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Concept
Roman surface
Summary
In mathematics, the Roman surface or Steiner surface 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. Its name arises because it was discovered by Jakob Steiner when he was in Rome in 1844.
The simplest construction is as the image of a sphere centered at the origin under the map This gives an implicit formula of
Also, taking a parametrization of the sphere in terms of longitude (θ) and latitude (φ), gives 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 coordinate axis which terminate in six 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 Veronese surface.
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.
- 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.
- Suppose that W is 0. From (*) this implies and hence at least one of U, V must be 0 also. This shows that is it impossible for exactly one of U, V, W to be 0.
- Suppose that exactly two of U, V, W are 0. Without loss of generality we assume () It follows that (since implies that and hence contradicting ().
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