Descartes' theorem

Definition of curvature

Kissing circles. Given three mutually tangent circles (black), what radius can a fourth tangent circle have? There are in general two possible answers (red).

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 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 k_i (for i = 1,…,4), Descartes' theorem says:

(1)

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

(2)

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, then one k_i, say k₃, is zero and drops out of equation (1). Equation (2) then becomes much simpler:

(3)

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.

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