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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 
,
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).
See also
Curvature, Evolute, Osculating Curves, Radius of Curvature, Tangent
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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." ยง5.6 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/.
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Cite this as:
Weisstein, Eric W. "Osculating Circle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OsculatingCircle.html