The osculating circle of a curve at a given point is the circle that has the same tangent as at point as well as the same curvature. Just as the tangent line is the line best approximating a curve at a point , the osculating circle is the best circle that approximates the curve at (Gray 1997, p. 111).

Ignoring degenerate curves such as straight lines, the osculating circle of a given curve at a given point is unique.

Given a plane curve with parametric equations and parameterized by a variable , the radius of the osculating circle is simply the radius of curvature

(1)

where is the curvature, and the center is just the point on the evolute corresponding to ,

			(2)
			(3)

Here, derivatives are taken with respect to the parameter .

In addition, let denote the circle passing through three points on a curve with . Then the osculating circle is given by

(4)

(Gray 1997).

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 with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 111-115, 1997.

Trott, M. The Mathematica GuideBook for Graphics. New York: Springer-Verlag, pp. 24-25, 2004. http://www.mathematicaguidebooks.org/.

CITE THIS AS:

Weisstein, Eric W. "Osculating Circle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/OsculatingCircle.html