## CGAL::Convex_hull_d<R>

### 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 both1. In the sequel we reserve the word simplex for the simplices of dimension dcur. For each simplex there is a handle of type Simplex_handlex 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).

### Types

Convex_hull_d<R>::R
| the representation class.
Convex_hull_d<R>::Point_d
| the point type.
Convex_hull_d<R>::Hyperplane_d
| the hyperplane type.
Convex_hull_d<R>::Simplex_handle
| handle for simplices.
Convex_hull_d<R>::Facet_handle
| handle for facets.
Convex_hull_d<R>::Vertex_handle
| handle for vertices.
Convex_hull_d<R>::Simplex_iterator
| iterator for simplices.
Convex_hull_d<R>::Facet_iterator
| iterator for facets.
Convex_hull_d<R>::Vertex_iterator
| iterator for vertices.
Convex_hull_d<R>::Hull_vertex_iterator
| iterator for vertices that are part of the convex hull.

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 the convex hull object is referenced as constant. The Hull_vertex_iterator is convertible to Vertex_iterator and Vertex_handle.

Convex_hull_d<R>::Point_const_iterator
| const iterator for all inserted points.
Convex_hull_d<R>::Hull_point_const_iterator
| const iterator for all points that are part of the convex hull.

### Creation

Convex_hull_d<R> C ( 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).

The data type Convex_hull_d offers neither copy constructor nor assignment operator.

### Requirements

R is a model of the concept ConvexHullTraits_d .

### Operations

All operations below that take a point x as argument have the common precondition that x is a point of ambient space.

int | C.dimension () | returns the dimension of ambient space
int | C.current_dimension ()
|  | returns the affine dimension dcur of S.
Point_d | C.associated_point ( Vertex_handle v)
|  | returns the point associated with vertex v.
Vertex_handle | C.vertex_of_simplex ( Simplex_handle s, int i)
|  | returns the vertex corresponding to the i-th vertex of s. Precondition: 0 i dcur.
Point_d | C.point_of_simplex ( Simplex_handle s, int i)
|  | same as C.associated_point(C.vertex_of_simplex(s,i)).
Simplex_handle | C.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 i dcur.

int

C.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 i dcur and there is a simplex opposite to the i-th vertex of s.
Simplex_handle | C.simplex ( Vertex_handle v)
|  | returns a simplex of which v is a node. Note that this simplex is not unique.
int | C.index ( Vertex_handle v)
|  | returns the index of v in simplex(v).
Vertex_handle | C.vertex_of_facet ( Facet_handle f, int i)
|  | returns the vertex corresponding to the i-th vertex of f. Precondition: 0 i < dcur.
Point_d | C.point_of_facet ( Facet_handle f, int i)
|  | same as C.associated_point(C.vertex_of_facet(f,i)).
Facet_handle | C.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 i < dcur and dcur > 1.

int

C.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 i < dcur and dcur > 1.
Hyperplane_d | C.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.
Vertex_handle | C.insert ( 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 Forward_iterator>

void

C.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].
bool | C.is_dimension_jump ( Point_d x)
|  | returns true if x is not contained in the affine hull of S.
std::list<Facet_handle>
| C.facets_visible_from ( Point_d x)
|  | returns the list of all facets that are visible from x. Precondition: x is contained in the affine hull of S.
Bounded_side | C.bounded_side ( 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.
void | C.clear ( int d) | reinitializes C to an empty hull in d-dimensional space.
int | C.number_of_vertices ()
|  | returns the number of vertices of C.
int | C.number_of_facets ()
|  | returns the number of facets of C.
int | C.number_of_simplices ()
|  | returns the number of bounded simplices of C.
void | C.print_statistics ()
|  | gives information about the size of the current hull and the number of visibility tests performed.
bool | C.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.

### Lists and Iterators

Vertex_iterator | C.vertices_begin ()
|  | an iterator of C to start the iteration over all vertices of the complex.
Vertex_iterator | C.vertices_end () | the past the end iterator for vertices.
Simplex_iterator | C.simplices_begin ()
|  | an iterator of C to start the iteration over all simplices of the complex.
Simplex_iterator | C.simplices_end () | the past the end iterator for simplices.
Facet_iterator | C.facets_begin () | an iterator of C to start the iteration over all facets of the complex.
Facet_iterator | C.facets_end () | the past the end iterator for facets.
Hull_vertex_iterator
| C.hull_vertices_begin ()
|  | an iterator to start the iteration over all vertices of C that are part of the convex hull.
Hull_vertex_iterator
| C.hull_vertices_end ()
|  | the past the end iterator for hull vertices.
Point_const_iterator
| C.points_begin () | returns the start iterator for all points that have been inserted to construct C.
Point_const_iterator
| C.points_end () | returns the past the end iterator for points.
Hull_point_const_iterator
| C.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.
Hull_point_const_iterator
| C.hull_points_end ()
|  | returns the past the end iterator for points in the convex hull.
template <typename Visitor>
void | C.visit_all_facets ( 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
std::list<Point_d>
| C.all_points () | returns a list of all points that have been inserted to construct C.
std::list<Vertex_handle>
| C.all_vertices () | returns a list of all vertices of C (also interior ones).
std::list<Simplex_handle>
| C.all_simplices () | returns a list of all simplices in C.
std::list<Facet_handle>
| C.all_facets () | returns a list of all facets of C.

### 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 [CMS93] and [BMS94]. 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 C1, C2, C3, ...be the sequence of hulls constructed and for a point x let ki be the number of facets of Ci that are visible from x and that are not already facets of Ci-1. Then the time for inserting x is O(dim i ki) 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 · dim bytes times the number of points.

### Low Dimensional Conversion Routine

include <CGAL/Convex_hull_d_to_polyhedron_3.h>

template <class R, class T, class HDS>

void

convex_hull_d_to_polyhedron_3 ( | C, Polyhedron_3<T,HDS>& P)

|  | converts the convex hull C to polyedral surface stored in P. Precondition: dim == 3 and dcur == 3.

### Low Dimensional Output Routines

include <CGAL/IO/Convex_hull_d_window_stream.h>

template <class R>
void | d2_show ( C, CGAL::Window_stream& W)
|  | draws the convex hull C in window W. Precondition: dim == 2.
template <class R>

void

d3_surface_map ( | C, GRAPH< typename ::Point_d ,int>& G)

|  | constructs the representation of the surface of C as a bidirected LEDA graph G. Precondition: dim == 3.

### Footnotes

1 | The empty set if a facet of every simplex.