Roman Surface

Roman Surface is a topological surface, given by a parametrisation, immersed with singularities, with Td symmetry (a point group).

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aperiodic closed closed-nonorientable def-parametric implemented non-orientable singular topological tradition-classical tradition-physical-model

About

Steiner's Roman surface, another image of the projective plane, named for the city he was visiting when he found it in 1844. Where Boy's surface is smooth, this one has six pinch points and three lines of self-intersection meeting at a triple point, which makes it far easier to write down -- a quartic -- and far less well behaved.

Formula

x(u, v)

\sin\left(u\right) \cos\left(v\right) \sin\left(v\right)

y(u, v)

\cos\left(u\right) \cos\left(v\right) \sin\left(v\right)

z(u, v)

\sin\left(u\right) \cos\left(u\right) \cos\left(v\right)^{2}

Properties

Family: topological
Given by: a parametrisation
Curvature: no condition imposed
Periodicity: not periodic
Ends: 0
Embedding: immersed with singularities
Orientable: no
Symmetry: Td
Symmetry kind: point group
Fidelity: exact
Exactness: elementary

Definition

Steiner's Roman surface: the unit sphere pushed through (x, y, z) -> (yz, zx, xy)

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.73 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.01739).

J. Steiner, discovered in Rome in 1844; published posthumously by K. Weierstrass (1863).