 TOPICS # Villarceau Circles  Four circles may be drawn through an arbitrary point on a torus. The first two circles are obvious: one is in the plane of the torus and the second perpendicular to it. The third and fourth circles (which are inclined with respect to the torus) are much more unexpected and are known as the Villarceau circles (Villarceau 1848, Schmidt 1950, Coxeter 1969, Melzak 1983).

To see that two additional circles exist, consider a coordinate system with origin at the center of torus, with pointing up. Specify the position of by its angle measured around the tube of the torus. Define for the circle of points farthest away from the center of the torus (i.e., the points with ), and draw the x-axis as the intersection of a plane through the z-axis and passing through with the -plane. Rotate about the y-axis by an angle , where (1)

In terms of the old coordinates, the new coordinates are   (2)   (3)

So in coordinates, equation (◇) of the torus becomes (4)

Expanding the left side gives (5)

But (6)

so (7)

In the plane, plugging in (◇) and factoring gives (8)

This gives the circles (9)

and (10)

in the plane. Written in matrix form with parameter , these are   (11)   (12)

In the original coordinates,   (13)   (14)   (15)   (16)

The point must satisfy (17)

so (18)

Plugging this in for and gives the angle by which the circle must be rotated about the z-axis in order to make it pass through , (19)

The four circles passing through are therefore   (20)   (21)   (22)   (23)

Torus

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## References

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 132-133, 1969.Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, p. 125, 2002.Melzak, Z. A. Invitation to Geometry. New York: Wiley, pp. 63-72, 1983.Schmidt, H. Die Inversion und ihre Anwendungen. Munich, Germany: Oldenbourg, 1950.Villarceau, M. "Théorème sur le tore." Nouv. Ann. Math. 7, 345-347, 1848.

## Referenced on Wolfram|Alpha

Villarceau Circles

## Cite this as:

Weisstein, Eric W. "Villarceau Circles." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VillarceauCircles.html