Roman surface
In mathematics, the Roman surface or Steiner surface 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.
[1] The simplest construction is as the image of a sphere centered at the origin under the map
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: The origin is a triple point, and each of the xy-, yz-, and xz-planes are tangential to the surface there.
In the case where none of U, V, W is 0, we can set (Note that (*) guarantees that either all three of U, V, W are positive, or else exactly two are negative.
(each of which is a noncompact portion of a coordinate axis, in two pieces) do not correspond to any point on the Roman surface.
Let a sphere have radius r, longitude φ, and latitude θ.
The sphere, before being transformed, is not homeomorphic to the real projective plane, RP2.
Because some distinct pairs of antipodes are all taken to identical points in the Roman surface, it is not homeomorphic to RP2, but is instead a quotient of the real projective plane RP2 = S2 / (x~-x).
Furthermore, the map T (above) from S2 to this quotient has the special property that it is locally injective away from six pairs of antipodal points.
The Roman surface has four bulbous "lobes", each one on a different corner of a tetrahedron.
A Roman surface can be constructed by splicing together three hyperbolic paraboloids and then smoothing out the edges as necessary so that it will fit a desired shape (e.g. parametrization).
On the west-southwest and east-northeast directions in Figure 2 there are a pair of openings.
When the openings are closed up, the result is the Roman surface shown in Figure 3.
A pair of lobes can be seen in the West and East directions of Figure 3.
Another pair of lobes are hidden underneath the third (z = xy) paraboloid and lie in the North and South directions.
One of the lobes of the Roman surface is seen frontally in Figure 5, and its bulbous – balloon-like—shape is evident.
Between each pair of lobes there is a locus of double points corresponding to a coordinate axis.
The Roman surface shown at the top of this article also has three lobes in sideways view.
So the ant moves to the North and falls off the edge of the world, so to speak.
It now finds itself on the northern lobe, hidden underneath the third paraboloid of Figure 3.
It will climb a slope (upside-down) until it finds itself "inside" the Western lobe.
Now let the ant move in a Southeastern direction along the inside of the Western lobe towards the z = 0 axis, always above the x-y plane.
As soon as it passes through the z = 0 axis the ant will be on the "outside" of the Eastern lobe, standing rightside-up.
Then let it move Northwards, over "the hill", then towards the Northwest so that it starts sliding down towards the x = 0 axis.
As soon as the ant crosses this axis it will find itself "inside" the Northern lobe, standing right side up.
The ant is back on the third hyperbolic paraboloid, but this time under it and standing upside-down.
The surface has a total of three lines of double points, which lie (in the parametrization given earlier) on the coordinate axes.
One might expect from the preceding statements that there could be up to eight lobes, one in each octant of space which has been divided by the coordinate planes.
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 points happens to be a Whitney singularity.
