Osculating Circle
The Circle which shares the same Tangent as a curve at a given point. The Radius of Curvature ofthe osculating circle is
 | (1) |
is the Curvature, and the center is |  |  | (2) |
 | |  | (3) |
i.e., the centers of the osculating circles to a curve form the Evolute to that curve.
In addition, let 
denote the Circle passing through three points on a curve 
with
. Then the osculating circle 
is given by
 | (4) |
References
Gardner, M. ``The Game of Life, Parts I-III.'' Chs. 20-22 in Wheels, Life, and other Mathematical Amusements. New York: W. H. Freeman, pp. 221, 237, and 243, 1983. Gray, A. ``Osculating Circles to Plane Curves.'' §5.6 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 90-95, 1993.
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