Radius, Diameter, Incenter, Circumcenter, Width and Minimum Enclosing Cylinder for Some Polyhedral Distance Functions

Abstract

In this paper we present some efficient and a few optimal algorithms to compute the radius, diameter, incenter, circumcenter, width and k-dimensional enclosing cylinder for convex polyhedral and convex polyhedral offset distance functions in plane and in \(\mathfrak {R}^d\). The radius, incenter and circumcenter of a convex polygon can be optimally computed in linear time, i.e. \(O(n+m)\), in \(\mathfrak {R}^2\) and in time \(O(n+m+|R|)\), if the incenter or circumcenter is additionally constrained in a convex polygon R, where n is the size of input polygon and m is the size of polygon in convex polygonal or convex polygonal offset distance functions. In \(\mathfrak {R}^d\), the radius, incenter and circumcenter of a convex polyhedron as well as of a set of convex polyhedra, unconstrained or constrained, can be computed in O(nm) and \(O(nm+|R|)\) time respectively, for convex constraint polyhedron R. The diameter of a convex polygon in plane can be computed in \(O(n+m)\) time. The diameter of a polyhedron in \(\mathfrak {R}^d\) can be computed in O(nm) time by our methods. The width of a convex polyhedron can be computed in \(O(n + m)\) time and O(nm) time, for \(\mathfrak {R}^2\) and \(\mathfrak {R}^d\), respectively. The diameter and width of a set of convex polyhedra can be computed in O(nm)-time. We also show how the k-dimensional minimum enclosing cylinder can be computed in \(O(n^{d-k+1}nm^2)\) in \(\mathfrak {R}^d\) and in \(O(n^{d}m(n+m))\) in \(\mathfrak {R}^d\) for \(k=1\). The k-dimensional minimum enclosing cylinder for a set of convex polyhedra in \(\mathfrak {R}^d\) can be computed in \(O(n^{d-k+1}nm^2)\)-time. The diameter and width problems can also be solved for a set of points in \(\mathfrak {R}^d\), either in time O(nm) or in time O(mT(n)), where T(n) is the time complexity of the best convex hull computation algorithm for the given set of points. We also compute the minimum stabbing sphere for a set of convex polyhedra, unconstrained or constrained, for the above mentioned distance functions in O(nm)-time or \(O(nm+|R|)\)-time respectively for convex constraint polyhedron R.