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Osculating Circle

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).

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

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 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/.

Referenced on Wolfram|Alpha

Osculating Circle

Cite this as:

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