Torus

Torus is a surface of revolution, given by a parametrisation, genus 1, embedded or immersed depending on parameters, with a continuous symmetry group.

Open Torus in the interactive viewer →

aperiodic closed closed-orientable def-parametric implemented revolution tradition-classical

About

The surface of a doughnut, and the simplest closed surface after the sphere. It is flat in the sense that matters to a geometer -- it can carry a geometry with no curvature at all, which is why a video game can wrap its world around one -- even though the doughnut sitting in space is visibly curved.

Formula

x(u, v)

\left(1 + 0.35 \cos\left(v\right)\right) \cos\left(u\right)

y(u, v)

\left(1 + 0.35 \cos\left(v\right)\right) \sin\left(u\right)

z(u, v)

0.35 \sin\left(v\right)

Properties

Family: revolution
Given by: a parametrisation
Curvature: no condition imposed
Periodicity: not periodic
Genus: 1
Ends: 0
Embedding: embedded or immersed depending on parameters
Orientable: yes
Symmetry kind: continuous symmetry group
Fidelity: exact
Exactness: elementary

Definition

centre-circle radius 1 and tube radius 0.35 fixed to the operator defaults

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.28 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.03435).

Classical.