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osculating circle[¦äs·kyə‚lād·iŋ ′sər·kəl]
(or circle of curvature). The osculating circle of a curve l at a point M is the circle having contact of order n ≥ 2 with l at M. If l has zero curvature at M, the osculating circle degenerates to a line.
Since the order of contact of the osculating circle with l is at least 2, the osculating circle reproduces the shape of l to within infinitesimals of the third order relative to the dimensions of a portion of the curve. Figure 1 shows the usual relative positions of a curve and its osculating circle; the order of contact is 2. The curve penetrates the osculating circle at M.
The radius of curvature of l at M is the radius of the osculating circle, and the center of curvature is the center of the osculating circle. If l is a plane curve defined by the equation y = f(x), the radius of the osculating circle is given by the formula
If l is a twisted curve defined by the equations x = x(u), y = y(u), and z = z(u), the radius of the osculating circle is given by the formula
Here, the primes indicate differentiation with respect to the parameter u. (See alsoDIFFERENTIAL GEOMETRY.)