# Osculating Circle

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## osculating circle

[¦äs·kyə‚lād·iŋ ′sər·kəl]*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*.

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 also*DIFFERENTIAL GEOMETRY.)

### REFERENCE

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