On an approximate minimax circle closest to a set of points

We show how the Chebychev minimax criterion for finding a circle closest to a set of points can be approximated well by standard linear programming procedures. Scope and purpose Problems that arise in location theory and in the quality control of manufactured parts (drilled holes, shaped spheres) call for finding an annulus of minimum width that encompasses a set of points. For the two-dimensional case, this is equivalent to determining a closest “deviation” circle with center (x0,y0) and radius r0 such that the maximum radial distance of the points to the circumference of the deviation circle is minimized. The required annulus (narrowest ring) is formed by two circles, centered at (x0,y0), that inscribe and circumscribe the given set of points. We suggest that our linear-programming procedure be used to approximate this annulus as, unlike exact methods, it is stable, fast, and generalizes readily to higher-dimensional point sets.