Math Art
Bianchi-Pinkall Flat Torus

Bianchi-Pinkall Flat Torus

Bianchi-Pinkall Flat Torus is a surface, of zero Gaussian curvature, given by a parametrisation, genus 1, embedded.

Open Bianchi-Pinkall Flat Torus in the interactive viewer →

aperiodic closed closed-orientable def-parametric embedded flat implemented misc tradition-classical

Formula

x(u, v)
cos⁡(π4+17.5⁢π180⁢sin⁡(3⁢u))⁢cos⁡(u+v)1−sin⁡(π4+17.5⁢π180⁢sin⁡(3⁢u))⁢sin⁡(v)\frac{\cos\left(\frac{\pi}{4} + \frac{17.5 \pi}{180} \sin\left(3 u\right)\right) \cos\left(u + v\right)}{1 - \sin\left(\frac{\pi}{4} + \frac{17.5 \pi}{180} \sin\left(3 u\right)\right) \sin\left(v\right)}
y(u, v)
cos⁡(π4+17.5⁢π180⁢sin⁡(3⁢u))⁢sin⁡(u+v)1−sin⁡(π4+17.5⁢π180⁢sin⁡(3⁢u))⁢sin⁡(v)\frac{\cos\left(\frac{\pi}{4} + \frac{17.5 \pi}{180} \sin\left(3 u\right)\right) \sin\left(u + v\right)}{1 - \sin\left(\frac{\pi}{4} + \frac{17.5 \pi}{180} \sin\left(3 u\right)\right) \sin\left(v\right)}
z(u, v)
sin⁡(π4+17.5⁢π180⁢sin⁡(3⁢u))⁢cos⁡(v)1−sin⁡(π4+17.5⁢π180⁢sin⁡(3⁢u))⁢sin⁡(v)\frac{\sin\left(\frac{\pi}{4} + \frac{17.5 \pi}{180} \sin\left(3 u\right)\right) \cos\left(v\right)}{1 - \sin\left(\frac{\pi}{4} + \frac{17.5 \pi}{180} \sin\left(3 u\right)\right) \sin\left(v\right)}

Properties

Family
misc
Given by
a parametrisation
Curvature
zero Gaussian curvature
Periodicity
not periodic
Genus
1
Ends
0
Embedding
embedded
Orientable
yes
Fidelity
exact
Exactness
elementary

Definition

the 3D-XplorMath Bianchi-Pinkall profile at the operator defaults: alpha(u) = pi/4 + (17.5 deg) sin 3u, colatitude beta = 2 alpha, lifted along the Hopf fibres and stereographically projected. Flat in the induced metric because Hopf tori over curves of constant enclosed area are; same builder normalisation (and fit) as hopf-torus

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.52 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.007408).