CGAL 6.1 - 3D Convex Hulls

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User Manual

Authors
  Susan Hert and Stefan Schirra

Figure 13.1 The convex hull of a model made of 192135 points.

# Introduction

A subset \( S \subseteq \mathbb{R}^3\) 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 \( P \in \mathbb{R}^3\) 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 consists of only extreme points.

This chapter describes the functions provided in CGAL for producing convex hulls in three dimensions as well as functions for checking if sets of points are strongly convex are not. One can compute the convex hull of a set of points in three dimensions in two ways in CGAL: using a static algorithm or using a triangulation to get a fully dynamic computation.

# Static Convex Hull Construction

The function `convex_hull_3()` provides an implementation of the quickhull algorithm [1]. There are two versions of this function available, one that can be used when it is known that the output will be a polyhedron (i.e., there are more than three points and they are not all collinear) and one that handles all degenerate cases and returns an `Object`, which may be a point, a segment, a triangle, or a polyhedron. Both versions accept a range of input iterators defining the set of points whose convex hull is to be computed and a traits class defining the geometric types and predicates used in computing the hull.

## Traits Class

The function `convex_hull_3()` is parameterized by a traits class, which specifies the types and geometric primitives to be used in the computation. As the function constructs 3D planes from three input points, we cannot simply pass a kernel with inexact constructions as optional argument for the traits class.

If input points from a kernel with exact predicates and non-exact constructions are used, and a certified result is expected, the class `Convex_hull_traits_3<R>` should be used (`R` being the input kernel). If the constructions from a kernel are exact this kernel can be used directly as a traits class.

Note that the default traits class takes this into account, that is the above considerations are only important for custom traits classes.

### Example

The following program reads points from an input file and computes their convex hull. We assume that the points are not all identical and not all collinear, thus we directly use a polyhedron as output. In the example you see that the convex hull function can write in any model of the concept `MutableFaceGraph`.

File Convex_hull_3/quickhull_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Polyhedron_3.h>
#include <CGAL/Surface_mesh.h>
#include <CGAL/convex_hull_3.h>
#include <vector>
#include <fstream>
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef CGAL::Polyhedron_3<K> Polyhedron_3;
typedef K::Point_3 Point_3;
typedef CGAL::Surface_mesh<Point_3> Surface_mesh;
int main(int argc, char* argv[])
{
std::ifstream in( (argc>1)? argv[1] : CGAL::data_file_path("points_3/cube.xyz"));
std::vector<Point_3> points;
Point_3 p;
while(in >> p){
points.push_back(p);
}
// define polyhedron to hold convex hull
Polyhedron_3 poly;
// compute convex hull of non-collinear points
CGAL::convex_hull_3(points.begin(), points.end(), poly);
std::cout << "The convex hull contains " << poly.size_of_vertices() << " vertices" << std::endl;
Surface_mesh sm;
CGAL::convex_hull_3(points.begin(), points.end(), sm);
std::cout << "The convex hull contains " << num_vertices(sm) << " vertices" << std::endl;
return 0;
}
CGAL::Exact_predicates_inexact_constructions_kernel
CGAL::Polyhedron_3
CGAL::Surface_mesh
CGAL::convex_hull_3
void convex_hull_3(InputIterator first, InputIterator last, PolygonMesh &pm, const Traits &ch_traits=Default_traits)
computes the convex hull of the set of points in the range [first, last).

### Example for Lower Dimensional Results

The following program reads points from an input file and computes their convex hull. Depending on the dimension of the result, we will get a point, a segment, a triangle, or a polyhedral surface. Note that the latter may also be planar polygon with a border.

File Convex_hull_3/quickhull_any_dim_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Polyhedron_3.h>
#include <CGAL/convex_hull_3.h>
#include <vector>
#include <fstream>
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef CGAL::Polyhedron_3<K> Polyhedron_3;
typedef K::Point_3 Point_3;
typedef K::Segment_3 Segment_3;
typedef K::Triangle_3 Triangle_3;
int main(int argc, char* argv[])
{
std::ifstream in( (argc>1)? argv[1] : CGAL::data_file_path("points_3/cube.xyz"));
std::vector<Point_3> points;
Point_3 p;
while(in >> p){
points.push_back(p);
}
CGAL::Object obj;
// compute convex hull of non-collinear points
CGAL::convex_hull_3(points.begin(), points.end(), obj);
if(const Point_3* p = CGAL::object_cast<Point_3>(&obj)){
std::cout << "Point " << *p << std::endl;
}
else if(const Segment_3* s = CGAL::object_cast<Segment_3>(&obj)){
std::cout << "Segment " << *s << std::endl;
}
else if(const Triangle_3* t = CGAL::object_cast<Triangle_3>(&obj)){
std::cout << "Triangle " << *t << std::endl;
}
else if(const Polyhedron_3* poly = CGAL::object_cast<Polyhedron_3>(&obj)){
std::cout << "Polyhedron\n " << *poly << std::endl;
std::cout << "The convex hull contains " << poly->size_of_vertices() << " vertices" << std::endl;
}
else {
std::cout << "something else"<< std::endl;
}
return 0;
}
CGAL::Object

## Extreme points

In addition to the `convex_hull_3()` function, the function `extreme_points_3()` is also provided in case only the points on the convex hull are required (without the connectivity information). In addition the traits class adapter `CGAL::Extreme_points_traits_adapter_3` is also provided in order to get the indices or more generally any given entity that is associated a 3D point that is on the convex hull.

The following program reads a set of points from an OFF file and outputs the indices of the points that are on the convex hull.
File Convex_hull_3/extreme_indices_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/convex_hull_3.h>
#include <CGAL/Extreme_points_traits_adapter_3.h>
#include <CGAL/IO/read_points.h>
#include <boost/iterator/counting_iterator.hpp>
#include <vector>
#include <fstream>
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef K::Point_3 Point_3;
int main(int argc, char* argv[])
{
const std::string filename = (argc>1) ? argv[1] : CGAL::data_file_path("meshes/star.off");
std::vector<Point_3> points;
if(!CGAL::IO::read_points(filename, std::back_inserter(points)))
{
std::cerr<< "Cannot open input file." <<std::endl;
return 1;
}
//This will contain the extreme vertices
std::vector<std::size_t> extreme_point_indices;
//call the function with the traits adapter for vertices
CGAL::extreme_points_3(CGAL::make_range(boost::counting_iterator<std::size_t>(0),
boost::counting_iterator<std::size_t>(points.size())),
std::back_inserter(extreme_point_indices),
CGAL::make_extreme_points_traits_adapter(CGAL::make_property_map(points)));
//print the number of extreme vertices
std::cout << "Indices of points on the convex hull are:\n";
std::copy(extreme_point_indices.begin(), extreme_point_indices.end(), std::ostream_iterator<std::size_t>(std::cout, " "));
std::cout << "\n";
return 0;
}
CGAL::extreme_points_3
OutputIterator extreme_points_3(InputRange range, OutputIterator out, const Traits &traits)
copies in out the points on the convex hull of the points in range.
CGAL::make_extreme_points_traits_adapter
Extreme_points_traits_adapter_3< PointPropertyMap, Base_traits > make_extreme_points_traits_adapter(const PointPropertyMap &pmap, Base_traits traits)
Returns Extreme_points_traits_adapter_3<PointPropertyMap, Base_traits>(pmap, traits).

The following program reads and builds a mesh from an OFF file, and then collects the vertices that are on the convex hull of the mesh.
File Convex_hull_3/extreme_points_3_sm.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Surface_mesh.h>
#include <CGAL/Extreme_points_traits_adapter_3.h>
#include <CGAL/convex_hull_3.h>
#include <vector>
#include <fstream>
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef K::Point_3 Point_3;
typedef CGAL::Surface_mesh<Point_3> Mesh;
int main(int argc, char* argv[])
{
const std::string filename = (argc>1) ? argv[1] : CGAL::data_file_path("meshes/star.off");
Mesh sm;
if(!CGAL::IO::read_polygon_mesh(filename, sm))
{
std::cerr<< "Cannot open input file." <<std::endl;
return 1;
}
//This will contain the extreme vertices
std::vector<Mesh::Vertex_index> extreme_vertices;
//call the function with the traits adapter for vertices
CGAL::extreme_points_3(vertices(sm), std::back_inserter(extreme_vertices),
CGAL::make_extreme_points_traits_adapter(sm.points()));
//print the number of extreme vertices
std::cout << "There are " << extreme_vertices.size() << " extreme vertices in this mesh." << std::endl;
return 0;
}
CGAL::IO::read_polygon_mesh
bool read_polygon_mesh(const std::string &fname, Graph &g, const NamedParameters &np=parameters::default_values())

## Halfspace Intersection

The functions `halfspace_intersection_3()` and `halfspace_intersection_with_constructions_3()` uses the convex hull algorithm and the duality to compute the intersection of a list of halfspaces. The first version does not explicitly compute the dual points: the traits class handles this issue. The second one constructs these points and hence is less robust but the computation is faster.

In order to compute the intersection an interior point is needed. It can be either given by the user or computed using linear programming. Notice that the second approach is slower due to the resolution of a linear program.

### Example

The following example computes the intersection of halfspaces defined by tangent planes to a sphere.

File Convex_hull_3/halfspace_intersection_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Convex_hull_3/dual/halfspace_intersection_3.h>
#include <CGAL/point_generators_3.h>
#include <CGAL/Surface_mesh.h>
#include <list>
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef K::Plane_3 Plane;
typedef K::Point_3 Point;
typedef CGAL::Surface_mesh<Point> Surface_mesh;
// compute the tangent plane of a point
template <typename K>
typename K::Plane_3 tangent_plane (typename K::Point_3 const& p) {
typename K::Vector_3 v(p.x(), p.y(), p.z());
v = v / sqrt(v.squared_length());
typename K::Plane_3 plane(v.x(), v.y(), v.z(), -(p - CGAL::ORIGIN) * v);
return plane;
}
int main (void) {
// number of generated planes
int N = 200;
// generates random planes on a sphere
std::list<Plane> planes;
CGAL::Random_points_on_sphere_3<Point> g;
for (int i = 0; i < N; i++) {
planes.push_back(tangent_plane<K>(*g++));
}
// define polyhedron to hold the intersection
Surface_mesh chull;
// compute the intersection
// if no point inside the intersection is provided, one
// will be automatically found using linear programming
CGAL::halfspace_intersection_3(planes.begin(),
planes.end(),
chull );
std::cout << "The convex hull contains " << num_vertices(chull) << " vertices" << std::endl;
return 0;
}
CGAL::sqrt
NT sqrt(const NT &x)
CGAL::halfspace_intersection_3
void halfspace_intersection_3(PlaneIterator begin, PlaneIterator end, PolygonMesh &pm, std::optional< Kernel_traits< std::iterator_traits< PlaneIterator >::value_type >::Kernel::Point_3 > > origin=std::nullopt)
computes robustly the intersection of the halfspaces defined by the planes contained in the range [be...
CGAL::ORIGIN
const CGAL::Origin ORIGIN

## Convexity Checking

The function `is_strongly_convex_3()` implements the algorithm of Mehlhorn et al. [2] to determine if the vertices of a given polytope constitute a strongly convex point set or not. This function is used in postcondition testing for `convex_hull_3()`.

# Dynamic Convex Hull Construction

Fully dynamic maintenance of a convex hull can be achieved by using the class `Delaunay_triangulation_3`. This class supports insertion and removal of points (i.e., vertices of the triangulation) and the convex hull edges are simply the finite edges of infinite faces. The following example illustrates the dynamic construction of a convex hull. First, random points from a sphere of a certain radius are generated and are inserted into a triangulation. Then the number of points of the convex hull are obtained by counting the number of triangulation vertices incident to the infinite vertex. Some of the points are removed and then the number of points remaining on the hull are determined. Notice that the vertices incident to the infinite vertex of the triangulation are on the convex hull but it may be that not all of them are vertices of the hull.

## Example

The following example shows how to compute a convex hull with a triangulation. The vertices incident to the infinite vertex are on the convex hull.

The function `convex_hull_3_to_face_graph()` can be used to obtain a polyhedral surface that is model of the concept `MutableFaceGraph`, e.g. `Polyhedron_3` and `Surface_mesh`.

File Convex_hull_3/dynamic_hull_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/point_generators_3.h>
#include <CGAL/Delaunay_triangulation_3.h>
#include <CGAL/Surface_mesh.h>
#include <CGAL/algorithm.h>
#include <CGAL/convex_hull_3_to_face_graph.h>
#include <list>
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef K::Point_3 Point_3;
typedef CGAL::Delaunay_triangulation_3<K> Delaunay;
typedef Delaunay::Vertex_handle Vertex_handle;
typedef CGAL::Surface_mesh<Point_3> Surface_mesh;
int main()
{
CGAL::Random_points_in_sphere_3<Point_3> gen(100.0);
std::list<Point_3> points;
// generate 250 points randomly in a sphere of radius 100.0
// and insert them into the triangulation
std::copy_n(gen, 250, std::back_inserter(points));
Delaunay T;
T.insert(points.begin(), points.end());
std::list<Vertex_handle> vertices;
T.incident_vertices(T.infinite_vertex(), std::back_inserter(vertices));
std::cout << "This convex hull of the 250 points has "
<< vertices.size() << " points on it." << std::endl;
// remove 25 of the input points
std::list<Vertex_handle>::iterator v_set_it = vertices.begin();
for (int i = 0; i < 25; i++)
{
T.remove(*v_set_it);
v_set_it++;
}
//copy the convex hull of points into a polyhedron and use it
//to get the number of points on the convex hull
Surface_mesh chull;
CGAL::convex_hull_3_to_face_graph(T, chull);
std::cout << "After removal of 25 points, there are "
<< num_vertices(chull) << " points on the convex hull." << std::endl;
return 0;
}
CGAL::Delaunay_triangulation_3
CGAL::convex_hull_3_to_face_graph
void convex_hull_3_to_face_graph(const Triangulation &T, PolygonMesh &pm)
fills a polyhedron with the convex hull of a set of 3D points contained in a 3D triangulation of CGAL...

# Performance

In the following, we compare the running times of the two approaches to compute 3D convex hulls. For the static version (using `convex_hull_3()`) and the dynamic version (using `Delaunay_triangulation_3` and `convex_hull_3_to_face_graph()`), the kernel used was `Exact_predicates_inexact_constructions_kernel`.

To compute the convex hull of a million of random points in a unit ball the static approach needed 1.63s, while the dynamic approach needed 9.50s. To compute the convex hull of the model of Figure 13.1 featuring 192135 points, the static approach needed 0.18s, while the dynamic approach needed 1.90s.

The measurements have been performed using CGAL 3.9, using the GNU C++ compiler version 4.3.5, under Linux (Debian distribution), with the compilation options `-O3 -DCGAL_NDEBUG`. The computer used was equipped with a 64bit Intel Xeon 2.27GHz processor and 12GB of RAM.