# Chapter 9

dD Convex Hulls and Delaunay Triangulations

*Susan Hert and Michael Seel*

## Table of Contents

## 9.1 Introduction

A subset $$*S *^{d} is convex if for any two points $$*p* and $$*q*
in the set the line segment with endpoints $$*p* and $$*q* is contained
in $$*S*. The convex hull
of a set $$*S*
is the smallest convex set containing
$$*S*. The convex hull of a set of points $$*P* is a convex
polytope with vertices in $$*P*. A point in $$*P* is an extreme point
(with respect to $$*P*)
if it is a vertex of
the convex hull of $$*P*. A set of points is said to be strongly convex
if it consist of only extreme points.

This chapter describes the class provided in CGAL for producing convex
hull in arbitrary dimensions. There is an intimate relationship between
the Delaunay triangulation of a point set $$*S* and the convex hull of
*lift(S)*: The nearest site
Delaunay triangulation is the projection of the lower hull and the
furthest site Delaunay triangulation is the upper hull. Here we also
describe the companion class to the convex hull class that computes
nearest and furthest site Delaunay triangulations.

## 9.2 dD Convex Hull

The class *CGAL::Convex_hull_d<R>* is used to represent the
convex hull of a set of points in $$*d*-dimensional space. This class
supports incremental construction of hulls, and provides a rich
interface for exploration. There are also output routines for hulls
of dimension 2 and 3.

The convex hull class is parameterized by a traits class that provides
$$*d*-dimensional data types and predicates. The class
*Convex_hull_d_traits_3* adapts any low-dimensional standard kernel
model *e.g.*, *Homogeneous<RT>* or *Cartesian<FT>* for use
with *Convex_hull_d*, where the dimension is fixed to three.
The validity of the computed convex hull can be checked using the
member function *is_valid*, which implements the algorithm
of Mehlhorn *et al.*[MNS$$^{+}96] to determine if
the vertices of a given polytope constitute a strongly convex point
set or not.

The implementation follows the papers [CMS93] and
[BMS94].

## 9.3 Delaunay Triangulation

There is a class type with a thorough interface providing the
construction and exploration of closest and furthest site Delaunay
simplicial complexes in arbitrary higher dimension. The class
*CGAL::Delaunay_d<R, Lifted_R>*
provides an
implementation via the lifting map to higher dimensional
convex hulls.
.
The class supports incremental construction of Delaunay triangulations
and various kind of query operations (in particular, nearest and furthest
neighbor queries and range queries with spheres and simplices).