Steiner Surface

Roman surface is topologically equivalent to Boy's surface Apery, Boys (Bryant-Kusner) surface, Cross-cap. It has tetrhedron symmetry.

Algebraci equation: x^2* y^2 + y^2* z^2 + z^2* x^2 - r^2* x* y* z = 0.

Parametric formula: {Cos[u] Cos[v] Sin[v], Sin[u] Cos[v] Sin[v], Cos[u] Sin[u] Cos[v]^2}*r^2

Here is a smooth deformation from Boy's surface to Roman surface, with b from 0 to 1. (formula from Peter Wang)

{Sqrt[2]*Cos[2*u]*Cos(v)^2+Cos[u]*Sin[2*v]/(2-(b*Sqrt[2]*Sin[3*u]*Sin[2*v])),
 Sqrt[2]*Sin[2*u]*Cos(v)^2-Sin[u]*Sin[2*v]/(2-(b*Sqrt[2]*Sin[3*u]*Sin[2*v])),
 3*[Cos[v]]^2/(2-(b*Sqrt[2]*Sin[3*u]*Sin[2*v]))-1}