Fitting a two-joint orthogonal chain to a point set

We study the problem of fitting a two-joint orthogonal polygonal chain to a set S of n points in the plane, where the objective function is to minimize the maximum orthogonal distance from S to the chain. We show that this problem can be solved in Θ(n) time if the orientation of the chain is fixed, and in time when the orientation is not a priori known. Moreover, our algorithm can be used to maintain the rectilinear convex hull of S while rotating the coordinate system in time and O(n) space, improving on a recent result (Bae et al., 2009 [4]). We also consider some variations of the problem in three-dimensions where a polygonal chain is interpreted as a configuration of orthogonal planes. In this case we obtain O(n), , and O(n2) time algorithms depending on which plane orientations are fixed.