Toroid

A toroid is a doughnut-shaped object, typically referring to a torus or a torus-like form. In mathematics, a torus is the surface obtained by revolving a circle in its plane about an axis that does not intersect the circle. The resulting surface encloses a hole and can be described parametrically by x = (R + r cos v) cos u, y = (R + r cos v) sin u, z = r sin v, where R is the distance from the center of the circle to the axis and r is the circle’s radius, with R > r for a standard ring torus.

Variants of the ring torus include the horn torus (R = r), where the inner edge meets at the axis, and the spindle torus (R < r), which self-intersects in three-dimensional space. A solid torus refers to the three-dimensional volume bounded by the torus surface.

In topology, the torus is a compact 2-dimensional manifold of genus 1, with fundamental group isomorphic to Z × Z and Euler characteristic 0. The term can also describe products of circles, as in the 2-torus S^1 × S^1, or higher-dimensional generalizations such as T^n.

Outside pure mathematics, “toroid” is used for ring-shaped objects in engineering and physics, such as toroidal inductors and transformers that confine magnetic flux, and toroidal confinement devices in plasma physics. The torus thus appears in geometry, topology, and various applied fields.