V xz xy yz12/7/2023 ![]() Let there be these three hyperbolic paraboloids: The Roman surface has four bulbous "lobes", each one on a different corner of a tetrahedron.Ī 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. Or from RP 2 the resulting map making this an immersion of RP 2 - minus six points - into 3-space. Furthermore, the map T (above) from S 2 to this quotient has the special property that it is locally injective away from six pairs of antipodal points. Because some distinct pairs of antipodes are all taken to identical points in the Roman surface, it is not homeomorphic to RP 2, but is instead a quotient of the real projective plane RP 2 = S 2 / (x~-x). Since this is true of all points of S 2, then it is clear that the Roman surface is a continuous image of a "sphere modulo antipodes". The simplest construction is as the image of a sphere centered at the origin under the map f ( x, y, z ) = ( y z, x z, x y ). Its name arises because it was discovered by Jakob Steiner when he was in Rome in 1844. This mapping is not an immersion of the projective plane however, the figure resulting from removing six singular points is one. 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. ( March 2018) ( Learn how and when to remove this template message) Please help to improve this article by introducing more precise citations. ![]() ![]() This article includes a list of general references, but it lacks sufficient corresponding inline citations.
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