The function convex_hull_3() computes the convex hull of a given set of three-dimensional points.

Two versions of this function are available. The first can be used when it is known that the result will be a polyhedron and the second when a degenerate hull may also be possible.

Functions
template<class InputIterator , class PolygonMesh , class Traits >
void	CGAL::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`).

template<class VertexListGraph , class PolygonMesh , class NamedParameters = parameters::Default_named_parameters>
void	CGAL::convex_hull_3 (const VertexListGraph &g, PolygonMesh &pm, const NamedParameters &np=parameters::default_values())
	computes the convex hull of the points associated to the vertices of `g`.

template<class InputIterator , class Traits >
void	CGAL::convex_hull_3 (InputIterator first, InputIterator last, Object &ch_object, const Traits &ch_traits=Default_traits)
	computes the convex hull of the set of points in the range [`first`, `last`).

template<class InputIterator , class PointRange , class PolygonRange class Traits>
void	CGAL::convex_hull_3 (InputIterator first, InputIterator last, PointRange &vertices, PolygonRange &faces, const Traits &ch_traits=Default_traits)
	computes the convex hull of the set of points in the range [`first`, `last`).

template<class InputRange , class OutputIterator , class Traits >
OutputIterator	CGAL::extreme_points_3 (InputRange range, OutputIterator out, const Traits &traits)
	copies in `out` the points on the convex hull of the points in `range`.

template<class PlaneIterator , class PolygonMesh >
void	CGAL::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 [`begin`, `end`) without constructing the dual points.

template<class PlaneIterator , class PolygonMesh , class Traits >
void	CGAL::halfspace_intersection_with_constructions_3 (PlaneIterator pbegin, PlaneIterator pend, PolygonMesh &pm, std::optional< Kernel_traits< std::iterator_traits< PlaneIterator >::value_type >::Kernel::Point_3 > > origin=std::nullopt, const Traits &ch_traits=Default_traits)
	computes the intersection of the halfspaces defined by the planes contained in the range [`begin`, `end`).

template<class Triangulation , class PolygonMesh >
void	CGAL::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.

template<class PointPropertyMap , class Base_traits >
Extreme_points_traits_adapter_3< PointPropertyMap, Base_traits >	CGAL::make_extreme_points_traits_adapter (const PointPropertyMap &pmap, Base_traits traits)
	Returns `Extreme_points_traits_adapter_3<PointPropertyMap, Base_traits>(pmap, traits)`.

template<class PlaneIterator >
std::optional< typename Kernel_traits< typename std::iterator_traits< PlaneIterator >::value_type >::Kernel::Point_3 >	CGAL::halfspace_intersection_interior_point_3 (PlaneIterator begin, PlaneIterator end)
	computes a point belonging to the intersection of the halfspaces defined by the planes contained in the range `[begin, end)`.

Function Documentation

◆ convex_hull_3() [1/4]

template<class VertexListGraph , class PolygonMesh , class NamedParameters = parameters::Default_named_parameters>

void CGAL::convex_hull_3	(	const VertexListGraph &	g,
		PolygonMesh &	pm,
		const NamedParameters &	np = `parameters::default_values()`
	)

#include <CGAL/convex_hull_3.h>

computes the convex hull of the points associated to the vertices of g.

The polygon mesh pm is cleared, then the convex hull is stored in pm. Note that the convex hull will be triangulated, that is pm will contain only triangular faces. if the convex hull is a point or a segment, endpoints will be added in pm as isolated vertices.

Template Parameters

VertexListGraph	a model of `VertexListGraph`.
PolygonMesh	must be a model of `MutableFaceGraph`. an internal property map for `CGAL::vertex_point_t` must be available.
NamedParameters	a sequence of named parameters

Parameters

g	the graph
pm	the `PolygonMesh` that will contain the convex hull
np	an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

a property map associating points to the vertices of g
Type: a class model of ReadablePropertyMap with boost::graph_traits<VertexListGraph>::vertex_descriptor as key type and Point_3 as value type
Default: boost::get(CGAL::vertex_point, g)
Extra: If this parameter is omitted, an internal property map for CGAL::vertex_point_t must be available in VertexListGraph.

Attention: The user must include the header file of the PolygonMesh and VertexListGraph types.

◆ convex_hull_3() [2/4]

template<class InputIterator , class Traits >

void CGAL::convex_hull_3	(	InputIterator	first,
		InputIterator	last,
		Object &	ch_object,
		const Traits &	ch_traits = `Default_traits`
	)

#include <CGAL/convex_hull_3.h>

computes the convex hull of the set of points in the range [first, last).

The result, which may be a point, a segment, a triangle, or a polygon mesh, is stored in ch_object. In the case the result is a polygon mesh, the convex hull will be triangulated, that is the polygon mesh will contain only triangular facets.

Template Parameters

InputIterator	must be an input iterator with a value type equivalent to `Traits::Point_3`.
Traits	must be model of the concept `ConvexHullTraits_3`. For the purposes of checking the postcondition that the convex hull is valid, `Traits` must also be a model of the concept `IsStronglyConvexTraits_3`. Furthermore, `Traits` must define a type `PolygonMesh` that is a model of `MutableFaceGraph`.

If the kernel R of the points determined by the value type of InputIterator is a kernel with exact predicates but inexact constructions (in practice we check R::Has_filtered_predicates_tag is Tag_true and R::FT is a floating point type), then the default traits class of convex_hull_3() is Convex_hull_traits_3<R>, and R otherwise.

Attention: The user must include the header file of the PolygonMesh type.

◆ convex_hull_3() [3/4]

template<class InputIterator , class PointRange , class PolygonRange class Traits>

void CGAL::convex_hull_3	(	InputIterator	first,
		InputIterator	last,
		PointRange &	vertices,
		PolygonRange &	faces,
		const Traits &	ch_traits = `Default_traits`
	)

#include <CGAL/convex_hull_3.h>

computes the convex hull of the set of points in the range [first, last).

The result, which may be a point, a segment, a triangle, or a triangle mesh, is stored as an indexed triangle soup in a vector of points and a vector of index triples.

Template Parameters

InputIterator	must be an input iterator with a value type equivalent to `Traits::Point_3`.
PointRange	a model of the concept `RandomAccessContainer` whose value type is the point type.
PolygonRange	a model of the concepts `RandomAccessContainer` and `BackInsertionSequence` whose `value_type` is itself a model of the concepts `RandomAccessContainer` and `BackInsertionSequence` whose `value_type` is an unsigned integer type convertible to `std::size_t`
Traits	must be model of the concept `ConvexHullTraits_3`. For the purposes of checking the postcondition that the convex hull is valid, `Traits` must also be a model of the concept `IsStronglyConvexTraits_3`.

If the kernel R of the points determined by the value type of InputIterator is a kernel with exact predicates but inexact constructions (in practice we check R::Has_filtered_predicates_tag is Tag_true and R::FT is a floating point type), then the default traits class of convex_hull_3() is Convex_hull_traits_3<R>, and R otherwise.

◆ convex_hull_3() [4/4]

template<class InputIterator , class PolygonMesh , class Traits >

void CGAL::convex_hull_3	(	InputIterator	first,
		InputIterator	last,
		PolygonMesh &	pm,
		const Traits &	ch_traits = `Default_traits`
	)

#include <CGAL/convex_hull_3.h>

computes the convex hull of the set of points in the range [first, last).

The polygon mesh pm is cleared, then the convex hull is stored in pm. Note that the convex hull will be triangulated, that is pm will contain only triangular facets. if the convex hull is a point or a segment, endpoints will be added in pm as isolated vertices.

Template Parameters

InputIterator	must be an input iterator with a value type equivalent to `Traits::Point_3`.
PolygonMesh	must be a model of `MutableFaceGraph`.
Traits	must be a model of the concept `ConvexHullTraits_3`. For the purposes of checking the postcondition that the convex hull is valid, `Traits` must also be a model of the concept `IsStronglyConvexTraits_3`.

If the kernel R of the points determined by the value type of InputIterator is a kernel with exact predicates but inexact constructions (in practice we check R::Has_filtered_predicates_tag is Tag_true and R::FT is a floating point type), then the default traits class of convex_hull_3() is Convex_hull_traits_3<R>, and R otherwise.

Attention: The user must include the header file of the PolygonMesh type.

Implementation

The algorithm implemented by these functions is the quickhull algorithm of Barber et al. [1].

Examples: Convex_hull_3/quickhull_3.cpp, Convex_hull_3/quickhull_OM_3.cpp, Convex_hull_3/quickhull_any_dim_3.cpp, and Convex_hull_3/quickhull_indexed_triangle_set_3.cpp.

◆ convex_hull_3_to_face_graph()

template<class Triangulation , class PolygonMesh >

void CGAL::convex_hull_3_to_face_graph	(	const Triangulation &	T,
		PolygonMesh &	pm
	)

#include <CGAL/convex_hull_3_to_face_graph.h>

fills a polyhedron with the convex hull of a set of 3D points contained in a 3D triangulation of CGAL.

The polyhedron pm is cleared and the convex hull of the set of 3D points is stored in pm.

Precondition: T.dimension()==3.

Template Parameters

Triangulation	must be a CGAL 3D triangulation
PolygonMesh	must be a model of the concept `MutableFaceGraph`

See also: convex_hull_3(); link_to_face_graph()

Examples: Convex_hull_3/dynamic_hull_3.cpp, and Convex_hull_3/dynamic_hull_OM_3.cpp.

◆ extreme_points_3()

template<class InputRange , class OutputIterator , class Traits >

OutputIterator CGAL::extreme_points_3	(	InputRange	range,
		OutputIterator	out,
		const Traits &	traits
	)

#include <CGAL/convex_hull_3.h>

copies in out the points on the convex hull of the points in range.

Template Parameters

InputRange	a range of `Traits::Point_3`, model of `ConstRange`. Its iterator type is `InputIterator`.
OutputIterator	must be an output iterator where points of type `Traits::Point_3` can be put.
Traits	must be model of the concept `ConvexHullTraits_3`.

If the kernel R of the points from InputRange is a kernel with exact predicates but inexact constructions (in practice we check R::Has_filtered_predicates_tag is Tag_true and R::FT is a floating point type), then the default traits class used is Convex_hull_traits_3<R>, and R otherwise.

Parameters

range	the range of input points.
out	an output iterator where the extreme points will be put.
traits	an instance of `Traits`.

See also: CGAL::Extreme_points_traits_adapter_3

Examples: Convex_hull_3/extreme_indices_3.cpp, and Convex_hull_3/extreme_points_3_sm.cpp.

◆ halfspace_intersection_3()

template<class PlaneIterator , class PolygonMesh >

void CGAL::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`
	)

#include <CGAL/Convex_hull_3/dual/halfspace_intersection_3.h>

computes robustly the intersection of the halfspaces defined by the planes contained in the range [begin, end) without constructing the dual points.

The result is stored in the polyhedron pm. If origin is given then it must be a point strictly inside the polyhedron. If an interior point is not given then it is computed using the function halfspace_intersection_interior_point_3() based on solving a linear program and thus is slower.

This version does not construct the dual points explicitly but uses a special traits class for the function CGAL::convex_hull_3() to handle predicates on dual points without constructing them.

Halfspaces are considered as lower halfspaces, that is if the plane equation is \( a\, x +b\, y +c\, z + d = 0 \) then the corresponding halfspace is defined by \( a\, x +b\, y +c\, z + d \le 0 \) .

Precondition: The point type of origin and the point type of the vertices of PolygonMesh must come from the same CGAL Kernel.; if provided, origin is inside the intersection of halfspaces defined by the range [begin, end).; The computed intersection must be a bounded convex polyhedron.

Template Parameters

PlaneIterator	must be an input iterator where the value type is a model of the concept `Kernel::Plane_3` and this plane type must come from the same kernel as the point type.
PolygonMesh	must be a model of `MutableFaceGraph`.

See also: halfspace_intersection_with_constructions_3()

Examples: Convex_hull_3/halfspace_intersection_3.cpp.

◆ halfspace_intersection_interior_point_3()

template<class PlaneIterator >

std::optional< typename Kernel_traits< typename std::iterator_traits< PlaneIterator >::value_type >::Kernel::Point_3 > CGAL::halfspace_intersection_interior_point_3	(	PlaneIterator	begin,
		PlaneIterator	end
	)

#include <CGAL/Convex_hull_3/dual/halfspace_intersection_interior_point_3.h>

computes a point belonging to the intersection of the halfspaces defined by the planes contained in the range [begin, end).

If the intersection is empty, std::nullopt is returned.

Attention: Halfspaces are considered as lower halfspaces that is to say if the plane's equation is \( a\, x +b\, y +c\, z + d = 0 \) then the corresponding halfspace is defined by \( a\, x +b\, y +c\, z + d \le 0 \) .

Template Parameters

PlaneIterator must be an input iterator with the value type being a Plane_3 object from CGAL Kernel

◆ halfspace_intersection_with_constructions_3()

template<class PlaneIterator , class PolygonMesh , class Traits >

void CGAL::halfspace_intersection_with_constructions_3	(	PlaneIterator	pbegin,
		PlaneIterator	pend,
		PolygonMesh &	pm,
		std::optional< Kernel_traits< std::iterator_traits< PlaneIterator >::value_type >::Kernel::Point_3 >	,
		origin	= `std::nullopt`,
		const Traits &	ch_traits = `Default_traits`
	)

#include <CGAL/Convex_hull_3/dual/halfspace_intersection_with_constructions_3.h>

computes the intersection of the halfspaces defined by the planes contained in the range [begin, end).

The result is stored in the polyhedron pm. If origin is given then it must be a point strictly inside the polyhedron. If an interior point is not given then it is computed using the function halfspace_intersection_interior_point_3() based on solving a linear program and thus is slower. This version constructs explicitly the dual points using the convex hull algorithm parametrized with the given traits class.

Halfspaces are considered as lower halfspaces, that is if the plane equation is \( a\, x +b\, y +c\, z + d = 0 \) then the corresponding halfspace is defined by \( a\, x +b\, y +c\, z + d \le 0 \) .

Precondition: The value type of PlaneIterator and the point type of origin must come from the same CGAL Kernel.; if provided, origin is inside the intersection of halfspaces defined by the range [begin, end).; The computed intersection must be a bounded convex polyhedron.

Template Parameters

PlaneIterator	must be an input iterator where the value type is a model of the concept `Kernel::Plane_3` and this plane type must come from the same kernel as the point type.
PolygonMesh	must be a model of `MutableFaceGraph`.
Traits	must be a model of the concept `ConvexHullTraits_3`.

See also: halfspace_intersection_3()

Functions

Function Documentation

◆ convex_hull_3() [1/4]

◆ convex_hull_3() [2/4]

◆ convex_hull_3() [3/4]

◆ convex_hull_3() [4/4]

◆ convex_hull_3_to_face_graph()

◆ extreme_points_3()

◆ halfspace_intersection_3()

◆ halfspace_intersection_interior_point_3()

◆ halfspace_intersection_with_constructions_3()