Roman surface

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The Roman surface (so called because Jakob Steiner was in Rome when he thought of it) is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; however the figure resulting from removing six singular points is one.

The simplest construction is as the image of a sphere centered at the origin under the map f(x,y,z) = (yz,xz,xy). This gives us an implicit formula of

Also, taking a parametrization of the sphere in terms of longitude (θ) and latitude (φ), we get parametric equations for the Roman surface as follows:

The origin is a triple point, and each of the xy-, yz-, and xz-planes are tangential to the surface there. The other places of self-intersection are double points, defining segments along each axis which terminate in pinch points. The entire surface has tetrahedral symmetry. It is a particular type (called type 1) of Steiner surface, that is, a 3-dimensional linear projection of the 5-dimensional Veronese surface, which is a natural embedding of projective space in 5-dimensional space. It contains four pinch-points or cross-caps.


Derivation of implicit formula

For simplicity we consider only the case r = 1. Given the sphere defined by the points (x, y, z) such that

we apply to these points the transformation T defined by


But then we have

and so

as desired.

Conversely, suppose we are given (U, V, W) satisfying

(*) U^2 V^2 + V^2 W^2 + W^2 U^2 - U V W = 0.\,

We prove that there exists (x,y,z) such that

(**) x^2 + y^2 + z^2 = 1,\,

for which

with one exception: In case 3.b. below, we show this cannot be proved.

1. In the case where none of U, V, W is 0, we can set

(Note that (*) guarantees that either all three of U, V, W are positive, or else exactly two are negative. So these square roots are of positive numbers.)

It is easy to use (*) to confirm that (**) holds for x, y, z defined this way.

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