 |  |
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).
