gravelBox

For showing off, this is intentionally not automated. Its an adaptation from Ana Moura Santos and João Pedro Pargana 's "Curves Through Given Points In The Plane". Assuming that your CDF player is running you will first notice that the file is a little small. Working on that.

Start with two points and drag them to the edge of the outline (horizontal if possible). Then switch to 3 points. The third data point can go either on the curved tip or above the other two, which I find easier. The equation at the bottom is a dynamic fit to the circle defined by the three points. Although I have not taken the next obvious step, the solution could also be described simply with a radius value.

The more interesting option involves all five data points. Leave the original two points in position while dragging two more to trace the outer inclusive angle. The fifth point is most useful and revealing if pulled below the outline. Clearly a hyperbola is the more appropriate likelihood function for the displayed curvature. Again, a reformatting of the equation would reveal the form similar to:

with all of the details such as eccentricity, foci, directrices, etc.

The sketch is actually not a sketch but a microscope grayscale image of a laser lensed fiber that went through edge detection (Canny), scaled to microns and then color inverted for humans. The fit is solved on the fly. The equation would be then related to parametric equations typical of Radius of Curvature and Hyperbolic features, if these were of interest.

Additional features would allow overlaying a circle and hyperboloid together for visually comparing the fits (zoom would be good too).

Ultimately a maximum likelihood goodness of fit provided for each solution. Curiously, the hyperbolic shape is a direct result of how the lens was created as it is a sliced cylinder after all.