Expressing the surface area and the volume by the distance p of an outermost point on the surface of the torus to the center, and the distance q of an innermost point to the center (so that R = p + q/2 and r = p − q/2), yields

As a torus is the product of two circles, a modified version of the spherical coordinate system is sometimes used. In traditional spherical coordinates there are three measures, R, the distance from the center of the coordinate system, and θ and φ, angles measured from the center point. As a torus has, effectively, two center points, the centerpoints of the angles are moved; φ measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of θ is moved to the center of r, and is known as the "poloidal" direction. These terms were first used in a discussion of the Earths magnetic field, where "poloidal" was used to denote "the direction toward the poles".In modern use, toroidal and poloidal are more commonly used to discuss magnetic confinement fusion devices. Topology
Topologically, a torus is a closed surface defined as the product of two circles: S1 × S1. This can be viewed as lying in C2 and is a subset of the 3-sphere S3 of radius √2. This topological torus is also often called the Clifford torus.
See also
Notes
Nociones de Geometría Analítica y Álgebra Lineal, ISBN 978-970-10-6596-9, Author: Kozak Ana Maria, Pompeya Pastorelli Sonia, Verdanega Pedro Emilio, Editorial: McGraw-Hill, Edition 2007, 744 pages, language: Spanish
Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 0-521-79540-0. V. V. Nikulin, I. R. Shafarevich. Geometries and Groups. Springer, 1987. ISBN 3-540-15281-4, ISBN 978-3-540-15281-1. "Tore (notion géométrique)" at Encyclopédie des Formes Mathématiques Remarquables

References
External links

Creation of a torus at cut-the-knot
"4D torus" Fly-through cross-sections of a four-dimensional torus
"Relational Perspective Map" Visualizing high dimensional data with flat torus
Polydoes, doughnut-shaped polygons
Archived at Ghostarchive and the Wayback Machine: Séquin, Carlo H (27 January 2014). "Topology of a Twisted Torus – Numberphile" (video). Brady Haran. Anders Sandberg (4 February 2014). "Torus Earth". Retrieved 24 July 2019.