Toroidisk

Toroidisk is a term occasionally used in mathematics and engineering to denote a torus-like solid with a disk-shaped cross-section. The name is a portmanteau of torus and disk, and its usage is not standardized; some authors apply it to the general class of solids formed by sweeping a disk of radius r along a closed circular path of radius R, with R > r.

Geometrically, a typical toroidisk can be described as a solid torus. A common parameterization uses two angles, u and v, with R > r > 0: 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. This represents a three-dimensional region whose boundary is a torus surface and whose interior is a thickened ring.

Topologically, the solid torus is a 3D manifold whose boundary is a two-dimensional torus. Its fundamental group is isomorphic to the integers, reflecting a single fundamental loop around the central hole. The term is sometimes used interchangeably with solid torus, though some authors reserve toroidisk for specific cross-sections or manufacturing contexts.

Applications of toroidisk concepts appear in topology as standard examples, in computer graphics for donut-shaped models, and in engineering for toroidal magnetic cores, gaskets, or flywheels where a circular cross-section is advantageous. The concept emphasizes the combination of toroidal symmetry with a disk-like cross-section in a single, volumetric shape.

See also: torus, solid torus, toroidal coordinates, toroidal magnet, toroidal inductors.