## Osculating Circle

The Circle which shares the same Tangent as a curve at a given point. The Radius of Curvature of the osculating circle is

 (1)

where 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)

(Gray 1993).

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.