In several disciplines, as diverse as shape analysis, location theory, quality control, archaeology, and psychometrics, it can be of interest to fit a circle through a set of points. We use the result that it suffices to locate a center for which the variance of the distances from the center to a set of given points is minimal. In this paper, we propose a new algorithm based on iterative majorization to locate the center. This algorithm is guaranteed to yield a series nonincreasing variances until a stationary point is obtained. In all practical cases, the stationary point turns out to be a local minimum. Numerical experiments show that the majorizing algorithm is stable and fast. In addition, we extend the method to fit other shapes, such as a square, an ellipse, a rectangle, and a rhombus by making use of the class of $l_p$ distances and dimension weighting. In addition, we allow for rotations for shapes that might be rotated in the plane. We illustrate how this extended algorithm can be used as a tool for shape recognition.

iterative majorization, location, optimization, shape analysis
hdl.handle.net/1765/944
Econometric Institute Research Papers
Erasmus School of Economics

van Deun, K, & Groenen, P.J.F. (2003). Majorization algorithms for inspecting circles, ellipses, squares, rectangles, and rhombi (No. EI 2003-35). Econometric Institute Research Papers. Retrieved from http://hdl.handle.net/1765/944