The Veronese Surface
The projection of the Veronese surface as Steiner's Roman surface.
The Veronese surface is an embedding of the real projective plane which starts with the hemispherex^{2} + y^{2} + z^{2} = 1 ,z < 0 and maps each point (x,y,z) to 6space. The projection of this surface(x^{2},y^{2},z2,Ö2 xy,Ö2 yz,Ö2 zx) from 6space into 4dimensional space given by(Ö2 xz,Ö2 yz, (1/Ö2)(x^{2}  y^{2}), Ö2 xy) is again an embedding and we examine a family of projections of this surface into 3dimensional subspaces (all of which must have local singularities). This will appear in a paper on normal Euler classes by the author.The paper referred to was published, in expanded form, as a collaboration with Ockle Johnson [11] twenty years after the Helsinki Congress! In the meantime, several other papers of the author have used the fact that the normal Euler class of a surface embedded in fourspace can be obtained as an indexed sum of singularities of any generic orthogonal projection into a hyperplane. Since the normal Euler class of an embedded real projective plane is nonzero, there must be singular points for almost any such projection.
The projection into the last three coordinates gives a crosscap with two pinch points (Whitney umbrella points). The linear interpolation of the left hemisphere into the crosscap is a regular homotopy right up to the last instant when opposite points on the equator are identified, forming a segment of double points.Deforming a hemisphere into a crosscap is a another remarkable use of linear interpolation between surfaces with the same parametrization. It is a challenge, however, to position the surfaces so that the intermediate stages are all embedded with twofold symmetry. One way is to interpolate between the crosscap
and the hemisphere given by (^{1}/_{Ö2}) (sinu sin2v,cos2u(1+cos2v),sin2u(1+cos2v))(sinv,sinucosv,cosu cosv) where u goes form 0 to p and v goes from p/2 to p/2.
Rotating in the plane of the first and third coordinates gives a deformation from the crosscap to Steiner's Roman surface (Ö2 xz,Ö2 yz, Ö2 xy) with tetrahedral symmetry. This projection has six pinch points that are the endpoints of three double point segments intersecting in a triple point. These examples are described in the classical book, "Geometry and the Imagination" by Hilbert and CohnVossen. The embedding in 4space is tight (i.e. almost every height function when restricted to the surface has exactly one maximum and one minimum) and this property is shared by the images in 3dimensional subspaces. These examples lead to the conjecture that any stable tight mapping of the real projective plane into 3space must have either two pinch points or six pinch points.This conjecture was established in the Ph.D. thesis of
Leslie Coghlan , under the author's direction [21,22].

