The Roman surface or Steiner surface (so called because Jakob Steiner was in Rome when he thought of it) 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.
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 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:
- 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 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.
Read more about Roman Surface: Derivation of Implicit Formula, Derivation of Parametric Equations, Relation To The Real Projective Plane, Structure of The Roman Surface, One-sidedness, Double, Triple, and Pinching Points
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