A more accurate method consists in more measurements than three. Do as may measurements as possible. Then a elliptic regression (least squares fitting) will provide an accurate approximation of the center of the ellipse and of the length of axes. Nevertheless, the method works also with three points only.
A direct method (non iterative) is given page 16 of the paper "Régressions coniques, quadriques,..." :
Note : In general, at least five points are necessary to fully determine the parameters of an ellipse. If you measure only three points, you need more information. For example if you know the location of the directions of the axes : Then, the general method given in the paper referenced above has to be slightly modified, wich is easy : the matrix will be simpler (size 3x3 instead of 5x5).