Osculating circle

The osculating circle of a curve \(C\) at point \(P=\left(x\left(t_{0}\right),y\left(t_{0}\right)\right)\) is the tangent circle to the curve at \(P\) that best approximates the curve in the neighbourhood of \(P\); more precisely, it is the circle that has the same tangent in \(P\) as \(C\) as well as the same curvature. Therefore, the radius of such circle is equal to the inverse of the curvature at \(P\), that is, \(r=\left|\frac{1}{k(t)}\right|\).