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.
Referenced By
Boy's surface | List of geometric topology topics | List of mathematical topics (P-R) | Projective plane | Real projective plane | Roman | Romans | Steiner surface
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