Generalizations for 2D
In two dimensions, a curve can be expressed with the parametric equations and . Similarly, the formulas for the radius of curvature and center of curvature can be derived using limits. At the point the center and radius of the circle of convergence is
Remark. The absolute value is necessary, otherwise the formula would only work for a curve that is positively oriented.
Now let Mathematica find the limit of the collocation circles and get the osculating circle.
(c) John H. Mathews 2004