Math Art
Clifford Torus

Clifford Torus

Clifford Torus is a topological surface, given by a parametrisation, genus 1, immersed.

Open Clifford Torus in the interactive viewer →

aperiodic closed closed-orientable def-parametric immersed implemented topological tradition-classical

Formula

x(u, v)
cos⁡(u+v)2−sin⁡(v)\frac{\cos\left(u + v\right)}{\sqrt{2} - \sin\left(v\right)}
y(u, v)
sin⁡(u+v)2−sin⁡(v)\frac{\sin\left(u + v\right)}{\sqrt{2} - \sin\left(v\right)}
z(u, v)
cos⁡(v)2−sin⁡(v)\frac{\cos\left(v\right)}{\sqrt{2} - \sin\left(v\right)}

Properties

Family
topological
Given by
a parametrisation
Curvature
no condition imposed
Periodicity
not periodic
Genus
1
Ends
0
Embedding
immersed
Orientable
yes
Fidelity
exact
Exactness
elementary

Definition

the CIRCLE preset at its default mean colatitude of 90 degrees, i.e. the Hopf preimage of a great circle -- the Clifford torus S1(1/sqrt 2) x S1(1/sqrt 2) in S^3 -- stereographically projected to a torus of revolution; same builder normalisation (and fit) as hopf-torus

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.31 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.00772).