Example 1.  Consider the parabola   and the point {0,f[0]} = (0,0) on the curve.  Find the radius of curvature and the circle of curvature.

Solution 1.

At  x=0,  we have  

We know the shape of this parabola and from symmetry we can conclude that the circle of curvature will have center   and radius  .

Discussion. What do other kinds of "tangent circles" look like?

It looks like the "circle of curvature" is the "best fitting" circle.

(c) John H. Mathews 2004