Osculating Circle

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osculating circle

[¦äs·kyə‚lād·iŋ ′sər·kəl]
(mathematics)
For a plane curve C at a point p, the limiting circle obtained by taking the circle that is tangent to C at p and passes through a variable point q on C, and then letting q approach p.

Osculating Circle

 

(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.

Figure 1

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.)

REFERENCE

Rashevskii, P. K. Kurs differentsial’not geometrii, 4th ed. Moscow, 1956.