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Villarceau Circles

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Four circles may be drawn through an arbitrary point P 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 z^^ pointing up. Specify the position of P by its angle phi measured around the tube of the torus. Define phi=0 for the circle of points farthest away from the center of the torus (i.e., the points with x^2+y^2=R^2), and draw the x-axis as the intersection of a plane through the z-axis and passing through P with the xy-plane. Rotate about the y-axis by an angle theta, where

theta=sin^(-1)(a/c).
(1)

In terms of the old coordinates, the new coordinates are

x=x_1costheta-z_1sintheta
(2)
z=x_1sintheta+z_1costheta.
(3)

So in (x_1,y_1,z_1) coordinates, equation (◇) of the torus becomes

[sqrt((x_1costheta-z_1sintheta)^2+y_1^2)-c]^2+(x_1sintheta+z_1costheta)^2=a^2.
(4)

Expanding the left side gives

(x_1costheta-z_1sintheta)^2+y_1^2+c^2 -2csqrt((x_1costheta-z_1sintheta)^2+y_1^2)+(x_1sintheta+z_1costheta)^2=a^2.
(5)

But

(x_1costheta-z_1sintheta)^2+(x_1sintheta+z_1costheta)^2=x_1^2+z_1^2,
(6)

so

x_1^2+y_1^2+z_1^2+c^2-2csqrt((x_1costheta-z_1sintheta)^2+y_1^2)=a^2.
(7)

In the z_1=0 plane, plugging in (◇) and factoring gives

[x_1^2+(y_1-a)^2-c^2][x_1^2+(y_1+a)^2-c^2]=0.
(8)

This gives the circles

x_1^2+(y_1-a)^2=c^2
(9)

and

x_1^2+(y_1+a)^2=c^2
(10)

in the z_1 plane. Written in matrix form with parameter t in [0,2pi), these are

C_1=[ccost; csint+a; 0]
(11)
C_2=[ccost; csint-a; 0].
(12)

In the original (x,y,z) coordinates,

C_1=[costheta 0 -sintheta; 0 1 0; -sintheta 0 costheta][ccost; csint+a; 0]
(13)
=[ccosthetacost; csint+a; -csinthetacost]
(14)
C_2=[costheta 0 sintheta; 0 1 0; -sintheta 0 costheta][ccost; csint-a; 0]
(15)
=[ccosthetacost; csint-a; -csinthetacost].
(16)

The point P must satisfy

z=asinphi=csinthetacost,
(17)

so

cost=(asinphi)/(csintheta).
(18)

Plugging this in for x_1 and y_1 gives the angle psi by which the circle must be rotated about the z-axis in order to make it pass through P,

psi=tan^(-1)(y/x)=(csint+a)/(ccosthetacost)=(csqrt(1-cos^2t)+a)/(ccosthetacost).
(19)

The four circles passing through P are therefore

C_1=[cospsi sinpsi 0; -sinpsi cospsi 0; 0 0 1][ccosthetacost; csint+a; -csinthetacost]
(20)
C_2=[cospsi sinpsi 0; -sinpsi cospsi 0; 0 0 1][ccosthetacost; csint-a; -csinthetacost]
(21)
C_3=[(c+acosphi)cost; (c+acosphi)sint; asinphi]
(22)
C_4=[c+acost; 0; asint].
(23)

See also

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

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