#include <CGAL/Convex_hull_d.h>

Definition

An instance C of type Convex_hull_d<R> is the convex hull of a multi-set S of points in \( d\)-dimensional space.

We call S the underlying point set and \( d\) or dim the dimension of the underlying space. We use dcur to denote the affine dimension of S. The data type supports incremental construction of hulls.

The closure of the hull is maintained as a simplicial complex, i.e., as a collection of simplices. The intersection of any two is a face of both (The empty set if a facet of every simplex). In the sequel we reserve the word simplex for the simplices of dimension dcur. For each simplex there is a handle of type Simplex_handle and for each vertex there is a handle of type Vertex_handle. Each simplex has 1 + dcur vertices indexed from 0 to dcur; for a simplex s and an index i, C.vertex(s,i) returns the i-th vertex of s. For any simplex s and any index i of s there may or may not be a simplex t opposite to the i-th vertex of s. The function C.opposite_simplex(s,i) returns t if it exists and returns Simplex_handle() (the undefined handle) otherwise. If t exists then s and t share dcur vertices, namely all but the vertex with index i of s and the vertex with index C.index_of_vertex_in_opposite_simplex(s,i) of t. Assume that t exists and let j = C.index_of_vertex_in_opposite_simplex(s,i). Then s = C.opposite_simplex(t,j) and i = C.index_of_vertex_in_opposite_simplex(t,j).

The boundary of the hull is also a simplicial complex. All simplices in this complex have dimension dcur - 1. For each boundary simplex there is a handle of type Facet_handle. Each facet has dcur vertices indexed from 0 to dcur - 1. If dcur> 1 then for each facet f and each index i, 0 <= i < dcur, there is a facet g opposite to the i-th vertex of f. The function C.opposite_facet(f,i) returns g. Two neighboring facets f and g share dcur - 1 vertices, namely all but the vertex with index i of f and the vertex with index C.index_of_vertex_in_opposite_facet(f,i) of g. Let j = C.index_of_vertex_in_opposite_facet(f,i). Then f = C.opposite_facet(g,j) and i =C.index_of_vertex_in_opposite_facet(g,j).

Template Parameters

R	must be a model of the concept `ConvexHullTraits_d`.

Iteration Statements

forall_ch_vertices( \( v,C\)) \( \{\) the vertices of \( C\) are successively assigned to \( v\) \( \}\)

forall_ch_simplices( \( s,C\)) \( \{\) the simplices of \( C\) are successively assigned to \( s\) \( \}\)

forall_ch_facets( \( f,C\)) \( \{\) the facets of \( C\) are successively assigned to \( f\) \( \}\)

Implementation

The implementation of type Convex_hull_d is based on [2] and [1]. The details of the implementation can be found in the implementation document available at the download site of this package.

The time and space requirements are input dependent. Let \(C_1\), \(C_2\), \(C_3\), \(\ldots\) be the sequence of hulls constructed and for a point \( x\) let \( k_i\) be the number of facets of \( C_i\) that are visible from \( x\) and that are not already facets of \( C_{i-1}\).

Then the time for inserting \( x\) is \( O(dim \sum_i k_i)\) and the number of new simplices constructed during the insertion of \(x\) is the number of facets of the hull which were not already facets of the hull before the insertion.

The data type Convex_hull_d is derived from Regular_complex_d. The space requirement of regular complexes is essentially \( 12(dim +2 )\) bytes times the number of simplices plus the space for the points. Convex_hull_d needs an additional \( 8 + (4 + x)dim\) bytes per simplex where \( x\) is the space requirement of the underlying number type and an additional \( 12\) bytes per point. The total is therefore \( (16 + x)dim + 32\) bytes times the number of simplices plus \( 28 + x \cdot dim\) bytes times the number of points.

Related Functions
(Note that these are not member functions.)
template<class R , class T , class HDS >
void	convex_hull_d_to_polyhedron_3 (const Convex_hull_d< R > &C, Polyhedron_3< T, HDS > &P)
	converts the convex hull `C` to polyhedral surface stored in `P`. More...

template<class R >
void	d3_surface_map (const Convex_hull_d< R > &C, GRAPH< typename Convex_hull_d< R >::Point_d, int > &G)
	constructs the representation of the surface of `C` as a bidirected LEDA graph `G`. More...

Types
Note that each iterator fits the `Handle` concept, i.e. iterators can be used as handles. Note also that all iterator and handle types come also in a const flavor, e.g., `Vertex_const_iterator` is the constant version of `Vertex_iterator`. Const correctness requires to use the const version whenever the convex hull object is referenced as constant. The `Hull_vertex_iterator` is convertible to `Vertex_iterator` and `Vertex_handle`.
typedef unspecified_type	R
	the representation class.

typedef unspecified_type	Point_d
	the point type.

typedef unspecified_type	Hyperplane_d
	the hyperplane type.

typedef unspecified_type	Simplex_handle
	handle for simplices.

typedef unspecified_type	Facet_handle
	handle for facets.

typedef unspecified_type	Vertex_handle
	handle for vertices.

typedef unspecified_type	Simplex_iterator
	iterator for simplices.

typedef unspecified_type	Facet_iterator
	iterator for facets.

typedef unspecified_type	Vertex_iterator
	iterator for vertices.

typedef unspecified_type	Hull_vertex_iterator
	iterator for vertices that are part of the convex hull.

typedef unspecified_type	Point_const_iterator
	const iterator for all inserted points.

typedef unspecified_type	Hull_point_const_iterator
	const iterator for all points that are part of the convex hull.

Creation
The data type `Convex_hull_d` offers neither copy constructor nor assignment operator.
	Convex_hull_d (int d, R Kernel=R())
	creates an instance `C` of type `Convex_hull_d`. More...

Operations
All operations below that take a point `x` as argument have the common precondition that `x` is a point of ambient space.
int	dimension ()
	returns the dimension of ambient space.

int	current_dimension ()
	returns the affine dimension `dcur` of \( S\).

Point_d	associated_point (Vertex_handle v)
	returns the point associated with vertex \( v\).

Vertex_handle	vertex_of_simplex (Simplex_handle s, int i)
	returns the vertex corresponding to the \( i\)-th vertex of \( s\). More...

Point_d	point_of_simplex (Simplex_handle s, int i)
	same as `C.associated_point(C.vertex_of_simplex(s,i))`.

Simplex_handle	opposite_simplex (Simplex_handle s, int i)
	returns the simplex opposite to the \( i\)-th vertex of \( s\) (`Simplex_handle()` if there is no such simplex). More...

int	index_of_vertex_in_opposite_simplex (Simplex_handle s, int i)
	returns the index of the vertex opposite to the \( i\)-th vertex of \( s\). More...

Simplex_handle	simplex (Vertex_handle v)
	returns a simplex of which \( v\) is a node. More...

int	index (Vertex_handle v)
	returns the index of \( v\) in `simplex(v)`.

Vertex_handle	vertex_of_facet (Facet_handle f, int i)
	returns the vertex corresponding to the \( i\)-th vertex of \( f\). More...

Point_d	point_of_facet (Facet_handle f, int i)
	same as `C.associated_point(C.vertex_of_facet(f,i))`.

Facet_handle	opposite_facet (Facet_handle f, int i)
	returns the facet opposite to the \( i\)-th vertex of \( f\) (`Facet_handle()` if there is no such facet). More...

int	index_of_vertex_in_opposite_facet (Facet_handle f, int i)
	returns the index of the vertex opposite to the \( i\)-th vertex of \( f\). More...

Hyperplane_d	hyperplane_supporting (Facet_handle f)
	returns a hyperplane supporting facet `f`. More...

Vertex_handle	insert (const Point_d &x)
	adds point `x` to the underlying set of points. More...

template<typename Forward_iterator >
void	insert (Forward_iterator first, Forward_iterator last)
	adds `S = set [first,last)` to the underlying set of points. More...

bool	is_dimension_jump (const Point_d &x)
	returns true if \( x\) is not contained in the affine hull of `S`.

std::list< Facet_handle >	facets_visible_from (const Point_d &x)
	returns the list of all facets that are visible from `x`. More...

Bounded_side	bounded_side (const Point_d &x)
	returns `ON_BOUNDED_SIDE` (`ON_BOUNDARY`,`ON_UNBOUNDED_SIDE`) if `x` is contained in the interior (lies on the boundary, is contained in the exterior) of `C`. More...

void	clear (int d)
	re-initializes `C` to an empty hull in \( d\)-dimensional space.

int	number_of_vertices ()
	returns the number of vertices of `C`.

int	number_of_facets ()
	returns the number of facets of `C`.

int	number_of_simplices ()
	returns the number of bounded simplices of `C`.

void	print_statistics ()
	gives information about the size of the current hull and the number of visibility tests performed.

bool	is_valid (bool throw_exceptions=false)
	checks the validity of the data structure. More...

Lists and Iterators
Vertex_iterator	vertices_begin ()
	an iterator of `C` to start the iteration over all vertices of the complex.

Vertex_iterator	vertices_end ()
	the past the end iterator for vertices.

Simplex_iterator	simplices_begin ()
	an iterator of `C` to start the iteration over all simplices of the complex.

Simplex_iterator	simplices_end ()
	the past the end iterator for simplices.

Facet_iterator	facets_begin ()
	an iterator of `C` to start the iteration over all facets of the complex.

Facet_iterator	facets_end ()
	the past the end iterator for facets.

Hull_vertex_iterator	hull_vertices_begin ()
	an iterator to start the iteration over all vertices of `C` that are part of the convex hull.

Hull_vertex_iterator	hull_vertices_end ()
	the past the end iterator for hull vertices.

Point_const_iterator	points_begin ()
	returns the start iterator for all points that have been inserted to construct `C`.

Point_const_iterator	points_end ()
	returns the past the end iterator for points.

Hull_point_const_iterator	hull_points_begin ()
	returns an iterator to start the iteration over all points in the convex hull `C`. More...

Hull_point_const_iterator	hull_points_end ()
	returns the past the end iterator for points in the convex hull.

template<typename Visitor >
void	visit_all_facets (const Visitor &V)
	each facet of `C` is visited by the visitor object `V`. More...

const std::list< Point_d > &	all_points ()
	returns a list of all points that have been inserted to construct `C`.

std::list< Vertex_handle >	all_vertices ()
	returns a list of all vertices of `C` (also interior ones).

std::list< Simplex_handle >	all_simplices ()
	returns a list of all simplices in `C`.

std::list< Facet_handle >	all_facets ()
	returns a list of all facets of `C`.

Constructor & Destructor Documentation

template<typename R>

CGAL::Convex_hull_d< R >::Convex_hull_d	(	int	d,
		R	Kernel = `R()`
	)

creates an instance C of type Convex_hull_d.

The dimension of the underlying space is \( d\) and S is initialized to the empty point set. The traits class R specifies the models of all types and the implementations of all geometric primitives used by the convex hull class. The default model is one of the \( d\)-dimensional representation classes (e.g., Homogeneous_d).

Member Function Documentation

template<typename R>

Bounded_side CGAL::Convex_hull_d< R >::bounded_side ( const Point_d & x)

returns ON_BOUNDED_SIDE (ON_BOUNDARY,ON_UNBOUNDED_SIDE) if x is contained in the interior (lies on the boundary, is contained in the exterior) of C.

Precondition: x is contained in the affine hull of S.

template<typename R>

std::list<Facet_handle> CGAL::Convex_hull_d< R >::facets_visible_from ( const Point_d & x)

returns the list of all facets that are visible from x.

Precondition: x is contained in the affine hull of S.

template<typename R>

Hull_point_const_iterator CGAL::Convex_hull_d< R >::hull_points_begin ( )

returns an iterator to start the iteration over all points in the convex hull C.

Included are points in the interior of facets.

template<typename R>

Hyperplane_d CGAL::Convex_hull_d< R >::hyperplane_supporting ( Facet_handle f)

returns a hyperplane supporting facet f.

The hyperplane is oriented such that the interior of C is on the negative side of it.

Precondition: f is a facet of C and dcur > 1.

template<typename R>

int CGAL::Convex_hull_d< R >::index_of_vertex_in_opposite_facet	(	Facet_handle	f,
		int	i
	)

returns the index of the vertex opposite to the \( i\)-th vertex of \( f\).

Precondition: \( 0 \leq i < dcur\) and dcur > 1.

template<typename R>

int CGAL::Convex_hull_d< R >::index_of_vertex_in_opposite_simplex	(	Simplex_handle	s,
		int	i
	)

returns the index of the vertex opposite to the \( i\)-th vertex of \( s\).

Precondition: \( 0 \leq i \leq dcur\) and there is a simplex opposite to the \( i\)-th vertex of \( s\).

template<typename R>

Vertex_handle CGAL::Convex_hull_d< R >::insert ( const Point_d & x)

adds point x to the underlying set of points.

If \( x\) is equal to (the point associated with) a vertex of the current hull this vertex is returned and its associated point is changed to \( x\). If \( x\) lies outside the current hull, a new vertex v with associated point \( x\) is added to the hull and returned. In all other cases, i.e., if \( x\) lies in the interior of the hull or on the boundary but not on a vertex, the current hull is not changed and Vertex_handle() is returned. If CGAL_CHECK_EXPENSIVE is defined then the validity check is_valid(true) is executed as a post condition.

template<typename R>

template<typename Forward_iterator >

void CGAL::Convex_hull_d< R >::insert	(	Forward_iterator	first,
		Forward_iterator	last
	)

adds S = set [first,last) to the underlying set of points.

If any point S[i] is equal to (the point associated with) a vertex of the current hull its associated point is changed to S[i].

template<typename R>

bool CGAL::Convex_hull_d< R >::is_valid ( bool throw_exceptions = false)

checks the validity of the data structure.

If throw_exceptions == thrue then the program throws the following exceptions to inform about the problem. chull_has_center_on_wrong_side_of_hull_facet the hyperplane supporting a facet has the wrong orientation.

chull_has_local_non_convexity a ridge is locally non convex.

chull_has_double_coverage the hull has a winding number larger than 1.

template<typename R>

Facet_handle CGAL::Convex_hull_d< R >::opposite_facet	(	Facet_handle	f,
		int	i
	)

returns the facet opposite to the \( i\)-th vertex of \( f\) (Facet_handle() if there is no such facet).

Precondition: \( 0 \leq i < dcur\) and dcur > 1.

template<typename R>

Simplex_handle CGAL::Convex_hull_d< R >::opposite_simplex	(	Simplex_handle	s,
		int	i
	)

returns the simplex opposite to the \( i\)-th vertex of \( s\) (Simplex_handle() if there is no such simplex).

Precondition: \( 0 \leq i \leq dcur\).

template<typename R>

Simplex_handle CGAL::Convex_hull_d< R >::simplex ( Vertex_handle v)

returns a simplex of which \( v\) is a node.

Note that this simplex is not unique.

template<typename R>

Vertex_handle CGAL::Convex_hull_d< R >::vertex_of_facet	(	Facet_handle	f,
		int	i
	)

returns the vertex corresponding to the \( i\)-th vertex of \( f\).

Precondition: \( 0 \leq i < dcur\).

template<typename R>

Vertex_handle CGAL::Convex_hull_d< R >::vertex_of_simplex	(	Simplex_handle	s,
		int	i
	)

returns the vertex corresponding to the \( i\)-th vertex of \( s\).

Precondition: \( 0 \leq i \leq dcur\).

template<typename R>

template<typename Visitor >

void CGAL::Convex_hull_d< R >::visit_all_facets ( const Visitor & V)

each facet of C is visited by the visitor object V.

V has to have a function call operator: void operator()(Facet_handle) const

Friends And Related Function Documentation

template<class R , class T , class HDS >

void convex_hull_d_to_polyhedron_3	(	const Convex_hull_d< R > &	C,
		Polyhedron_3< T, HDS > &	P
	)

Precondition: dim == 3 and dcur == 3.

template<class R >

void d3_surface_map	(	const Convex_hull_d< R > &	C,
		GRAPH< typename Convex_hull_d< R >::Point_d, int > &	G
	)

Precondition: dim == 3.

Definition

Related Functions

Types

Creation

Operations

Lists and Iterators

Constructor & Destructor Documentation

Member Function Documentation

Friends And Related Function Documentation