Editing Roman surface (section)

==Relation to the real projective plane==
The sphere, before being transformed, is not [[homeomorphism|homeomorphic]] to the real projective plane, ''RP<sup>2</sup>''. But the sphere centered at the origin has this property, that if point ''(x,y,z)'' belongs to the sphere, then so does the antipodal point ''(-x,-y,-z)'' and these two points are different: they lie on opposite sides of the center of the sphere.

The transformation ''T'' converts both of these antipodal points into the same point,
:<math> T : (x, y, z) \rightarrow (y z, z x, x y), </math>
:<math> T : (-x, -y, -z) \rightarrow ((-y) (-z), (-z) (-x), (-x) (-y)) = (y z, z x, x y). </math>

Since this is true of all points of S<sup>2</sup>, then it is clear that the Roman surface is a continuous image of a "sphere modulo antipodes". Because some distinct pairs of antipodes are all taken to identical points in the Roman surface, it is not homeomorphic to ''RP<sup>2</sup>'', but is instead a quotient of the real projective plane ''RP<sup>2</sup> = S<sup>2</sup> / (x~-x)''. Furthermore, the map T (above) from S<sup>2</sup> to this quotient has the special property that it is locally injective away from six pairs of antipodal points. Or from RP<sup>2</sup> the resulting map making this an immersion of RP<sup>2</sup> — minus six points — into 3-space.