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Torus

Anton2008-11-28 08:28:35

TorusA torus has the shape of a doughnut. Formally, a torus is a surface of revolution generated by revolving a circle in three dimensional space about a line which does not intersect the circle.

Torus surface area and volume are calculated by the Pappus's centroid theorems:

A = s d, where s - the arc length of the curve being rotated, d is the distance traveled by the centroid of the region in one rotation,
V = A d, where A - area of the region being rotated, d is the distance traveled by the centroid of the region in one rotation,

So surface area of torus: S = (2\pi r)(2\pi R) = 4 \pi^2 Rr
Torus volume: V = (\pi r^2)(2\pi R)=2 \pi^2 R r^2

TorusCreative Commons Attribution/Share-Alike License 3.0 (Unported)
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Standard torus to horn torus and spindle torus transformationThe three different classes of standard tori correspond to the three possible relative sizes of r and R. When R > r, the surface will be the standard ring torus. The case R = r corresponds to the horn torus, which in effect is a torus with no "hole". The case R < r describes a self-intersecting surface called a spindle torus.

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