Parallel Tangency in R{sup}3

摘要

In this paper we examine the mathematics behind pairs of surface points with parallel tangent planes in R{sup}3. We look at the case of disjoint surface pieces, exploring the maps linking the parameters at the two points of tangency and some of their singularities. We go on to consider the same issues for the local case, considering pairs of points with parallel tangent planes in the neighbourhood of a parabolic point on a single surface piece. For the latter case we also consider the effect of the parabolic point being a cusp of Gauss and go on to describe the arrangement of various special curves on the surface in a neighbourhood of the cusp of Gauss. In the second part of the paper we consider surfaces constructed from the chords joining parallel tangent pairs. Giblin and Zakalukin [3] have investigated the envelope of such chords, but here we consider the "equidistants" which are surfaces formed by points at a fixed proportion along the chords. These affinely invariant surfaces are a type of symmetry construction and as such we pay particular attention to the half way equidistant or Mid-Point Tangent Surface (MPTS). We describe its structure in the disjoint surface pieces and local cases and show how it behaves quite differently from other equidistants.