The Hausdorff Nearest Circle to a Convex Compact Set in the Plane

Abstract

The problem of finding the nearest in the Hausdorff metric circle to a non-empty convex compact set $T$ in the plane is considered from geometrical point of view. The consideration is based on the equivalence of this problem with the Chebyshevian best approximation of $2\pi$-periodic functions by trigonometric polynomials of first order, whence it follows that the Hausdorif nearest circle to a convex compact set in the plane exists and is unique. It can be characterized by a geometric Chebyshevian alternance. As a consequence, in the particular case of a polygon the centre of the circle is described as an intersection of a midline between some two vertices and a bisectrix of some two sides. In the general case, geometrical algorithms corresponding to the one and the four point exchange Remez algorithms are described. They assure correspondingly linear and superlinear convergence. Following the idea, in the case of a polygon to get the exact solution in finite number of steps, a modified two-point exchange algorithm is suggested and illustrated by a numerical example. An application is given to estimate the Hausdorif distance between an arbitrary convex set and its Hausdorif nearest circle. The considered problem arises as a practical problem by measuring and pattern recognition in the production of circular machine parts.