Fitting a set of points by a circle

Abstract

Given a set of points S = {pi,...,pn} in the euclidean d-dimensional space, we address the problem of computing the d-dimensional annulus of smallest width containing the set. We give a complete characterization of the centers of annuli which are locally minimal in arbitrary dimension and we show that, for d = 2, a locally minimal annulus has two points on the inner circle and two points on the outer circle that interlace angle-wise as seen from the center of the annulus. Using this characterization, we show that, given a circular order of the points, there is at most one locally minimal annulus consistent with that order and it can be computed in time O(n log n) using a simple algorithm: it involves the computation of furthest and closest point Voronoi diagrams and a pruneand-search in the former one.

Location Science

“…In Balas and Ng (1989b) it is shown that each of these facets can be obtained using a lifting procedure from an inequality with only three non-zero coefficients that is valid in a lower dimensional polytope. Sánchez- García et al (1998) do a similar study for the case of facets with coefficients in f0; 1; 2; 3g and right-hand side equal to 3.…”

“…The theorem was shown for the unweighted case independently in many papers, among others in Rivlin (1979), Ebara et al (1989, García-López et al (1998) and it was generalized to the weighted case in Brimberg et al (2009a). The result can be interpreted in different ways:…”

Approximation Algorithms for Minimum-Width Annuli and Shells

“…For example, the circularity of a two-dimensional object O in the plane is measured by sampling a set S of points on the surface of O (e.g., using coordinate measurement machines) and computing the width of the thinnest shell containing S [14]. Motivated by this and other applications, the problem of computing A * (S) in the plane has been studied extensively [2]- [5], [11], [12], [15], [18], [20], [22]- [27], [29]. Ebara et al [11] noticed that in the planar case the center of A * (S) is a vertex of the overlay of the nearest-and farthest-neighbor Voronoi diagrams of S. This property was later refined and extended in [24] and [27].…”

“…Very little has been done so far in this direction, and previous work [9], [15] solved the problem under additional assumptions concerning the input data.…”

Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus

“…[1] unnecessary. Variants and special cases of the problem have also been addressed arising from practical consideration[8,18,20,32]. Approximation Algorithms.…”