Torus - Mathematics
Torus - Mathematics
In geometry, a torus (pl. tori) is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle. Examples of tori include the surfaces of doughnuts, inner tubes and particle accelerators used to examine the fundamental components of atoms. A circle rotated about a chord of the circle is called a torus in some contexts, but this is not a common usage in mathematics. The shape produced when a circle is rotated about a chord resembles a round cushion. Torus was the Latin word for a cushion of this shape.
Geometry
A torus can be defined parametrically by:
where
u, v are in the interval [0, 2π),
R is the distance from the center of the tube to the center of the torus,
r is the radius of the tube.
An equation in Cartesian coordinates for a torus radially symmetric about the z-axis is
and clearing the square root produces a quartic:
The surface area and interior volume of this torus are given by
According to a broader definition, the generator of a torus need not be a circle but could also be an ellipse or any other conic section.
Information Courtesy of Wikipeda
