Editing Roman surface (section)

{{short description|Self-intersecting, highly symmetrical mapping of the real projective plane into 3D space}}
{{research paper|date=March 2018}}
{{more footnotes|date=March 2018}}
[[Image:Steiner's Roman Surface.gif|thumb|An animation of the Roman surface]]

In [[mathematics]], the '''Roman surface''' or '''Steiner surface''' is a self-intersecting [[Map (mathematics)|mapping]] of the [[real projective plane]] into [[three-dimensional space]], with an unusually high degree of [[symmetry]]. This mapping is not an [[immersion (mathematics)|immersion]] of the projective plane; however, the figure resulting from removing six [[Singular point of a curve|singular points]] is one. Its name arises because it was discovered by [[Jakob Steiner]] when he was in [[Rome]] in 1844.<ref name="Coffman">{{cite web|last1=Coffman|first1=Adam|title=Steiner Roman Surfaces|url=http://old.nationalcurvebank.org/romansurfaces/romansurfaces.htm |website=National Curve Bank|publisher=Indiana University - Purdue University Fort Wayne}}</ref>

The simplest construction is as the image of a [[sphere]] centered at the origin under the map <math>f(x,y,z)=(yz,xz,xy).</math> This gives an implicit [[formula]] of
<!--
:''x''<sup>2</sup>''y''<sup>2</sup> + ''y''<sup>2</sup>''z''<sup>2</sup> + ''x''<sup>2</sup>''z''<sup>2</sup> &minus; ''r''<sup>2</sup>''xyz'' = 0 -->
:<math> x^2 y^2 + y^2 z^2 + z^2 x^2 - r^2 x y z = 0. \,</math>

Also, taking a parametrization of the sphere in terms of [[longitude]]  ({{mvar|θ}}) and [[latitude]] ({{mvar|φ}}), gives [[Parametric equation|parametric equations]] for the Roman surface as follows:
:<math>x=r^{2} \cos \theta \cos \varphi \sin \varphi</math>
:<math>y=r^{2} \sin \theta \cos \varphi \sin \varphi</math>
:<math>z=r^{2} \cos \theta \sin \theta \cos^{2} \varphi </math>
The origin is a triple point, and each of the {{mvar|xy}}-, {{mvar|yz}}-, and {{mvar|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 [[tetrahedron|tetrahedral]] [[symmetry group|symmetry]]. It is a particular type (called type 1) of Steiner surface, that is, a 3-dimensional [[linear projection]] of the [[Veronese surface]].