Singularities and extrinsic geometry

We have shown how the geometry of an immersion of a surface in 3-space is reflected in the types of contact that occur between the immersion and affine lines or planes. This point of view has been developed recently be many people. In this chapter we describe briefly the kinds of work being done (see [Mc2] for more details).

The singularities which occur in the family of orthogonal projections to planes of a generic surface in 3-space have been classified by Arnold [A6], Lyashko [Ly1] [Ly2], Goryunov [Gor1] [Gor2] [Gor3], Kergosien [KeT] [Ke1] [Ke2] [Ke3], and Gaffney and Ruas [GafR]. (Kergosien's results were the subject of a lecture given by R. Thom at I.H.E.S. in November, 1979. Gaffney and Ruas' results were described in a letter to Wall in October, 1977.) Related work has been done by Platonova [Pl1] [Pl2] [Pl3] and Landis [Lan]. According to Arnold, most of our results on cusps of Gauss maps are contained in the papers of Landis and Platonova.

Projections of surfaces to planes have been analyzed from a visual
perspective by Koenderink and van Doorn [KoD]. We have obtained a classification of
apparent contours of surfaces which combines their work and the
classification of Arnold et al. [Mc1]. A
classification of germs of embeddings
**R**^{2} -> **R**^{3} according to
their apparent contours has been done by Kergosien [Ke3].

The relation between the geometry of convex hulls and the singularities of the Gauss map has been investigated by Zakalyukin [Z], Sedych [Se1] [Se2], Brisgalova [Bri1] [Bri2], and Romero Fuster [Rom1].

Bleeker and Wilson [BlW] found a relation between the number of cusps of the Gauss map of a closed surface and the Euler characteristic of the hyperbolic region. This result was refined by Banchoff and Thom [BaT].

Zakalyukin [66] and Romero Fuster have investigated the relation
between the number of Gaussian cusps on the boundary of the convex
hull of a closed surface and the number of triple tangent support
planes of the surface. We have found a relation between the total
number of Gaussian cusps of an immersion
M^{2} -> **R**^{3} and the number of
tritangent planes of the immersion [BaGM1].

Freedman has shown that for a generic simple closed space curve
with nonvanishing torsion, the number of tritangent planes is even [F]. We have shown that the torsion zeroes of a
generic simple closed space curve can be indexed so that the sum of
these indices is congruent mod 2 to the number of tritangent planes [BaGM1]. (Torsion zeroes of a curve
correspond to Gaussian cusps of its canal surface - see chapter 2,
example 5 above.) Morton and Mond have shown that a generic simple
closed space curve with no quadrisecants is unknotted [Mort].