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The Mathematics of Torus Triptych

A torus can be generated by rotating a circle around an axis in the same plane as the circle, but not intersecting it. We can produce parametric equations for such a torus of revolution as follows: if we consider a circle of radius b in the xz-plane, centered at the point (x,z) = (a,0) on the x-axis, then the points on the circle are given by (x,z) = (a + bcos q, bsin q). If we rotate this circle about the z-axis, then each point (x,z) on the original circle traces out a new circle in a plane parallel to the xy-plane; the radius of this new circle will be x (the distance of the original point from the z-axis), and the height of the plane containing the new circle will be z. This means the new circle can be parameterized by (xcos f, xsin f, z). As we let (x,y) vary over the entire original circle, we obtain a parameterization for the torus:

T(q,f) = ((a + bcos q) cos f, (a + bcos q) sin f,bsin q).

In "Torus Triptych" we used a = sqrt(2) and b = 1. This basic torus was rotated to three different positions and then sliced by a horizontal plane at various heights to obtain the three sequences presented. The lower sequence (in blue) has a particularly interesting slice in the second image. Here, the horizontal plane intersects the torus in two overlapping circles of equal radius. This sequence of slices was discussed recently in The College Math Journal.

Surfaces Beyond the Third Dimension
Last modified: 08 Oct 2000 08:19:30
Comments to: Thomas F. Banchoff
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