CGAL 5.1.2 - dD Convex Hulls and Delaunay Triangulations
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#include <CGAL/Delaunay_d.h>
CGAL::Convex_hull_d< Lifted_R >.
The package dD Triangulations should be used instead.
An instance DT
of type Delaunay_d< R, Lifted_R >
is the nearest and furthest site Delaunay triangulation of a set S
of points in some 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 Delaunay triangulations and various kind of query operations (in particular, nearest and furthest neighbor queries and range queries with spheres and simplices).
A Delaunay triangulation is a simplicial complex. All simplices in the Delaunay triangulation have dimension dcur
. In the nearest site Delaunay triangulation the circumsphere of any simplex in the triangulation contains no point of \( S\) in its interior. In the furthest site Delaunay triangulation the circumsphere of any simplex contains no point of \( S\) in its exterior. If the points in \( S\) are co-circular then any triangulation of \( S\) is a nearest as well as a furthest site Delaunay triangulation of \( S\). If the points in \( S\) are not co-circular then no simplex can be a simplex of both triangulations. Accordingly, we view DT
as either one or two collection(s) of simplices. If the points in \( S\) are co-circular there is just one collection: the set of simplices of some triangulation. If the points in \( S\) are not co-circular there are two collections. One collection consists of the simplices of a nearest site Delaunay triangulation and the other collection consists of the simplices of a furthest site Delaunay triangulation.
For each simplex of maximal dimension there is a handle of type Simplex_handle
and for each vertex of the triangulation there is a handle of type Vertex_handle
. Each simplex has 1 + dcur
vertices indexed from 0
to dcur
. For any simplex s
and any index i
, DT.vertex_of(s,i)
returns the i
-th vertex of s
. There may or may not be a simplex t
opposite to the vertex of s
with index i
. The function DT.opposite_simplex(s,i)
returns t
if it exists and returns Simplex_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 DT.index_of_vertex_in_opposite_simplex(s,i)
of t
. Assume that t = DT.opposite_simplex(s,i)
exists and let j = DT.index_of_vertex_in_opposite_simplex(s,i)
. Then s = DT.opposite_simplex(t,j)
and i = DT.index_of_vertex_in_opposite_simplex(t,j)
. In general, a vertex belongs to many simplices.
Any simplex of DT
belongs either to the nearest or to the furthest site Delaunay triangulation or both. The test DT.simplex_of_nearest(dt_simplex s)
returns true if s
belongs to the nearest site triangulation and the test DT.simplex_of_furthest(dt_simplex s)
returns true if s
belongs to the furthest site triangulation.
R | must be a model of the concept DelaunayTraits_d . |
Lifted_R | must be a model of the concept DelaunayLiftedTraits_d . |
Implementation
The data type is derived from Convex_hull_d
via the lifting map. For a point x
in d
-dimensional space let lift(x)
be its lifting to the unit paraboloid of revolution. 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. For implementation details we refer the reader to the implementation report available from the CGAL server.
The space requirement is the same as for convex hulls. The time requirement for an insert is the time to insert the lifted point into the convex hull of the lifted points.
Example
The abstract data type Delaunay_d
has a default instantiation by means of the d
-dimensional geometric kernel.
Traits Requirements
Delaunay_d< R, Lifted_R >
requires the following types from the kernel traits Lifted_R
:
and uses the following function objects from the kernel traits:
Construct_hyperplane_d
Construct_vector_d
Vector_to_point_d
/ Point_to_vector_d
Orientation_d
Orthogonal_vector_d
Oriented_side_d
/ Has_on_positive_side_d
Affinely_independent_d
Contained_in_simplex_d
Contained_in_affine_hull_d
Intersect_d
Lift_to_paraboloid_d
/ Project_along_d_axis_d
Component_accessor_d
Delaunay_d< R, Lifted_R >
requires the following types from the kernel traits R
:
Related Functions | |
(Note that these are not member functions.) | |
template<typename R , typename Lifted_R > | |
void | d2_map (const Delaunay_d< R, Lifted_R > &D, GRAPH< typename Delaunay_d< R, Lifted_R >::Point_d, int > &DTG, typename Delaunay_d< R, Lifted_R >::Delaunay_voronoi_kind k=Delaunay_d< R, Lifted_R >::NEAREST) |
constructs a LEDA graph representation of the nearest (kind = NEAREST or the furthest (kind = FURTHEST ) site Delaunay triangulation. More... | |
Related Functions inherited from CGAL::Convex_hull_d< Lifted_R > | |
void | convex_hull_d_to_polyhedron_3 (const Convex_hull_d< Lifted_R > &C, Polyhedron_3< T, HDS > &P) |
void | d3_surface_map (const Convex_hull_d< Lifted_R > &C, GRAPH< typename Convex_hull_d< Lifted_R >::Point_d, int > &G) |
Types | |
enum | Delaunay_voronoi_kind { NEAREST, FURTHEST } |
interface flags More... | |
typedef unspecified_type | Simplex_handle |
handles to the simplices of the complex. | |
typedef unspecified_type | Vertex_handle |
handles to vertices of the complex. | |
typedef unspecified_type | Point_d |
the point type More... | |
typedef unspecified_type | Sphere_d |
the sphere type More... | |
typedef unspecified_type | Point_const_iterator |
the iterator for points. | |
typedef unspecified_type | Vertex_iterator |
the iterator for vertices. | |
typedef unspecified_type | Simplex_iterator |
the iterator for simplices. | |
Creation | |
The data type | |
Delaunay_d (int d, R k1=R(), Lifted_R k2=Lifted_R()) | |
creates an instance DT of type Delaunay_d . More... | |
Operations | |
All operations below that take a point | |
int | dimension () |
returns the dimension of ambient space. | |
int | current_dimension () |
returns the affine dimension of the current point set, i.e., -1 is \( S\) is empty, 0 if \( S\) consists of a single point, 1 if all points of \( S\) lie on a common line, etc. | |
bool | is_simplex_of_nearest (Simplex_handle s) |
returns true if s is a simplex of the nearest site triangulation. | |
bool | is_simplex_of_furthest (Simplex_handle s) |
returns true if s is a simplex of the furthest site triangulation. | |
Vertex_handle | vertex_of_simplex (Simplex_handle s, int i) |
returns the vertex associated with the i -th node of s . More... | |
Point_d | associated_point (Vertex_handle v) |
returns the point associated with vertex v . | |
Point_d | point_of_simplex (Simplex_handle s, int i) |
returns the point associated with the i -th vertex of s . More... | |
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 the nearest site triangulation incident to v . | |
int | index (Vertex_handle v) |
returns the index of v in DT.simplex(v) . | |
bool | contains (Simplex_handle s, const Point_d &x) |
returns true if x is contained in the closure of simplex s . | |
bool | empty () |
decides whether DT is empty. | |
void | clear () |
re-initializes DT to the empty Delaunay triangulation. | |
Vertex_handle | insert (const Point_d &x) |
inserts point x into DT and returns the corresponding Vertex_handle . More... | |
Simplex_handle | locate (const Point_d &x) |
returns a simplex of the nearest site triangulation containing x in its closure (returns Simplex_handle() if x lies outside the convex hull of \( S\)). | |
Vertex_handle | lookup (const Point_d &x) |
if DT contains a vertex v with associated_point(v) = x the result is v otherwise the result is Vertex_handle() . | |
Vertex_handle | nearest_neighbor (const Point_d &x) |
computes a vertex v of DT that is closest to x , i.e., dist(x,associated_point(v)) = min{dist(x, associated_point(u) | u \(\in S\) } . | |
std::list< Vertex_handle > | range_search (const Sphere_d &C) |
returns the list of all vertices contained in the closure of sphere \( C\). | |
std::list< Vertex_handle > | range_search (const std::vector< Point_d > &A) |
returns the list of all vertices contained in the closure of the simplex whose corners are given by A . More... | |
std::list< Simplex_handle > | all_simplices (Delaunay_voronoi_kind k=NEAREST) |
returns a list of all simplices of either the nearest or the furthest site Delaunay triangulation of S . | |
std::list< Vertex_handle > | all_vertices (Delaunay_voronoi_kind k=NEAREST) |
returns a list of all vertices of either the nearest or the furthest site Delaunay triangulation of S . | |
std::list< Point_d > | all_points () |
returns \( S\). | |
Point_const_iterator | points_begin () |
returns the start iterator for points in DT . | |
Point_const_iterator | points_end () |
returns the past the end iterator for points in DT . | |
Simplex_iterator | simplices_begin (Delaunay_voronoi_kind k=NEAREST) |
returns the start iterator for simplices of DT . | |
Simplex_iterator | simplices_end () |
returns the past the end iterator for simplices of DT . | |
typedef unspecified_type CGAL::Delaunay_d< R, Lifted_R >::Point_d |
the point type
typedef unspecified_type CGAL::Delaunay_d< R, Lifted_R >::Sphere_d |
the sphere type
enum CGAL::Delaunay_d::Delaunay_voronoi_kind |
CGAL::Delaunay_d< R, Lifted_R >::Delaunay_d | ( | int | d, |
R | k1 = R() , |
||
Lifted_R | k2 = Lifted_R() |
||
) |
creates an instance DT
of type Delaunay_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 Delaunay class. The traits class Lifted_R
specifies the models of all types and the implementations of all geometric primitives used by the base class of Delaunay_d< R, Lifted_R >
. The second template parameter defaults to the first: Delaunay_d<R> = Delaunay_d<R, Lifted_R = R >
.
int CGAL::Delaunay_d< R, Lifted_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
.
0 <= i <= dcur
. Vertex_handle CGAL::Delaunay_d< R, Lifted_R >::insert | ( | const Point_d & | x | ) |
inserts point x
into DT
and returns the corresponding Vertex_handle
.
More precisely, if there is already a vertex v
in DT
positioned at x
(i.e., associated_point(v)
is equal to x
) then associated_point(v)
is changed to x
(i.e., associated_point(v)
is made identical to x
) and if there is no such vertex then a new vertex v
with associated_point(v) = x
is added to DT
. In either case, v
is returned.
Simplex_handle CGAL::Delaunay_d< R, Lifted_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).
0 <= i <= dcur
. Point_d CGAL::Delaunay_d< R, Lifted_R >::point_of_simplex | ( | Simplex_handle | s, |
int | i | ||
) |
returns the point associated with the i
-th vertex of s
.
0 <= i <= dcur
. std::list<Vertex_handle> CGAL::Delaunay_d< R, Lifted_R >::range_search | ( | const std::vector< Point_d > & | A | ) |
returns the list of all vertices contained in the closure of the simplex whose corners are given by A
.
A
must consist of d+1
affinely independent points in base space. Vertex_handle CGAL::Delaunay_d< R, Lifted_R >::vertex_of_simplex | ( | Simplex_handle | s, |
int | i | ||
) |
returns the vertex associated with the i
-th node of s
.
0 <= i <= dcur
.
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related |
constructs a LEDA graph representation of the nearest (kind = NEAREST
or the furthest (kind = FURTHEST
) site Delaunay triangulation.
dim() == 2
.