# Osculating circle

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at a given point of a curve

The circle that has contact of order with at (see Osculation). If the curvature of at is zero, then the osculating circle degenerates into a straight line. The radius of the osculating circle is called the radius of curvature of at , and its centre the centre of curvature (see Fig.). If is the plane curve given by an equation , then the radius of the osculating circle is given by

Figure: o070540a

If is the spatial curve given by equations

then the radius of the osculating circle is given by

(where the primes denote differentiation with respect to ).