Example 4.  Consider the parabola   and the point  (0,f(0)) = (0,0)  on the curve.  
Find collocation circle to go through the three points  ,  ,  and  , and explore the situation for h = 1,.1,.01.

Solution 4.

Start with the equation of a circle  

    .  

Then write down three equations that force the collocation circle to go through the three points  ,  ,  and  .  
Enter the equations into Mathematica

Expand the equations and get

Solve the equations for and extract the formula for the radius of the collocation circle.  
Since it depends on we will store it as the function .    

Animation.
Draw the circle of curvature for various values

We can conjecture that the limit of the collocation polynomial is the circle of curvature.

(c) John H. Mathews 2004