# Radius, diameter, incenter, circumcenter, width and minimum enclosing cylinder for some polyhedral distance functions

## Article Type

Research Article

## Publication Title

Discrete Applied Mathematics

## Abstract

In this paper we present efficient 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 ℜd for several types of inputs. We get optimal algorithms when the time complexities are linear. Let the size of input (convex polyhedron P/set S of points/convex polyhedra) be n/N, respectively, size of distance function convex polygon/polyhedron Q be m and size of constraint convex polyhedral region R be r in plane or ℜd. The radius, incenter and circumcenter of the convex polygon P in ℜ2 can be optimally computed in linear time, i.e. O(n+m) and in time O(n+m+r), if the incenter or circumcenter is additionally constrained in a convex polygon/polyhedron R. In ℜd, 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 polyhedral region R. 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 polyhedral region R in ℜd. The diameter of a convex polygon in plane can be computed in O(n+m) time and the diameter of a convex polyhedron in ℜd can be computed in O(nm) time. The width of a convex polyhedron can be computed in O(n+m) time and O(n2m) time, for ℜ2 and ℜd, respectively. The diameter and width of a set of convex polyhedra can be computed in O(Nm)-time and (N2m), respectively, in any dimensions. We also show how the k dimensional minimum enclosing/stabbing cylinder for the convex polyhedron P can be computed in O(nd−k+2m2) in ℜd and in O(ndm(n+m)) in ℜd for k=1. The k dimensional minimum enclosing/stabbing cylinder for the set of convex polyhedra S in ℜd can be computed in O(Nd−k+2m2)-time. The diameter and width problems can also be solved for a set of points in ℜd, 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.

## First Page

311

## Last Page

328

## DOI

10.1016/j.dam.2020.10.021

## Publication Date

12-31-2021

## Recommended Citation

Das, Sandip; Nandy, Ayan; and Sarvottamananda, Swami, "Radius, diameter, incenter, circumcenter, width and minimum enclosing cylinder for some polyhedral distance functions" (2021). *Journal Articles*. 1648.

https://digitalcommons.isical.ac.in/journal-articles/1648