Torus

A torus is a two-dimensional surface of revolution formed by revolving a circle of radius r around an axis in the same plane that does not intersect the circle. The axis is located at a distance R from the center of the circle, and R is greater than r for the standard ring torus. The resulting doughnut-shaped surface has a hollow center and a circular cross-section.

Parametrically, a torus can be described by

x(u,v) = (R + r cos v) cos u,

y(u,v) = (R + r cos v) sin u,

z(u,v) = r sin v,

where u and v range from 0 to 2π. In implicit form, the torus satisfies (x^2 + y^2 + z^2 + R^2 − r^2)^2 = 4 R^2 (x^2 + y^2). The standard torus is embedded in three-dimensional Euclidean space and has a circular cross-section of radius r.

Topology and geometry: a torus is a compact, orientable surface of genus 1, equivalent to the Cartesian product of two circles, S^1 × S^1. Its fundamental group is Z × Z, and its Euler characteristic is 0. The surface features rotational symmetry about the central axis and a central hole corresponding to the circular tube.

Variants and applications: when R > r, the surface is a ring torus with a clear hole; when R = r, a horn torus occurs, touching itself at a single point; when R < r, a spindle torus self-intersects. Tori appear in geometry and topology, in computer graphics and 3D modeling, and in physics and engineering contexts such as magnetic confinement devices (tokamaks) and circular accelerators.