), 97-118

Received: 18 March 2018. Received in revised form: 26 December 2018

DOI: 10.5427/jsing.2020.21f

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Abstract:

For a smooth surface in R^3 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 R^4 to R^3. 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.

2010 Mathematical Subject Classification:

57R45, 53A05

Key words and phrases:

affine equidistant, surface family in 3-space, critical set, map germ 4-space to 3-space

Author(s) information:

Peter Giblin	Graham Reeve
Department of Mathematical Sciences	Department of Mathematics
The University of Liverpool	and Computer Science
Liverpool L69 7ZL, UK	Liverpool Hope University
	Liverpool L16 9JD, UK
email: pjgiblin@liv.ac.uk	email: reeveg@hope.ac.uk