Best Fit Circle: find radius given center

Given a center (a, b) of a circle, we want to find the best fit radius, r, to a set of given points P_i = (x_i , y_i), for i = 1 to n. By addressing the relationship between best radius for a given set of points, we will be able to make r dependent on a and b, rather than being something we guess at independently. (This is ground work for another post.)

In other words, we have a set of points and a candidate center for what we think is a good approximation to a circle (or a circular arc). We want to find the best radius for that center with those points and have a means of quantifying how good of a fit we have. If we can find this, we can decide which of a set of candidate centers is best.

The usual thing to do when looking for a best fit shape is to minimize the sum of squares of the errors. If we knew the radius we might calculate the sum of squares of the errors as

So, for each point, we find out how far it is from the candidate center, find the difference between that distance and the radius to get the error, and then square the error. And then we add all of those squared errors together to get SSE(r). We want to minimize SSE(r), which is a simple problem in differential calculus. When SSE’(r) = 0, SSE(r) is at a minimum, maximum, or perhaps a point of inflection. Then

Observe that the best fit radius is just the average distance from the candidate center to each point. Seems sensible. Also, observe that

and so, by the second derivative test, SSE(r) is at a minimum.

To see it all put together into a practical solution, see Best Fit Circle: find the center using Excel.