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* + *b* cos ,
*b* sin ).
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
(*x* cos ,
*x* sin ,
*z*).
As we let (*x*, *y*) vary over the entire original circle, we
obtain a parameterization for the torus:

*T*(,
) =
((*a* + *b* cos )
cos ,
(*a* + *b* cos )
sin ,
*b* sin ).

In "Torus Triptych" we used *a* = 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.