Osculating Circle - Mathematical Description

Mathematical Description

Let γ(s) be a regular parametric plane curve, where s is the arc length, or natural parameter. This determines the unit tangent vector T, the unit normal vector N, the signed curvature k(s) and the radius of curvature at each point:

Suppose that P is a point on C where k ≠ 0. The corresponding center of curvature is the point Q at distance R along N, in the same direction if k is positive and in the opposite direction if k is negative. The circle with center at Q and with radius R is called the osculating circle to the curve C at the point P.

If C is a regular space curve then the osculating circle is defined in a similar way, using the principal normal vector N. It lies in the osculating plane, the plane spanned by the tangent and principal normal vectors T and N at the point P.

The plane curve can also be given in a different regular parametrization where regular means that for all . Then the formulas for the signed curvature k(t), the normal unit vector N(t), the radius of curvature R(t), and the center Q(t) of the osculating cicle are

,