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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 
,
 |  |  |
(2)
|
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(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).
