Steiner's Roman Surface

A realization of the real projective plane

Click for VRML

Introduction:

Steiner's Roman surface is one realization of a mathematical object known as the real projective plane. The real projective plane is defined as "the set that results when antipodal points of S²(a) are identified; thus

RP²(a) = {{p,-p} | ||p|| = a}." (Gray 330)

Two points are said to be antipodal if p₁ = -p₂. In order to realize the real projective plane (i.e., make it into a surface in R³), we need a map with the antipodal property. That is, a map which satisfies F(-p) = F(p). The so-called Roman map, defined by romanmap(x, y, z) = (xy, yz, zx), has this property. When we apply it to a sphere, we obtain the parameterization for Steiner's Roman surface given below. Note that, like the Möbius strip and the Klein bottle, Steiner's Roman surface is non-orientable. This is clear from an attempted front-back coloring of the polygons that approximate the surface:

For another realization of the real projective plane, see the cross cap.

Definition:

,

where a is the radius of the sphere to which the Roman map is applied.

Properties:

Tangent Planes: At u = u0, v = v0, the tangent plane to the surface is parameterized by: [click for formula]

Infinitesimal Area: The infinitesimal area of a patch on the surface is given by dA = [click for formula]

Gaussian Curvature:

Mean Curvature: H[u, v] = [click for formula]