toroids

A toroid is a donut-shaped object. In geometry and topology, it typically refers to either the surface formed by revolving a circle around an axis in the same plane (a ring torus) or to the solid region bounded by that surface (a solid torus). The common model uses a circle of radius r whose center is offset from the axis by distance R, with R > r.

In parametric form, the standard torus is described by x = (R + r cos v) cos u, y = (R + r cos v) sin u, z = r sin v, where u and v range from 0 to 2π. An implicit equation is (x^2 + y^2 + z^2 + R^2 − r^2)^2 = 4 R^2 (x^2 + y^2). The surface has genus 1 and Euler characteristic 0.

Topologically, a torus is a genus-one surface whose fundamental group is isomorphic to Z × Z. The solid torus is a 3D object with a toroidal boundary; its interior is topologically different from a ball.

In science and engineering, toroids appear in magnetic confinement devices such as tokamaks, in toroidal inductors and transformers, and in network topologies that use a ring-like arrangement. In mathematics, toroids serve as basic examples in geometry, topology, and geometric modeling.