Math Art
Cross-Cap

Cross-Cap

Cross-Cap is a topological surface, given by a parametrisation, immersed with singularities.

Open Cross-Cap in the interactive viewer →

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

About

The projective plane rendered with a single segment of self-intersection ending in two pinch points. It is the crudest of the three standard pictures -- Boy's surface is smooth and the Roman surface is symmetric -- and the easiest to see through: a disc with its boundary glued to itself the wrong way round.

Formula

x(u, v)
0.5⁢sin⁡(u)⁢sin⁡(2⁢v)0.5 \sin\left(u\right) \sin\left(2 v\right)
y(u, v)
0.5⁢sin⁡(2⁢u)⁢cos⁡(v)20.5 \sin\left(2 u\right) \cos\left(v\right)^{2}
z(u, v)
0.5⁢cos⁡(2⁢u)⁢cos⁡(v)20.5 \cos\left(2 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
Fidelity
exact
Exactness
elementary

Definition

the standard cross-cap immersion of RP^2; antipodes (u + pi, -v) map to the same point, so the hemisphere v >= 0 covers the surface once

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.79 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.01661).