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.
image:romansurface.jpg
