Osculating Circle
Let P be a point on a curve C with parametrization r(t) and let ?P be the curvature at point P. If ?P 0, then there is a unique circle through P, called the osculating circle, denoted OscP, which has the following properties. OscP and C have the same tangent line and unit normal vectors at P. Both OscP and C have the same curvature kP at P. The osculating circle is the circle that "best fits" the curve at P, whose radius is R = kP-1. The value R is called the radius of curvature at P. The center of OscP should be located at a distance R from P in the normal direction, and so it is located at r(t0) + kP-1 N(t0). In addition, the osculating circle lies on the plane spanned by the unit tangent and unit normal vectors T(t) and N(t), respectively, named the osculating plane. Figure 1: The osculating circle of f(x) = x2 at x = 0. For r(t) = t2, t, t , find the following. The curvature ?(t). The center of the osculating circle. The equation of the osculating plane.