Osculating Circle

OsculatingCirclesDeltoidOsculating circles of a deltoid

The osculating circle of a curve C at a given point P is the circle that has the same tangent as C at point P as well as the same curvature. Just as the tangent line is the line best approximating a curve at a point P, the osculating circle is the best circle that approximates the curve at P (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 (f(t),g(t)) and parameterized by a variable t, the radius of the osculating circle is simply the radius of curvature


where kappa(t) is the curvature, and the center is just the point on the evolute corresponding to P,


Here, derivatives are taken with respect to the parameter t.


In addition, let C(t_1,t_2,t_3) denote the circle passing through three points on a curve (f(t),g(t)) with t_1<t_2<t_3. Then the osculating circle C is given by


(Gray 1997).

See also

Curvature, Evolute, Osculating Curves, Radius of Curvature, Tangent

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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.

Referenced on Wolfram|Alpha

Osculating Circle

Cite this as:

Weisstein, Eric W. "Osculating Circle." From MathWorld--A Wolfram Web Resource.

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