Slicing a Torus
or
How many ways can a torus be cut (with a single plane)
so that the resulting cross sections are perfect circles?
Written by Paul Bourke
November 2003
Update, June 2008: Appeared in The Scottish Mathematical Council
Journal #37, pp 62.
Answer: There are three ways to cut a torus (any torus) with a single
plane in such a way that the resulting cross section consist only of circles.
The first two solutions are fairly obvious, the third solution can be surprising.
An example torus is given below followed by images illustrating the three cuts.
The images here were all created using the following PovRay source code:
torus.ini and
torus.pov.
Torus, r1 = major radius = 2, r2 = minor radius = 0.75
1. "Horizontal" slice resulting in two concentric circles.
The radii of the two circles is given by
r1 + sqrt(r2^{2}  h^{2})
and
r1  sqrt(r2^{2}  h^{2})
where h is the distance of the cutting plane above the plane of the torus.
Note that when the cutting plane is at a distance equal to the
minor radius r2 then there is only one solution, at greater distances
there are no solutions (the plane doesn't cut the torus).
2. "Vertical" slice resulting in two nonintersecting circles.
Radius of circles = r2
3. "Angle" cut resulting in two overlapping circles.
Radius of circle = r1.
These are two of the so called Villarceau circles, the 4 Villarceau
are the circles that pass through an arbitrary point on the surface
of a torus.
