Toroide

A toroid, or torus, is a donut-shaped object. In geometry, it is the surface formed by rotating a circle of radius r about an axis in the same plane as the circle, at a distance R from the circle’s center. The most common form is the ring torus, which is defined for R > r and yields a closed, smooth surface in three-dimensional space.

A standard parameterization describes points on the surface as x = (R + r cos v) cos u, y = (R + r cos v) sin u, z = r sin v, where u and v range over 0 to 2π. This captures the circular cross-section (radius r) swept around the central circle (radius R). The geometry can produce several varieties depending on the relationship between R and r.

Toroids are typically classified by the relation of R to r: ring torus (R > r), horn torus (R = r), and spindle torus (R < r). The horn torus touches itself at the inner point when R = r, and a spindle torus exhibits self-intersection for R < r.

Topologically, a torus is a compact two-dimensional manifold of genus 1. It has an Euler characteristic of 0 and a fundamental group isomorphic to the product Z × Z, reflecting two independent cycles around the hole and through the hole.

Applications span mathematics, physics, and engineering. In electrical engineering, toroids are used as cores for inductors and transformers, offering efficient, closed magnetic paths and reduced leakage. In mathematics, the torus serves as a fundamental example of a genus-1 surface and as the product S^1 × S^1. The term toroid also appears in physics and design to describe torus-shaped magnets, reactors, and containment geometries.