Equidistants for families of surfaces

Abstract

For a smooth surface in R 3 this article contains local study of 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. MR Classification 57R45, 53A05

The Geometry of the Centre Symmetry Set of a Planar Curve

“…For instance, authors in [19,20,21,25] analyse the local properties of the CSS based on the theory of Lagrange and Legendre singularities. In [17,18] the authors study the Centre Symmetry Set of families of plane curves and of families of surfaces. The CSS is a part of the Global Centre Symmetry Set studied in [11].…”