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).