**Definition of Curvature**

Descartes' theorem is most easily stated in terms of the circles' curvatures. The **curvature** (or **bend**) 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 zero curvature (and thus infinite radius), 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.

Read more about this topic: Descartes' Theorem

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