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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... The Roman surface has four "lobes" ... The surface has a total of three lines of double points, which lie (in the parametrization given earlier) on the coordinate axes ... If the Roman surface were to be inscribed inside the tetrahedron with least possible volume, one would find that each edge of the tetrahedron is tangent to the Roman surface at a point, and that each of these six ...
... Steiner's Roman surface is a more degenerate map of the projective plane into 3-space, containing a cross-cap ... which has the same general form as Steiner's Roman Surface, shown to the right ...
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