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*=*r*2 cos θ cos φ sin φ*y*=*r*2 sin θ cos φ sin φ*z*=*r*2 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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