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Roman surface

The Roman surface (so called because Jakob Steiner[?] was in Rome when he thought of it) is a self-intersecting immersion of the real projective plane into three-dimensional space, with an unusually high degree of symmetry.

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

x2y2 + y2z2 + x2z2 - r2xyz = 0
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:
x = r2 cos θ cos φ sin φ
y = r2 sin θ cos φ sin φ
z = r2 cos θ sin θ cos2 φ

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.



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