The Roman surface (so called because Jakob Steiner was in Rome when he thought of it) is a selfintersecting mapping of the real projective plane into threedimensional 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 xzplanes are tangential to the surface there. The other places of selfintersection 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 3dimensional linear projection of the 5dimensional Veronese surface, which is a natural embedding of projective space in 5dimensional space. It contains four pinchpoints or crosscaps.
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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
say.
But then we have
and so
as desired.
Conversely, suppose we are given (U, V, W) satisfying
(*)
We prove that there exists (x,y,z) such that
(**)
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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