Descartes' theorem

In geometry, Descartes' theorem, named after René Descartes, establishes a relationship between four kissing, or mutually tangent, circles. The theorem can be used to construct a fourth circle tangent to three given, mutually tangent circles.

 Contents

History

Geometrical problems involving tangent circles have been pondered for millennia. In ancient Greece of the third century BC, Apollonius of Perga devoted an entire book to the topic. Unfortunately the book, which was called On Tangencies, is not among his surviving work.

René Descartes touched on the problem briefly in 1643, in a letter to Princess Elizabeth of Bohemia (as such things might go in those times). He came up with essentially the same solution as given in equation (1) below, and thus attached his name to the theorem.

Frederick Soddy rediscovered the equation in 1936. The kissing circles in this problem are sometimes known as Soddy circles, perhaps because Soddy chose to publish his version of the theorem in the form of a poem titled The Kiss Precise, which was printed in Nature (June 20, 1936). Soddy also extended the theorem to spheres.

Definition of curvature

Missing image
KissingCircles1.png
Kissing circles. Given three mutually tangent circles (black), what radius can a fourth tangent circle have? There are in general two possible answers (red). The numbers are the circles' curvatures.

Descartes' theorem is most easily stated in terms of the circles' curvature. The curvature of a circle is defined as k = ±1/r, where r is its radius. The larger a circle, the smaller is the magnitude of its curvature, and vice versa.

The plus sign in k = ±1/r applies to a circle that is externally tangent to the other circles, like the three black circles in the image. For an internally tangent circle like the big red circle, that circumscribes the other circles, the minus sign applies.

If a straight line is considered a degenerate circle with curvature k = 0, Descartes' theorem also applies to a line and two circles that are all three mutually tangent, giving the radius of a third circle tangent to the other two circles and the line.

Descartes' theorem

If four mutually tangent circles have curvature ki (for i = 1…4), Descartes' theorem says:

 [itex](1)[itex]
[itex](k_1+k_2+k_3+k_4)^2=2\,(k_1^2+k_2^2+k_3^2+k_4^2).[itex]

When trying to find the radius of a fourth circle tangent to three given kissing circles, the equation is best rewritten as:

 [itex](2)[itex]
[itex]k_4=k_1+k_2+k_3\pm2\sqrt{k_1k_2+k_2k_3+k_3k_1}.[itex]

The ± sign reflects the fact that there are in general two solutions. Other criteria may favor one solution over the other in any given problem.

Special cases

One of the circles is replaced by a straight line of zero curvature. Descartes' theorem still applies.

If one of the three circles is replaced by a straight line, k3 (say) is zero and drops out of equation (1). Equation (2) then becomes much simpler:

 [itex](3)[itex]
[itex]k_4=k_1+k_2\pm2\sqrt{k_1k_2}.[itex]

Descartes' theorem does not apply when two or all three circles are replaced by lines. Nor does the theorem apply when more than one circle is internally tangent, e.g. in the case of three nested circles all touching in one point.

Complex Descartes theorem

In order to determine a circle completely, not only its radius (or curvature), but also its center must be known. The relevant equation is expressed most clearly if the coordinates (xy) are interpreted as a complex number z = x + iy. The equation then looks similar to Descartes' theorem and is therefore called the complex Descartes theorem.

Given four circles with curvatures ki and centers zi (for i = 1…4), the following equality holds in addition to equation (1):

 [itex](4)[itex]
[itex](k_1z_1+k_2z_2+k_3z_3+k_4z_4)^2=2\,(k_1^2z_1^2+k_2^2z_2^2+k_3^2z_3^2+k_4^2z_4^2).[itex]

Once k4 has been found using equation (2), one may proceed to calculate z4 by rewriting equation (4) to a form similar to equation (2). Again, in general there will be two solutions for z4, corresponding to the two solutions for k4.

• Interactive applet (http://www.cut-the-knot.org/Curriculum/Geometry/Eppstein.shtml) demonstrating four mutually tangent circles.

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