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.

Details

Now let Mathematica find the limit of the collocation circles and get the osculating circle.

(c) John H. Mathews 2004