songsenand
/
cgal
mirror of https://github.com/CGAL/cgal
1
0
Fork 0
Code Issues Packages Projects Releases Wiki Activity

corrections afetr Francois' reading

This commit is contained in:
Monique Teillaud 1999-06-21 07:25:19 +00:00
parent 63688284a3
commit a65d65161c
4 changed files with 932 additions and 376 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

301
Packages/Triangulation_3/doc_tex/Triangulation_3/TDS3.tex
View File

@ -25,12 +25,28 @@
\section{Introduction}
\label{TDS3-sec-intro}
\subsection{Definition}
A three-dimensional triangulation is a three-dimensional simplicial
complex, pure connected and without singularities \cite{by-ag-98}. It
is a set of cells ($3$-faces, tetrahedra) such that two cells
either do not intersect or share a common facet ($2$-face), edge
($1$-face) or vertex ($0$-face).
A geometric triangulation has two aspects: the combinatorial structure which
gives the incidence and adjacency relations between faces, and the
geometric information related to the position of vertices.
\cgal\ proposes 3D geometric triangulations in which these
two aspects are clearly separated.
This chapter deals with 3D- triangulation data structures, meant to
maintain the combinatorial information for 3D-geometric
triangulations. The reader interested in geometric triangulations of
$\R^3$ is advised to read Chapter~\ref{chapter-Triangulation3}.
\subsection{Representation}
\label{TDS3-sec-def}
A 3D- triangulation data structure is meant to maintain the combinatorial
information (incidence and adjacency relations) for 3D-geometric
triangulations. In \cgal, a triangulation data structure is a
In \cgal, a triangulation data structure is a
container of cells ($3$-faces) and vertices ($0$-faces). Each cell gives
access to its four incident vertices and to its four adjacent
cells. Each vertex gives access to one of its incident cells.
@ -38,7 +54,21 @@ cells. Each vertex gives access to one of its incident cells.
The four vertices of a cell are indexed with 0, 1, 2 and 3. The
neighbors of a cell are also indexed with 0, 1, 2, 3
in such a way that the neighbor indexed by $i$ is opposite to the vertex
with the same index.
with the same index (see Figure~\ref{TDS3-fig-repres}).
\begin{ccTexOnly}
\begin{figure}
\begin{center}
\includegraphics{repres.eps}
\end{center}
\caption{Representation \label{TDS3-fig-repres}}
\end{figure}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<img border=0 src="./repres.gif" align=center
alt="Representation">
\end{ccHtmlOnly}
Edges ($1$-faces) and facets ($2$-faces) are not explicitely
represented: a facet is given by a cell and an index (the facet
@ -57,7 +87,7 @@ is not of full dimension.
As described in Chapter~\ref{chapter-Triangulation3}, a geometric
triangulation of a set of points in $\R^d$ is a partition of the
whole space $\R^d$ into cells having $d+1$ vertices: some of them
are infinite, they are obtained by liking an additional vertex at
are infinite, they are obtained by linking an additional vertex at
infinity to each facet of the convex hull of the points.
The underlying combinatorial triangulation of such a triangulation
without boundary of $\R^d$ can be seen as a triangulation of the
@ -76,15 +106,51 @@ the boundary of a 4-dimensional simplex, which has 5 vertices. A
geometric embedding consists in choosing one of these vertices to be
infinite, thus four of the five 3-cells becomes infinite: the geometric
triangulation has one finite tetrahedron remaining, each of its facets
being incident to the infinite cell.
being incident to the infinite cell. See Figure~\ref{TDS3-fig-topo-simplex4}.
\begin{ccTexOnly}
\begin{figure}
\begin{center}
\includegraphics{topo-simplex4.eps}
\end{center}
\caption{4D simplex and a 3D geometric embedding \label{TDS3-fig-topo-simplex4}}
\end{figure}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<img border=0 src="./topo-simplex4.gif" align=center
alt="4D simplex and a 3D geometric embedding">
\end{ccHtmlOnly}
\item \emph{dimension 2.} We have 4 vertices forming one 3-dimensional
simplex, i.e. the boundary of a tetrahedron. The geometric embedding in
the plane consists in choosing one of these vertices to be infinite,
then the geometric triangulation has one finite triangle whose edges are
adjacent to the infinite triangles. See Figure~\ref{TDS3-fig-topo-simplex}.
adjacent to the infinite triangles. See Figure~\ref{TDS3-fig-topo-simplex3}.
\begin{ccTexOnly}
\begin{figure}
\begin{center}
\includegraphics{topo-simplex3.eps}
\end{center}
\caption{3D simplex and a 2D geometric embedding \label{TDS3-fig-topo-simplex3}}
\end{figure}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<img border=0 src="./topo-simplex3.gif" align=center
alt="3D simplex and a 2D geometric embedding">
\end{ccHtmlOnly}
\item \emph{dimension 1.} A 2-dimensional simplex (a triangle) has 3
vertices. The geometric embedding is and edge whose vertices are linked
to an infinite point.
to an infinite point. See Figure~\ref{TDS3-fig-topo-simplex2}.
\begin{ccTexOnly}
\begin{figure}
\begin{center}
\includegraphics{topo-simplex2.eps}
\end{center}
\caption{2D simplex and a 1D geometric embedding \label{TDS3-fig-topo-simplex2}}
\end{figure}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<img border=0 src="./topo-simplex2.gif" align=center
alt="2D simplex and a 1D geometric embedding">
\end{ccHtmlOnly}
\end{itemize}
The last two cases are defined uniquely:
@ -105,20 +171,6 @@ triangulation data structure. The infinite vertex of the geometric
embedding is a vertex without any particularity in the
combinatorial triangulation.
\begin{ccTexOnly}
\begin{figure}
\begin{center}
\includegraphics{topo-simplex.eps}
\end{center}
\caption{3D simplex and its 2D geometric embedding \label{TDS3-fig-topo-simplex}}
\end{figure}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<img border=0 src="./topo-simplex.gif" align=center
alt="3D simplex and its 2D geometric embedding">
\end{ccHtmlOnly}
The implicit representation of facets (resp. edges) still holds
for degenerate ($< 3$) dimensions : in dimension~2, each cell has only one
facet of index 3, and 3 edges $(0,1)$, $(1,2)$ and $(2,0)$; in
@ -189,29 +241,144 @@ software design in this case.
<img border=0 src="./design.gif" align=center alt="Layers in the software design">
\end{ccHtmlOnly}
In the bottom
layer, the base classes store elementary geometric information as
well as any other information for the given application. The middle
layer class stores the triangulation data structure, which is purely
combinatorial. It provides operations such as insertion of a new
vertex in a given cell, and is responsible for the combinatorial
integrity of the triangulation. The upper layer is the geometric
triangulation class, providing operations such as location of a point
in the triangulation, insertion of a point, and is responsible for
the geometric validity.
In the bottom layer, the \cgal\ base classes store elementary
geometric information. These classes are parameterized by a geometric
traits class providing all the geometric types. A vertex has a pointer
to a cell, and a cell has four pointers to vertices. These pointers
are of type \ccc{void*}.
The triangulation data structure class is parameterized by the base
vertex and the base cell classes. It is up to the user to derive its
own base classes from the \cgal\ base classes, and to use the
triangulation data structure class proposed by \cgal.
The middle layer class stores the triangulation data structure, which
is purely combinatorial. A vertex of the triangulation data structure
has a pointer to a cell of the triangulation data structure, and a
cell has four pointers to vertices. These pointers are usual
\ccc{C++} pointers. The triangulation data structure provides
operations such as insertion of a new vertex in a given cell, on a
$1$ or $2$-face, or, if the dimension of the triangulation is smaller
than 3, outside the ``affine'' hull of the vertices, thus increasing
the dimension of the triangulation by one. The triangulation data
structure is responsible for the combinatorial integrity of the
triangulation.
It is up to the user to derive its own base classes from the \cgal\
base classes to add any other information the may need for his given
application, or to write his own base classes from scratch. In this
case, his base classes must be models for the concepts described in
Section~\ref{TDS3-sec-concept-Base}. As the triangulation data
structure class is parameterized by the base vertex and the base cell
classes. it will consider the user's bases classes instead of the
predefined base classes.
The upper layer, described in Chapter~\ref{chapter-Triangulation3}, is
the geometric triangulation class, providing operations such as
location of a point in the triangulation, insertion of a point, and is
responsible for the geometric validity. A vertex of the triangulation
has a pointer to a cell and a cell has four pointers to
vertices. These pointers are \cgal\ \emph{handles}. The triangulation
data structure class is one of the template parameters of the
geometric triangulation class. The user may choose to replace the
\cgal\ triangulation data structure class by its own triangulation
data structure, in this case, his class has to be a model of the
concept described in Section~\ref{TDS3-sec-concept}.
\section{Examples}
\label{TDS3-sec-examples}
The following example shows how to construct a 3D triangulation data
structure by inserting vertices.
\begin{verbatim}
#include <CGAL/basic.h>
#include <cstring>
#include <iostream>
#include <fstream>
#include <strstream.h>
#include <vector>
#include <CGAL/Cartesian.h>
#include <CGAL/Point_3.h>
#include <CGAL/Triangulation_geom_traits_3.h>
#include <CGAL/Triangulation_cell_base_3.h>
#include <CGAL/Triangulation_vertex_base_3.h>
#include <CGAL/Triangulation_data_structure_3.h>
// the definition of the geometric traits class is necessary to
// instanciate base vertices and cells but will in fact never be used
// in the program
typedef CGAL::Cartesian<double> Rep;
typedef CGAL::Triangulation_geom_traits_3<Rep> Gt;
typedef CGAL::Triangulation_vertex_base_3<Gt> Vb;
typedef CGAL::Triangulation_cell_base_3<Gt> Cb;
typedef CGAL::Triangulation_data_structure_3<Vb,Cb> Tds;
typedef typename Tds::Cell TDSCell;
typedef typename Tds::Vertex TDSVertex;
int main()
{
Tds T;
assert( T.number_of_vertices() == 0 );
assert( T.dimension() == -2 );
assert( T.is_valid() );
std::vector<TDSVertex> V(5);
std::vector<TDSVertex*> PV(7);
PV[0] = T.insert_outside_affine_hull(V[0]);
assert( T.number_of_vertices() == 1 );
assert( T.dimension() == -1 );
assert( T.is_valid() );
int i;
// each of the following insertions of vertices increases the dimension
for ( i=1; i<5; i++ ) {
PV[i] = T.insert_outside_affine_hull(V[i], PV[0]);
assert( T.number_of_vertices() == i+1 );
assert( T.dimension() == i-1 );
assert( T.is_valid() );
}
assert( T.number_of_cells() == 5 );
// we now have a simplex in dimension 4
// cell incident to PV[0]
TDSCell* c = PV[0]->cell();
int ind;
assert( c->has_vertex( PV[0], ind ) );
// PV[0] is the vertex of index ind in c
// insertion of a new vertex in the facet opposite to PV[0]
PV[5] = T.insert_in_facet(TDSVertex(), c, ind);
assert( T.number_of_vertices() == 6 );
assert( T.dimension() == 3 );
assert( T.is_valid() );
// insertion of a new vertex in c
PV[6] = T.insert_in_cell( TDSVertex(), c );
assert( T.number_of_vertices() == 7 );
assert( T.dimension() == 3 );
assert( T.is_valid() );
std::ofstream oFileT("output_tds",ios::out);
// writing file output_tds;
oFileT << T;
return 0;
}
\end{verbatim}
\clearpage
\section{Concepts}
This section describes the concepts for a 3D- triangulation data
structure, its vertices and cells.
\subsection{Concepts for a 3D- Triangulation Data Structure}
\label{TDS3-sec-concept}
\begin{ccClassTemplate}{Tds<Vb,Fb>}
@ -277,9 +444,12 @@ incident to a given edge. They are bidirectional and non mutable.
\ccMethod{Tds operator=(const Tds & tds1);}
{Assignment. All the vertices and cells are duplicated.}
The previous first two methods are equivalent.
\ccMethod{void swap(Tds & tds1);}
{Swaps \ccVar\ and \ccc{tds1}. Should be preferred to \ccVar=\ccc{tds1}
or \ccVar(\ccc{tds1}) when tds1 is deleted after that.}
or \ccVar(\ccc{tds1}) when tds1 is deleted after that. Indeed, there is no
copy of cells and vertices, thus this method runs in constant time.}
\ccMethod{Vertex* copy_tds(const Tds & tds1, Vertex* v = NULL);}
{\ccc{tds1} is copied into \ccVar. The vertex of \ccVar\
@ -288,7 +458,8 @@ corresponding to \ccc{v} is returned.
\ccc{tds1}.}}
\ccFunction{void \ccTilde\ccVar();}
{Destructor. All vertices and cells are deleted.}
{Destructor. All vertices and cells are deleted, and \ccVar\ itself is
deleted.}
\ccAccessFunctions
@ -424,11 +595,12 @@ operation is exactly reciprocal to \ccc{\ccVar.insert_in_cell(v)}.
\textit{not yet implemented}}
\ccMethod{void clear();}
{Deletes all cells and vertices.}
{Deletes all cells and vertices. \ccVar\ is reset as a triangulation
data structure constructed by the default constructor.}
In addition to these requirements, in order to be used as a triangulation
data structure by the class
\ccc{CGAL_Delaunay_triangulation_3<Traits,Tds>}, the triangulation
\ccc{Delaunay_triangulation_3<Traits,Tds>}, the triangulation
must offer the following method:
\ccMethod{void star_region( set<void*, less<void*> > & region,
@ -542,8 +714,6 @@ each cell given by their index in the preceding list of cells.
\ccCreationVariable{v}
The vertex stores a point and a pointer to an incident cell.
The vertex class of a 3D- triangulation data structure must define
the types and operations listed in this section. Some of these
requirements are of geometric nature, they are \textit{optional}
@ -725,8 +895,8 @@ a precise indication on the kind of invalidity encountered.}
\end{ccClass}
\begin{ccClassTemplate}{CGAL_Triangulation_data_structure_3<Vb,Cb>}
\section{The Triangulation Data Structure Class\\
\begin{ccClassTemplate}{Triangulation_data_structure_3<Vb,Cb>}
\section{A model of Triangulation Data Structure:\\
\protect \ccClassTemplateName}
\label{TDS3-sec-class}
@ -736,26 +906,24 @@ geometric functionalities. It is templated by base classes for
vertices and cells described in Section~\ref{TDS3-sec-concept-Base}.
\ccInheritsFrom
\ccc{CGAL_Triangulation_utils_3}
\ccc{Triangulation_utils_3}
\\This class defines basic computations on indices of vertices and
neighbors of cells (see Section~\ref{Triangulation3-sec-class-Utils} of
neighbors of cells (Details are presented in
Section~\ref{Triangulation3-sec-class-Utils} of
Chapter~\ref{chapter-Triangulation3}).
\ccInclude{CGAL/Triangulation_data_structure_3.h}
The class \ccc{CGAL_Triangulation_ds_vertex_3} is a model for the vertex
The class \ccc{Triangulation_ds_vertex_3} is a model for the vertex
concept described in Section~\ref{TDS3-sec-concept-Tds_Vertex}.
\ccInclude{CGAL/Triangulation_ds_vertex_3.h}
The class \ccc{CGAL_Triangulation_ds_cell_3} is a model for the
The class \ccc{Triangulation_ds_cell_3} is a model for the
cell concept described in Section~\ref{TDS3-sec-concept-Tds_Cell}.
\ccInclude{CGAL/Triangulation_ds_cell_3.h}
\ccInheritsFrom
\ccc{CGAL_Triangulation_utils_3}
\end{ccClassTemplate}
\section{Concepts for the Base Vertices and Cells}
@ -943,29 +1111,33 @@ a precise indication on the kind of invalidity encountered.}
\end{ccClass}
\section{The Base Classes for Vertices and Cells}
\section{Models of Base Vertices and Cells}
\label{TDS3-sec-class-Base}
\begin{ccClassTemplate}{CGAL_Triangulation_vertex_base_3<Traits>}
\begin{ccClassTemplate}{Triangulation_vertex_base_3<Traits>}
\subsection{The Base Class \protect \ccClassTemplateName}
\label{TDS3-sec-class-Base_Vertex}
This class is proposed by \cgal\ as a base vertex class.
It is templated by a geometric traits class. Using the same
geometric traits class \ccc{Traits} as the one used for
\ccc{CGAL_Triangulation_3<Traits,Tds>} is strongly advised. In this way, the
point type defined by \ccClassName\ is the same as the point type
defined by the geometric traits class.
This class is proposed by \cgal\ as a base vertex class. It is
templated by a geometric traits class \ccc{Traits} that provides the
geometric types. The user who uses the geometric layer (see
Section~\ref{TDS3-sec-design} and
Chapter~\ref{chapter-Triangulation3}) is strongly advised to use the
same geometric traits class \ccc{Traits} as the one used for
\ccc{Triangulation_3<Traits,Tds>}. In this way, the point type
defined by \ccClassName\ is the same as the point type defined by the
geometric traits class. The default geometric traits class proposed by
\cgal\ is presented in Section~\ref{Triangulation3-sec-Traits}.
This base class can be used directly or can serve as a base to derive
other base classes with some additionnal attribute (a color for example)
tuned for a specific application.
This base class can be used directly or can serve as a base to derive
other base classes with some additionnal attributes (a color for
example) tuned for a specific application.
\ccInclude{CGAL/Triangulation_vertex_base_3.h}
\end{ccClassTemplate}
\begin{ccClassTemplate}{CGAL_Triangulation_cell_base_3<Traits>}
\begin{ccClassTemplate}{Triangulation_cell_base_3<Traits>}
\subsection{The Base Class \protect \ccClassTemplateName}
This class is proposed by \cgal\ as a base cell class, and is similar
@ -975,4 +1147,3 @@ Section~\ref{TDS3-sec-class-Base_Vertex}).
\ccInclude{CGAL/Triangulation_cell_base_3.h}
\end{ccClassTemplate}

353
Packages/Triangulation_3/doc_tex/Triangulation_3/Triangulation3.tex
View File

@ -46,7 +46,7 @@ fourth vertex an auxiliary vertex called the \ccc{infinite vertex}. In
that way, each facet is incident to exactly two cells and special cases
at the boundary of the convex hull are simple to deal with.
The class \ccc{CGAL_Triangulation_3<Traits,Tds>} of \cgal\ implements this
The class \ccc{Triangulation_3<Traits,Tds>} of \cgal\ implements this
point of view and therefore considers the triangulation of the set
of points as a set of finite and infinite tetrahedra. Notice that the
infinite vertex has no significant coordinates and that no
@ -58,23 +58,45 @@ access to its four incident vertices and to its four adjacent
cells. Each vertex gives access to one of its incident cells.
The four vertices of a cell are indexed with 0, 1, 2 and 3 in positive
orientation. The neighbors of a cell are also indexed with 0, 1, 2, 3
in such a way that the neighbor indexed by $i$ is opposite to the vertex
with the same index.
orientation, the orientation being defined by the orientation of the
underlying space $\R^3$. The neighbors of a cell are also indexed with
0, 1, 2, 3 in such a way that the neighbor indexed by $i$ is opposite
to the vertex with the same index. See
Figure~\ref{Triangulation3-fig-orient}.
Edges ($1$-faces) and facets ($2$-faces) are not explicitely represented:
a facet is given by a cell and an index (the facet
\begin{ccTexOnly}
\begin{figure}[htbp]
\begin{center}
\includegraphics{orient.eps}
\end{center}
\caption{Orientation of a cell (3-dimensional case)
\label{Triangulation3-fig-orient}}
\end{figure}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<img border=0 src="./orient.gif" align=center alt="Orientation of a cell
(3-dimensional case)">
\end{ccHtmlOnly}
As in the underlying combinatorial triangulation (see
Chapter~\ref{chapter-TDS3}), edges ($1$-faces) and facets ($2$-faces)
are not explicitely
represented: a facet is given by a cell and an index (the facet
\ccc{i} of a cell \ccc{c} is the facet of \ccc{c} that is opposite to
the vertex of index \ccc{i}) and an edge is given by a cell and two
indices (the edge \ccc{(i,j)} of a cell \ccc{c} is the edge
whose endpoints are the vertices of indices \ccc{i} and \ccc{j} of
\ccc{c}).
indices (the edge \ccc{(i,j)} of a cell \ccc{c} is the edge whose
endpoints are the vertices of indices \ccc{i} and \ccc{j} of
\ccc{c}). See Figure~\ref{TDS3-fig-repres}.
\subsection{Degenerate Dimensions}
\label{Triangulation3-sec-degen_dim}
The class \ccc{CGAL_Triangulation_3<Traits,Tds>} can deal with degenerate
dimensions:
The class \ccc{Triangulation_3<Traits,Tds>} can deal with degenerate
dimensions. A triangulation of a set of points in $\R^d$ is a partition of the
whole space $\R^d$ into cells having $d+1$ vertices: some of them
are infinite, they are obtained by linking the additional infinite
vertex to each facet of the convex hull of the points.
\begin{itemize}
\item {} \emph{dimension 2:} when a triangulation only contains
coplanar points (which is the case when there are only three points),
@ -121,21 +143,6 @@ When all the points are collinear, this condition becomes:
{\bf (c-1D)} For any two adjacent edges $(u,v)$ and $(v,w)$, $u$ and
$w$ lie on opposite sides of the common vertex $v$.
\begin{ccTexOnly}
\begin{figure}[htbp]
\begin{center}
\includegraphics{orient.eps}
\end{center}
\caption{Orientation of a cell (3-dimensional case)
\label{Triangulation3-fig-orient}}
\end{figure}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<img border=0 src="./orient.gif" align=center alt="Orientation of a cell
(3-dimensional case)">
\end{ccHtmlOnly}
The \ccc{is_valid()} method provided by \cgal\ checks the local
validity of a given triangulation. This does not always
ensure global validity \cite{mnssssu-cgpvg-96,dlpt-ccpps-98} but it is
@ -144,7 +151,7 @@ sufficient for practical cases.
\section{Software Design}
\label{Triangulation3-sec-design}
The class \ccc{CGAL_Triangulation_3<Traits,Tds>} is designed to be used as
The class \ccc{Triangulation_3<Traits,Tds>} is designed to be used as
a layer upon a 3D-triangulation data structure as presented in
Section~\ref{TDS3-sec-design} of Chapter~\ref{chapter-TDS3}.
It provides high level geometric operations such as location of a point
@ -163,22 +170,114 @@ described in Chapter~\ref{chapter-TDS3}.
Delaunay triangulations as well as hierarchical Delaunay triangulations
\cite{d-iirdt-98} are also implemented in the package:
\ccc{CGAL_Delaunay_triangulation_3<Traits,Tds>} inherits from
\ccc{CGAL_Triangulation_3<Traits,Tds>} and
\ccc{CGAL_Delaunay_hierarchic_triangulation_3<Traits,Tds>} inherits from
\ccc{CGAL_Delaunay_triangulation_3<Traits,Tds>}.
\ccc{Delaunay_triangulation_3<Traits,Tds>} inherits from
\ccc{Triangulation_3<Traits,Tds>} and
\ccc{Delaunay_hierarchic_triangulation_3<Traits,Tds>} inherits from
\ccc{Delaunay_triangulation_3<Traits,Tds>}. \textit{(hierarchical
Delaunay triangulations are not yet implemented).}
\ccc{CGAL_Triangulation_3<Traits,Tds>} derives from
\ccc{CGAL_Triangulation_utils_3<Traits,Tds>}, which defines tools on
\ccc{Triangulation_3<Traits,Tds>} derives from
\ccc{Triangulation_utils_3<Traits,Tds>}, which defines tools on
the indices of vertices in cells (see
Section~\ref{Triangulation3-sec-class-Utils}).
\section{Examples}
\label{Triangulation3-sec-examples}
This example shows the incremental construction of a 3D triangulation,
the location of a point, and how to manipulate elementary operations
on indices in a cell.
\begin{verbatim}
#include <CGAL/basic.h>
#include <cstring>
#include <iostream>
#include <fstream>
#include <strstream.h>
#include <list>
#include <vector>
#include <CGAL/triangulation_assertions.h>
#include <CGAL/Cartesian.h>
#include <CGAL/Triangulation_cell_base_3.h>
#include <CGAL/Triangulation_vertex_base_3.h>
#include <CGAL/Triangulation_data_structure_3.h>
#include <CGAL/Triangulation_geom_traits_3.h>
#include <CGAL/Triangulation_3.h>
typedef CGAL::Cartesian<double> Rep;
typedef CGAL::Triangulation_geom_traits_3<Rep> Gt;
typedef CGAL::Triangulation_vertex_base_3<Gt> Vb;
typedef CGAL::Triangulation_cell_base_3<Gt> Cb;
typedef CGAL::Triangulation_data_structure_3<Vb,Cb> TDS;
typedef CGAL::Triangulation_3<Gt,TDS> Triangulation;
typedef typename Triangulation::Cell_handle Cell_handle;
typedef typename Triangulation::Vertex_handle Vertex_handle;
typedef typename Triangulation::Locate_type Locate_type;
typedef Gt::Point Point;
int main(int argc, char* argv[])
{
Triangulation T;
// insertion from a list :
std::list<Point> L;
L.push_front(Point(0,0,0));
L.push_front(Point(1,0,0));
L.push_front(Point(0,1,0));
int n = T.insert(L.begin(), L.end());
// insertion from a vector :
std::vector<Point> V(3);
V[0] = Point(0,0,1);
V[1] = Point(1,1,1);
V[2] = Point(2,2,2);
n = n + T.insert(V.begin(), V.end());
// 6 points have been inserted :
assert( n == 6 );
// checking validity of T :
assert( T.is_valid(false) );
Locate_type lt;
int li, lj;
Point p(0,0,0);
Cell_handle c = T.locate(p, lt, li, lj);
// p is the vertex of c of index li :
assert( lt == Triangulation::VERTEX );
assert( c->vertex(li)->point() == p );
Vertex_handle v = c->vertex( (li+1)&3 );
// v is another vertex of c
Cell_handle nc = c->neighbor(li);
// nc = neighbor of c opposite to the vertex associated with p
// nc must have vertex v :
int nli;
assert( nc->has_vertex( v, nli ) );
// nli is the index of v in nc
std::ofstream oFileT("output",ios::out);
// writing file output;
oFileT << T;
return 0;
}
\end{verbatim}
\clearpage
\begin{ccClass}{CGAL_Triangulation_utils_3}
\begin{ccClass}{Triangulation_utils_3}
\section{A Class of Tools \protect \ccClassTemplateName}
\label{Triangulation3-sec-class-Utils}
@ -223,7 +322,7 @@ Figure~\ref{Triangulation3-fig-utils}).
\end{ccClass}
\begin{ccClassTemplate}{CGAL_Triangulation_3<Traits,Tds>}
\begin{ccClassTemplate}{Triangulation_3<Traits,Tds>}
\section{The Triangulation Class \protect \ccClassTemplateName}
\ccDefinition
@ -237,7 +336,7 @@ and~\ref{Triangulation3-sec-class-Traits}.
\ccInclude{CGAL/Triangulation_3.h}
\ccInheritsFrom
\ccc{CGAL_Triangulation_utils_3}
\ccc{Triangulation_utils_3}
(see Section~\ref{Triangulation3-sec-class-Utils})
\ccTypes
@ -278,7 +377,8 @@ A handle is a type which supports the two dereference operators
bidirectionnel and non mutable. Circulators and iterators are
assignable to the corresponding handle types. Whenever a handle appears
in the parameter list of a function, an appropriate iterator or
circulator can be used as well \textit{not yet implemented}. The edges and facets of the
circulator can be used as well \textit{(not yet implemented)}. The
edges and facets of the
triangulation can also be visited through iterators and circulators
which are bidirectionnel and non mutable.
@ -314,7 +414,7 @@ OUTSIDE_AFFINE_HULL};}
{Introduces a triangulation \ccVar\ having only one vertex which is the
infinite vertex.}
\ccConstructor{\ccClassName(const CGAL_Triangulation_3<Traits,Tds> & tr);}
\ccConstructor{\ccClassName(const Triangulation_3<Traits,Tds> & tr);}
{Copy constructor. All the vertices and faces are duplicated.
\ccVar\ and \ccc{tr}\ refer to different triangulations. After the copy,
if \ccc{tr} is modified, \ccVar\ is not modified.}
@ -325,19 +425,22 @@ deleted.}
\ccHeading{Assignment}
\ccMethod{CGAL_Triangulation_3<Traits,Tds>
operator=(const CGAL_Triangulation_3<Traits,Tds> & tr);}
\ccMethod{Triangulation_3<Traits,Tds>
operator=(const Triangulation_3<Traits,Tds> & tr);}
{The triangulation is duplicated, and modifying one after the
copy does not modify the other.}
\ccMethod{copy_triangulation
(const CGAL_Triangulation_3<Traits,Tds> & tr);}
\ccMethod{void copy_triangulation
(const Triangulation_3<Traits,Tds> & tr);}
{The triangulation is duplicated.}
\ccMethod{void swap(CGAL_Triangulation_3<Traits,Tds> & tr);}
The previous first two methods are equivalent.
\ccMethod{void swap(Triangulation_3<Traits,Tds> & tr);}
{The triangulations \ccc{tr} and \ccVar\ are swapped.
\ccc{\ccVar.swap(tr)} should be preferred to \ccVar\ = \ccc{tr} or to
\ccc{t(tr)} if \ccc{tr} is deleted after that.}
\ccc{t(tr)} if \ccc{tr} is deleted after that. Indeed, there is no
copy of cells and vertices, thus this method runs in constant time.}
\ccAccessFunctions
\ccMethod{const Traits & geom_traits() const;}
@ -553,69 +656,69 @@ triangulation, \ccc{lt} is set to \ccc{OUTSIDE_AFFINE_HULL} and
{Same as the previous method, but \ccc{start} is used as a starting
place for the location.}
\ccMethod{CGAL_Bounded_side
\ccMethod{Bounded_side
side_of_cell(const Point & p,
Cell_handle c,
Locate_type & lt, int & li, int & lj) const;}
{Returns on which side of the oriented boundary of \ccc{c} lies
the point \ccc{p}. More precisely, it returns:\\
- \ccc{CGAL_ON_BOUNDED_SIDE} if \ccc{p} is inside the cell (for an infinite
- \ccc{ON_BOUNDED_SIDE} if \ccc{p} is inside the cell (for an infinite
cell this means that \ccc{p} lies strictly in the half space limited by
its finite facet) \\
- \ccc{CGAL_ON_BOUNDARY} if p on the boundary of the cell (for an infinite
- \ccc{ON_BOUNDARY} if p on the boundary of the cell (for an infinite
cell this means that \ccc{p} lies on the *finite* facet). Then
\ccc{lt} together with \ccc{li} and \ccc{lj} give the precise location
on the boundary (see the descriptions of \ccc{locate} methods)\\
- \ccc{CGAL_ON_UNBOUNDED_SIDE} if \ccc{p} lies outside the cell (for an
- \ccc{ON_UNBOUNDED_SIDE} if \ccc{p} lies outside the cell (for an
infinite cell this means that \ccc{p} is not in the preceding two
cases).
\ccPrecond{\ccc{\ccVar.dimension()} $=3$}}
\ccMethod{CGAL_Bounded_side
\ccMethod{Bounded_side
side_of_facet(const Point & p,
const Facet & f,
Locate_type & lt, int & li, int & lj) const;}
{Returns on which side of the oriented boundary of \ccc{f} lies
the point \ccc{p}:\\
- \ccc{CGAL_ON_BOUNDED_SIDE} if \ccc{p} is inside the facet (for an
- \ccc{ON_BOUNDED_SIDE} if \ccc{p} is inside the facet (for an
infinite facet this means that \ccc{p} lies strictly in the half plane
limited by its finite edge) \\
- \ccc{CGAL_ON_BOUNDARY} if \ccc{p} is on the boundary of the facet
- \ccc{ON_BOUNDARY} if \ccc{p} is on the boundary of the facet
(for an infinite facet this means that \ccc{p} lies on the finite
edge). \ccc{lt}, \ccc{li} and \ccc{lj} give the precise location of
\ccc{p} on the boundary of the facet. \ccc{li} and \ccc{lj} refer to
indices in the cell \ccc{c} storing \ccc{f}\\
- \ccc{CGAL_ON_UNBOUNDED_SIDE} if \ccc{p} lies outside the facet (for
- \ccc{ON_UNBOUNDED_SIDE} if \ccc{p} lies outside the facet (for
an infinite facet this means that \ccc{p} is not in the preceding two
cases)
\ccPrecond{\ccc{\ccVar.dimension()} $=2$ and \ccc{p} lies in the
plane containing the triangulation. \ccc{f.first} $=3$ (in dimension~2
there is only one facet per cell).}}
\ccGlue
\ccMethod{CGAL_Bounded_side
\ccMethod{Bounded_side
side_of_facet(const Point & p,
Cell_handle c,
Locate_type & lt, int & li, int & lj) const;}
{Same as the previous method for the facet \ccc{(c,3)}.}
\ccMethod{CGAL_Bounded_side
\ccMethod{Bounded_side
side_of_edge(const Point & p,
const Edge & e,
Locate_type & lt, int & li) const;}
{Returns on which side of the oriented boundary of \ccc{e} lies
the point \ccc{p}:\\
- \ccc{CGAL_ON_BOUNDED_SIDE} if \ccc{p} is inside the edge (for an
- \ccc{ON_BOUNDED_SIDE} if \ccc{p} is inside the edge (for an
infinite edge this means that \ccc{p} lies in the half line defined by
the vertex)\\
- \ccc{CGAL_ON_BOUNDARY} if \ccc{p} equals one of the vertices,
- \ccc{ON_BOUNDARY} if \ccc{p} equals one of the vertices,
\ccc{li} give the index of the vertex in the cell storing \ccc{e}\\
- \ccc{CGAL_ON_UNBOUNDED_SIDE} if \ccc{p} lies outside the edge (for
- \ccc{ON_UNBOUNDED_SIDE} if \ccc{p} lies outside the edge (for
an infinite edge this means that \ccc{p} lies on the other half line)\\
\ccPrecond{\ccc{\ccVar.dimension()} $=1$ and \ccc{p} is collinear
with the points of the triangulation. \ccc{e.first} $=0$ and
\ccc{e.second} $=1$ (in dimension~2 there is only one edge per cell).}}
\ccGlue
\ccMethod{CGAL_Bounded_side
\ccMethod{Bounded_side
side_of_edge(const Point & p,
Cell_handle c,
Locate_type & lt, int & li) const;}
@ -691,8 +794,8 @@ inside cell \ccc{c}.}}
{Inserts point \ccc{p} in facet \ccc{f}. In dimension~3, the 2
neighboring cells are split into 3 tetrahedra; in dimension~2, the facet
is split into 3 triangles.
\ccPrecond{\ccc{\ccVar.dimension()} $\geq 2$ and \ccc{p} lies inside
face \ccc{f}.}}
\ccPrecond{\ccc{\ccVar.dimension()} $\geq 2$ and \ccc{p} lies strictly
inside face \ccc{f}.}}
\ccGlue
\ccMethod{Vertex_handle insert_in_facet(const Point & p,
Cell_handle c, int i);}
@ -717,7 +820,7 @@ dimension~1.}}
\ccMethod{Vertex_handle insert_outside_convex_hull(const Point & p,
Cell_handle c,
int li, int lj=0);}
{\ccc{c}, together with \ccc{li} and possibly \ccc{lj}, give a
{The cell \ccc{c}, together with \ccc{li} and possibly \ccc{lj}, give a
separator (facet, edge or vertex, depending on the dimension) for
\ccc{p} from the triangulation (see the description of method
\ccc{locate()} for more details on the way the separator is represented).\\
@ -792,7 +895,7 @@ and \ccc{Vertex}. They are all invalidated by any change in the
triangulation.
\ccMethod{Vertex_iterator finite_vertices_begin() const;}
{Starts at an arbitrary finite vertex. Then \ccc{$++$} and \ccc{$--$} will
{Starts at an arbitrary finite vertex. Then \ccc{++} and \ccc{--} will
iterate on finite vertices. Returns \ccc{vertices_end()} when
\ccc{\ccVar.number_of_vertices()} $<1$.}
\ccGlue
@ -805,7 +908,7 @@ ones). Returns \ccc{vertices_end()} when
{Past-the-end iterator}
\ccMethod{Edge_iterator finite_edges_begin() const;}
{Starts at an arbitrary finite edge. Then \ccc{$++$} and \ccc{$--$} will
{Starts at an arbitrary finite edge. Then \ccc{++} and \ccc{--} will
iterate on finite edges. Returns \ccc{edges_end()} when
\ccc{\ccVar.dimension()} $<1$.}
\ccGlue
@ -817,7 +920,7 @@ ones). Returns \ccc{edges_end()} when \ccc{\ccVar.dimension()} $<1$.}
{Past-the-end iterator}
\ccMethod{Facet_iterator finite_facets_begin() const;}
{Starts at an arbitrary finite facet. Then \ccc{$++$} and \ccc{$--$} will
{Starts at an arbitrary finite facet. Then \ccc{++} and \ccc{--} will
iterate on finite facets. Returns \ccc{facets_end()} when
\ccc{\ccVar.dimension()} $<2$.}
\ccGlue
@ -830,7 +933,7 @@ ones). Returns \ccc{facets_end()} when
{Past-the-end iterator}
\ccMethod{Cell_iterator finite_cells_begin() const;}
{Starts at an arbitrary finite cell. Then \ccc{$++$} and \ccc{$--$} will
{Starts at an arbitrary finite cell. Then \ccc{++} and \ccc{--} will
iterate on finite cells. Returns \ccc{cells_end()} when
\ccc{\ccVar.dimension()} $<3$.}
\ccGlue
@ -907,7 +1010,7 @@ debugging.
\ccMethod{bool
is_valid(bool verbose = false) const;}
{Checks the combinatorial validity of the triangulation and also the
{Checks the combinatorial validity of the triangulation. Checks also the
validity of its geometric embedding (see
Section~\ref{Triangulation3-sec-Valid}). When \ccc{verbose} is set to true,
messages describing the first invalidity encountered are printed.\\
@ -936,7 +1039,7 @@ corresponding input operators of the vertex and the cell
classes. Assigns the resulting triangulation to \ccc{t}.}
\ccFunction{ostream& operator<<
(ostream& os, const CGAL_Triangulation_3<Traits, Tds> &t);}
(ostream& os, const Triangulation_3<Traits, Tds> &t);}
{Writes the triangulation \ccc{t} into \ccc{os}.}
The information in the \ccc{iostream} is: the dimension, the number of
@ -949,7 +1052,7 @@ of maximal dimension.)
\end{ccClassTemplate}
\begin{ccClassTemplate}{CGAL_Triangulation_vertex_3<Traits,Tds>}
\begin{ccClassTemplate}{Triangulation_vertex_3<Traits,Tds>}
\section{The Vertex Class of a Triangulation \protect \ccClassTemplateName}
\label{Triangulation3-sec-class-Vertex}
@ -973,9 +1076,9 @@ defined in the class \ccc{Triangulation_3<Traits,Tds>}:
\ccGlue
\ccTypedef{typedef typename Tds::Cell Ctds;}{cell of the triangulation data structure}
\ccGlue
\ccTypedef{typedef CGAL_Triangulation_vertex_handle_3<Traits,Tds> Vertex_handle;}{handle to a vertex}
\ccTypedef{typedef Triangulation_vertex_handle_3<Traits,Tds> Vertex_handle;}{handle to a vertex}
\ccGlue
\ccTypedef{typedef CGAL_Triangulation_cell_handle_3<Traits,Tds> Cell_handle;}{handle to a cell}
\ccTypedef{typedef Triangulation_cell_handle_3<Traits,Tds> Cell_handle;}{handle to a cell}
\begin{ccAdvanced}
\ccCreation
@ -1034,7 +1137,7 @@ of maximal dimension (Section~\ref{Triangulation3-sec-degen_dim}).}
\end{ccClassTemplate}
\begin{ccClassTemplate}{CGAL_Triangulation_cell_3<Traits,Tds>}
\begin{ccClassTemplate}{Triangulation_cell_3<Traits,Tds>}
\section{The Cell Class of a Triangulation \protect \ccClassTemplateName}
\label{Triangulation3-sec-class-Cell}
@ -1056,7 +1159,7 @@ dimension: (Section~\ref{Triangulation3-sec-degen_dim}).
\ccTypes
The class \ccClassTemplateName\ defines the same types as the
\ccc{CGAL_Triangulation_vertex_3<Traits,Tds>} class.
\ccc{Triangulation_vertex_3<Traits,Tds>} class.
\begin{ccAdvanced}
\ccCreation
@ -1168,7 +1271,7 @@ computes the index \ccc{i} of the neighbor.}
\label{Triangulation3-sec-Traits}
The first template parameter of the triangulation class
\ccc{CGAL_Triangulation_3<Traits,Tds>} of \cgal\ is the geometric traits class.
\ccc{Triangulation_3<Traits,Tds>} of \cgal\ is the geometric traits class.
The first subsection of this section describes the requirements
that the geometric traits class must fulfill. The second subsection
@ -1180,12 +1283,12 @@ presents a predefined geometric traits class available in \cgal.
\protect \ccClassName}
\label{Triangulation3-sec-concept-Traits}
\subsubsection{To be used by \protect \ccc{CGAL_Triangulation_3<Traits, Tds>}}
\subsubsection{To be used by \protect \ccc{Triangulation_3<Traits, Tds>}}
\ccCreationVariable{traits}
The geometric traits class \ccClassName\ of the triangulation
class \ccc{CGAL_Triangulation_3<Traits, Tds>} must define the geometric
class \ccc{Triangulation_3<Traits, Tds>} must define the geometric
objects (points, segments, triangles and tetrahedra) forming the
triangulation together with a few geometric predicates on these objects:
equality, coordinates comparison, orientation in the space, orientation
@ -1225,36 +1328,36 @@ can be provided.
\ccMethod{bool equal(const Point & p, const Point & q) const;}
{Equality test.}
\ccMethod{CGAL_Comparison_result compare_x(const Point & p, const Point
\ccMethod{Comparison_result compare_x(const Point & p, const Point
& q) const;}
{Comparison of \ccc{x}-coordinates. Returns \ccc{CGAL_LARGER}
(resp. \ccc{CGAL_EQUAL}, \ccc{CGAL_SMALLER}) when the \ccc{x}
{Comparison of \ccc{x}-coordinates. Returns \ccc{LARGER}
(resp. \ccc{EQUAL}, \ccc{SMALLER}) when the \ccc{x}
coordinate of \ccc{p} is larger than (resp. equal to, smaller than)
the \ccc{x} coordinate of \ccc{q}.}
\ccGlue
\ccMethod{CGAL_Comparison_result compare_y(const Point & p, const Point
\ccMethod{Comparison_result compare_y(const Point & p, const Point
& q) const;}
{Comparison of \ccc{y}-coordinates.}
\ccGlue
\ccMethod{CGAL_Comparison_result compare_z(const Point & p, const Point
\ccMethod{Comparison_result compare_z(const Point & p, const Point
& q) const;}
{Comparison of \ccc{z}-coordinates.}
\ccMethod{CGAL_Orientation orientation(const Point& p0,
\ccMethod{Orientation orientation(const Point& p0,
const Point& p1,
const Point& p2,
const Point& p3) const;}
{Orientation test in three dimensions.}
\ccMethod{CGAL_Orientation orientation_in_plane
\ccMethod{Orientation orientation_in_plane
(const Point & q,
const Point & r,
const Point & s,
const Point & test) const;}
{When \ccc{test, q, r, s} are coplanar, tests whether \ccc{test} is on the
same side of \ccc{(q, r)} as \ccc{s}. Returns \ccc{CGAL_COLLINEAR} if
\ccc{test, q, r} are collinear, \ccc{CGAL_POSITIVE} if \ccc{(q, r, test)}
and \ccc{(q, r, s)} have the same orientation, \ccc{CGAL_NEGATIVE} if
same side of \ccc{(q, r)} as \ccc{s}. Returns \ccc{COLLINEAR} if
\ccc{test, q, r} are collinear, \ccc{POSITIVE} if \ccc{(q, r, test)}
and \ccc{(q, r, s)} have the same orientation, \ccc{NEGATIVE} if
\ccc{(q, r, test)} and \ccc{(q, r, s)} have opposite orientations.
\ccPrecond{\ccc{test,q,r,s} are coplanar and \ccc{q,r,s} are not collinear.}}
@ -1264,7 +1367,7 @@ and \ccc{(q, r, s)} have the same orientation, \ccc{CGAL_NEGATIVE} if
{Collinearity test.}
\subsubsection{To be used by \protect
\ccc{CGAL_Delaunay_triangulation_3<Traits, Tds>}}
\ccc{Delaunay_triangulation_3<Traits, Tds>}}
In addition to the requirements described before, the geometric traits
class of a
@ -1272,7 +1375,7 @@ Delaunay triangulation must fulfill the following requirements:
\ccHeading{Predicates}
\ccMethod{CGAL_Oriented_side
\ccMethod{Oriented_side
side_of_oriented_sphere(const Point & p,
const Point & q,
const Point & r,
@ -1281,23 +1384,23 @@ Delaunay triangulation must fulfill the following requirements:
{Computes the side of point \ccc{test} with respect to the oriented
sphere circumscribing \ccc{p,q,r,s}.}
\ccMethod{CGAL_Oriented_side
\ccMethod{Oriented_side
side_of_oriented_circle(const Point & p,
const Point & q,
const Point & r,
const Point & test) const;}
{computes the side of \ccc{test} with respect to the oriented circle
{Computes the side of \ccc{test} with respect to the oriented circle
circumscribing \ccc{p,q,r}.
\ccPrecond{\ccc{p,q,r,test} are coplanar.}}
\subsubsection{To be used by \protect
\ccc{CGAL_Delaunay_hierarchic_triangulation_3<Traits, Tds>}}
\ccc{Delaunay_hierarchic_triangulation_3<Traits, Tds>}}
\textit{Not yet implemented}
\end{ccClass}
\begin{ccClassTemplate}{CGAL_Triangulation_geom_traits_3<R>}
\begin{ccClassTemplate}{Triangulation_geom_traits_3<R>}
\subsection{The Class \protect \ccClassTemplateName}
\label{Triangulation3-sec-class-Traits}
@ -1314,13 +1417,13 @@ Section~\ref{Triangulation3-sec-concept-Traits}.
\ccTypes
\ccTypedef{typedef CGAL_Point_3<R> Point;}{}
\ccTypedef{typedef Point_3<R> Point;}{}
\ccGlue
\ccTypedef{typedef CGAL_Segment_3<R> Segment;}{}
\ccTypedef{typedef Segment_3<R> Segment;}{}
\ccGlue
\ccTypedef{typedef CGAL_Triangle_3<R> Triangle;}{}
\ccTypedef{typedef Triangle_3<R> Triangle;}{}
\ccGlue
\ccTypedef{typedef CGAL_Tetrahedron_3<R> Tetrahedron;}{}
\ccTypedef{typedef Tetrahedron_3<R> Tetrahedron;}{}
\end{ccClassTemplate}
@ -1328,32 +1431,32 @@ Section~\ref{Triangulation3-sec-concept-Traits}.
\label{Triangulation3-sec-tds}
The second template parameter of the basic triangulation class
\ccc{CGAL_Triangulation_3<Traits,Tds>} is a triangulation data structure
\ccc{Triangulation_3<Traits,Tds>} is a triangulation data structure
class. This class can be seen as a container for the cells and
vertices maintaining incidence and adjacency relations.
The concept for the triangulation data structure is described in
Section~\ref{TDS3-sec-concept} of Chapter~\ref{chapter-TDS3}. Its optional
arguments related to geometry are compulsory for this use as a
template parameter of \ccc{CGAL_Triangulation_3<Traits,Tds>}.
template parameter of \ccc{Triangulation_3<Traits,Tds>}.
A model of this triangulation data structure is
\ccc{CGAL_Triangulation_data_structuree_3} presented in
\ccc{Triangulation_data_structuree_3} presented in
Section~\ref{TDS3-sec-class}.
\begin{ccClassTemplate}{CGAL_Delaunay_triangulation_3<Traits,Tds>}
\begin{ccClassTemplate}{Delaunay_triangulation_3<Traits,Tds>}
\section{The Delaunay Triangulation Class \protect
\ccClassTemplateName}
\ccDefinition
\ccInclude{CGAL/Delaunay_triangulation_3.h}
\ccInheritsFrom{\ccc{CGAL_Triangulation_3<Traits,Tds>}}
\ccInheritsFrom{\ccc{Triangulation_3<Traits,Tds>}}
\ccCreationVariable{dt}
\ccTypes
Inherits the types of \ccc{CGAL_Triangulation_3<Traits,Tds>}.
Inherits the types of \ccc{Triangulation_3<Traits,Tds>}.
\ccCreation
@ -1364,13 +1467,17 @@ Inherits the types of \ccc{CGAL_Triangulation_3<Traits,Tds>}.
{Creates an empty Delaunay triangulation with traits class
\ccc{traits}.}
\ccConstructor{\ccClassName(const CGAL_Delaunay_triangulation_3<Traits,Tds> & dt1)}
\ccConstructor{\ccClassName(const Delaunay_triangulation_3<Traits,Tds> & dt1)}
{Copy constructor.}
\ccModifiers
\ccHeading{Insertion}
The following methods, which already exist in triangulations, are
overloaded to ensure the empty sphere property of Delaunay
triangulations.
\ccMethod{Vertex_handle insert(const Point & p );}
{Inserts point \ccc{p} in the triangulation and returns the corresponding
vertex. Similar to the insertion in a triangulation, but insures in
@ -1408,45 +1515,45 @@ number of inserted points.
\ccHeading{Queries}
\ccMethod{CGAL_Bounded_side
\ccMethod{Bounded_side
side_of_sphere(Cell_handle c, const Point & p) const;}
{Returns on which side of the circumscribed sphere of \ccc{c} lies
the point \ccc{p}. More precisely, it returns:\\
- \ccc{CGAL_ON_BOUNDED_SIDE} if \ccc{p} is inside the sphere (for an infinite
- \ccc{ON_BOUNDED_SIDE} if \ccc{p} is inside the sphere (for an infinite
cell this means that \ccc{p} lies strictly either in the half space
limited by its finite facet, or in the interior of the disk
circumscribing the *finite* facet) \\
- \ccc{CGAL_ON_BOUNDARY} if p on the boundary of the sphere (for an infinite
- \ccc{ON_BOUNDARY} if p on the boundary of the sphere (for an infinite
cell this means that \ccc{p} lies on the circle circumscribing
the *finite* facet)\\
- \ccc{CGAL_ON_UNBOUNDED_SIDE} if \ccc{p} lies outside the sphere (for an
- \ccc{ON_UNBOUNDED_SIDE} if \ccc{p} lies outside the sphere (for an
infinite cell this means that \ccc{p} is not in the preceding two
cases).
\ccPrecond{\ccc{\ccVar.dimension()} $=3$.}}
\ccMethod{CGAL_Bounded_side
\ccMethod{Bounded_side
side_of_circle(const Facet & f, const Point & p) const;}
{Returns on which side of the circumscribed circle of \ccc{f} lies
the point \ccc{p}. More precisely, it returns:\\
- in dimension~3:\\
-- for a finite facet, \ccc{CGAL_ON_BOUNDARY} if \ccc{p} lies
on the circle, \ccc{CGAL_ON_UNBOUNDED_SIDE} when it lies in the exterior of
the disk, \ccc{CGAL_ON_BOUNDED_SIDE} when it lies in its interior.\\
-- for a finite facet, \ccc{ON_BOUNDARY} if \ccc{p} lies
on the circle, \ccc{ON_UNBOUNDED_SIDE} when it lies in the exterior of
the disk, \ccc{ON_BOUNDED_SIDE} when it lies in its interior.\\
-- for an infinite facet, it considers the plane defined by the finite
facet of the same cell, and does the same as in dimension~2 in this
plane.\\
- in dimension~2:\\
-- for a finite facet, \ccc{CGAL_ON_BOUNDARY} if \ccc{p} lies
on the circle, \ccc{CGAL_ON_UNBOUNDED_SIDE} when it lies in the exterior of
the disk, \ccc{CGAL_ON_BOUNDED_SIDE} when it lies in its interior.\\
-- for an infinite facet, \ccc{CGAL_ON_BOUNDARY} if the
-- for a finite facet, \ccc{ON_BOUNDARY} if \ccc{p} lies
on the circle, \ccc{ON_UNBOUNDED_SIDE} when it lies in the exterior of
the disk, \ccc{ON_BOUNDED_SIDE} when it lies in its interior.\\
-- for an infinite facet, \ccc{ON_BOUNDARY} if the
point lies on the finite edge of \ccc{f} (endpoints included),
\ccc{CGAL_ON_BOUNDED_SIDE} for a point in the open half-plane defined
by \ccc{f}, \ccc{CGAL_ON_UNBOUNDED_SIDE} elsewhere.
\ccc{ON_BOUNDED_SIDE} for a point in the open half-plane defined
by \ccc{f}, \ccc{ON_UNBOUNDED_SIDE} elsewhere.
\ccPrecond{\ccc{\ccVar.dimension()} $\geq 2$.}}
\ccMethod{CGAL_Bounded_side
\ccMethod{Bounded_side
side_of_circle(Cell_handle c, int i, const Point & p);}
{Same as the previous method.}
{Same as the previous method for facet \ccc{i} of cell \ccc{c}.}
\begin{ccAdvanced}
\ccHeading{Checking}
@ -1464,7 +1571,7 @@ This method is mainly a debugging help for the users of advanced features.
\end{ccClassTemplate}
\begin{ccClassTemplate}{CGAL_Delaunay_hierarchic_triangulation_3<Traits,Tds>}
\begin{ccClassTemplate}{Delaunay_hierarchic_triangulation_3<Traits,Tds>}
\section{The Hierarchic Delaunay Triangulation Class \protect
\ccClassTemplateName}
@ -1491,13 +1598,13 @@ To work around this problem, two functions have been defined to transform
\ccc{handles} into usual $C^{++}$ pointers for debugging purposes.
\ccFunction{template <class Traits, class Tds>
CGAL_Triangulation_vertex_3<Traits,Tds> *
CGAL_debug(const CGAL_Triangulation_vertex_handle_3<Traits,Tds> v);}
Triangulation_vertex_3<Traits,Tds> *
debug(const Triangulation_vertex_handle_3<Traits,Tds> v);}
{}
\ccFunction{template <class Traits, class Tds>
CGAL_Triangulation_cell_3<Traits,Tds> *
CGAL_debug(const CGAL_Triangulation_cell_handle_3<Traits,Tds>
Triangulation_cell_3<Traits,Tds> *
debug(const Triangulation_cell_handle_3<Traits,Tds>
c);}
{}

301
Packages/Triangulation_3/doc_tex/basic/Triangulation_3/TDS3.tex
View File

@ -25,12 +25,28 @@
\section{Introduction}
\label{TDS3-sec-intro}
\subsection{Definition}
A three-dimensional triangulation is a three-dimensional simplicial
complex, pure connected and without singularities \cite{by-ag-98}. It
is a set of cells ($3$-faces, tetrahedra) such that two cells
either do not intersect or share a common facet ($2$-face), edge
($1$-face) or vertex ($0$-face).
A geometric triangulation has two aspects: the combinatorial structure which
gives the incidence and adjacency relations between faces, and the
geometric information related to the position of vertices.
\cgal\ proposes 3D geometric triangulations in which these
two aspects are clearly separated.
This chapter deals with 3D- triangulation data structures, meant to
maintain the combinatorial information for 3D-geometric
triangulations. The reader interested in geometric triangulations of
$\R^3$ is advised to read Chapter~\ref{chapter-Triangulation3}.
\subsection{Representation}
\label{TDS3-sec-def}
A 3D- triangulation data structure is meant to maintain the combinatorial
information (incidence and adjacency relations) for 3D-geometric
triangulations. In \cgal, a triangulation data structure is a
In \cgal, a triangulation data structure is a
container of cells ($3$-faces) and vertices ($0$-faces). Each cell gives
access to its four incident vertices and to its four adjacent
cells. Each vertex gives access to one of its incident cells.
@ -38,7 +54,21 @@ cells. Each vertex gives access to one of its incident cells.
The four vertices of a cell are indexed with 0, 1, 2 and 3. The
neighbors of a cell are also indexed with 0, 1, 2, 3
in such a way that the neighbor indexed by $i$ is opposite to the vertex
with the same index.
with the same index (see Figure~\ref{TDS3-fig-repres}).
\begin{ccTexOnly}
\begin{figure}
\begin{center}
\includegraphics{repres.eps}
\end{center}
\caption{Representation \label{TDS3-fig-repres}}
\end{figure}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<img border=0 src="./repres.gif" align=center
alt="Representation">
\end{ccHtmlOnly}
Edges ($1$-faces) and facets ($2$-faces) are not explicitely
represented: a facet is given by a cell and an index (the facet
@ -57,7 +87,7 @@ is not of full dimension.
As described in Chapter~\ref{chapter-Triangulation3}, a geometric
triangulation of a set of points in $\R^d$ is a partition of the
whole space $\R^d$ into cells having $d+1$ vertices: some of them
are infinite, they are obtained by liking an additional vertex at
are infinite, they are obtained by linking an additional vertex at
infinity to each facet of the convex hull of the points.
The underlying combinatorial triangulation of such a triangulation
without boundary of $\R^d$ can be seen as a triangulation of the
@ -76,15 +106,51 @@ the boundary of a 4-dimensional simplex, which has 5 vertices. A
geometric embedding consists in choosing one of these vertices to be
infinite, thus four of the five 3-cells becomes infinite: the geometric
triangulation has one finite tetrahedron remaining, each of its facets
being incident to the infinite cell.
being incident to the infinite cell. See Figure~\ref{TDS3-fig-topo-simplex4}.
\begin{ccTexOnly}
\begin{figure}
\begin{center}
\includegraphics{topo-simplex4.eps}
\end{center}
\caption{4D simplex and a 3D geometric embedding \label{TDS3-fig-topo-simplex4}}
\end{figure}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<img border=0 src="./topo-simplex4.gif" align=center
alt="4D simplex and a 3D geometric embedding">
\end{ccHtmlOnly}
\item \emph{dimension 2.} We have 4 vertices forming one 3-dimensional
simplex, i.e. the boundary of a tetrahedron. The geometric embedding in
the plane consists in choosing one of these vertices to be infinite,
then the geometric triangulation has one finite triangle whose edges are
adjacent to the infinite triangles. See Figure~\ref{TDS3-fig-topo-simplex}.
adjacent to the infinite triangles. See Figure~\ref{TDS3-fig-topo-simplex3}.
\begin{ccTexOnly}
\begin{figure}
\begin{center}
\includegraphics{topo-simplex3.eps}
\end{center}
\caption{3D simplex and a 2D geometric embedding \label{TDS3-fig-topo-simplex3}}
\end{figure}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<img border=0 src="./topo-simplex3.gif" align=center
alt="3D simplex and a 2D geometric embedding">
\end{ccHtmlOnly}
\item \emph{dimension 1.} A 2-dimensional simplex (a triangle) has 3
vertices. The geometric embedding is and edge whose vertices are linked
to an infinite point.
to an infinite point. See Figure~\ref{TDS3-fig-topo-simplex2}.
\begin{ccTexOnly}
\begin{figure}
\begin{center}
\includegraphics{topo-simplex2.eps}
\end{center}
\caption{2D simplex and a 1D geometric embedding \label{TDS3-fig-topo-simplex2}}
\end{figure}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<img border=0 src="./topo-simplex2.gif" align=center
alt="2D simplex and a 1D geometric embedding">
\end{ccHtmlOnly}
\end{itemize}
The last two cases are defined uniquely:
@ -105,20 +171,6 @@ triangulation data structure. The infinite vertex of the geometric
embedding is a vertex without any particularity in the
combinatorial triangulation.
\begin{ccTexOnly}
\begin{figure}
\begin{center}
\includegraphics{topo-simplex.eps}
\end{center}
\caption{3D simplex and its 2D geometric embedding \label{TDS3-fig-topo-simplex}}
\end{figure}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<img border=0 src="./topo-simplex.gif" align=center
alt="3D simplex and its 2D geometric embedding">
\end{ccHtmlOnly}
The implicit representation of facets (resp. edges) still holds
for degenerate ($< 3$) dimensions : in dimension~2, each cell has only one
facet of index 3, and 3 edges $(0,1)$, $(1,2)$ and $(2,0)$; in
@ -189,29 +241,144 @@ software design in this case.
<img border=0 src="./design.gif" align=center alt="Layers in the software design">
\end{ccHtmlOnly}
In the bottom
layer, the base classes store elementary geometric information as
well as any other information for the given application. The middle
layer class stores the triangulation data structure, which is purely
combinatorial. It provides operations such as insertion of a new
vertex in a given cell, and is responsible for the combinatorial
integrity of the triangulation. The upper layer is the geometric
triangulation class, providing operations such as location of a point
in the triangulation, insertion of a point, and is responsible for
the geometric validity.
In the bottom layer, the \cgal\ base classes store elementary
geometric information. These classes are parameterized by a geometric
traits class providing all the geometric types. A vertex has a pointer
to a cell, and a cell has four pointers to vertices. These pointers
are of type \ccc{void*}.
The triangulation data structure class is parameterized by the base
vertex and the base cell classes. It is up to the user to derive its
own base classes from the \cgal\ base classes, and to use the
triangulation data structure class proposed by \cgal.
The middle layer class stores the triangulation data structure, which
is purely combinatorial. A vertex of the triangulation data structure
has a pointer to a cell of the triangulation data structure, and a
cell has four pointers to vertices. These pointers are usual
\ccc{C++} pointers. The triangulation data structure provides
operations such as insertion of a new vertex in a given cell, on a
$1$ or $2$-face, or, if the dimension of the triangulation is smaller
than 3, outside the ``affine'' hull of the vertices, thus increasing
the dimension of the triangulation by one. The triangulation data
structure is responsible for the combinatorial integrity of the
triangulation.
It is up to the user to derive its own base classes from the \cgal\
base classes to add any other information the may need for his given
application, or to write his own base classes from scratch. In this
case, his base classes must be models for the concepts described in
Section~\ref{TDS3-sec-concept-Base}. As the triangulation data
structure class is parameterized by the base vertex and the base cell
classes. it will consider the user's bases classes instead of the
predefined base classes.
The upper layer, described in Chapter~\ref{chapter-Triangulation3}, is
the geometric triangulation class, providing operations such as
location of a point in the triangulation, insertion of a point, and is
responsible for the geometric validity. A vertex of the triangulation
has a pointer to a cell and a cell has four pointers to
vertices. These pointers are \cgal\ \emph{handles}. The triangulation
data structure class is one of the template parameters of the
geometric triangulation class. The user may choose to replace the
\cgal\ triangulation data structure class by its own triangulation
data structure, in this case, his class has to be a model of the
concept described in Section~\ref{TDS3-sec-concept}.
\section{Examples}
\label{TDS3-sec-examples}
The following example shows how to construct a 3D triangulation data
structure by inserting vertices.
\begin{verbatim}
#include <CGAL/basic.h>
#include <cstring>
#include <iostream>
#include <fstream>
#include <strstream.h>
#include <vector>
#include <CGAL/Cartesian.h>
#include <CGAL/Point_3.h>
#include <CGAL/Triangulation_geom_traits_3.h>
#include <CGAL/Triangulation_cell_base_3.h>
#include <CGAL/Triangulation_vertex_base_3.h>
#include <CGAL/Triangulation_data_structure_3.h>
// the definition of the geometric traits class is necessary to
// instanciate base vertices and cells but will in fact never be used
// in the program
typedef CGAL::Cartesian<double> Rep;
typedef CGAL::Triangulation_geom_traits_3<Rep> Gt;
typedef CGAL::Triangulation_vertex_base_3<Gt> Vb;
typedef CGAL::Triangulation_cell_base_3<Gt> Cb;
typedef CGAL::Triangulation_data_structure_3<Vb,Cb> Tds;
typedef typename Tds::Cell TDSCell;
typedef typename Tds::Vertex TDSVertex;
int main()
{
Tds T;
assert( T.number_of_vertices() == 0 );
assert( T.dimension() == -2 );
assert( T.is_valid() );
std::vector<TDSVertex> V(5);
std::vector<TDSVertex*> PV(7);
PV[0] = T.insert_outside_affine_hull(V[0]);
assert( T.number_of_vertices() == 1 );
assert( T.dimension() == -1 );
assert( T.is_valid() );
int i;
// each of the following insertions of vertices increases the dimension
for ( i=1; i<5; i++ ) {
PV[i] = T.insert_outside_affine_hull(V[i], PV[0]);
assert( T.number_of_vertices() == i+1 );
assert( T.dimension() == i-1 );
assert( T.is_valid() );
}
assert( T.number_of_cells() == 5 );
// we now have a simplex in dimension 4
// cell incident to PV[0]
TDSCell* c = PV[0]->cell();
int ind;
assert( c->has_vertex( PV[0], ind ) );
// PV[0] is the vertex of index ind in c
// insertion of a new vertex in the facet opposite to PV[0]
PV[5] = T.insert_in_facet(TDSVertex(), c, ind);
assert( T.number_of_vertices() == 6 );
assert( T.dimension() == 3 );
assert( T.is_valid() );
// insertion of a new vertex in c
PV[6] = T.insert_in_cell( TDSVertex(), c );
assert( T.number_of_vertices() == 7 );
assert( T.dimension() == 3 );
assert( T.is_valid() );
std::ofstream oFileT("output_tds",ios::out);
// writing file output_tds;
oFileT << T;
return 0;
}
\end{verbatim}
\clearpage
\section{Concepts}
This section describes the concepts for a 3D- triangulation data
structure, its vertices and cells.
\subsection{Concepts for a 3D- Triangulation Data Structure}
\label{TDS3-sec-concept}
\begin{ccClassTemplate}{Tds<Vb,Fb>}
@ -277,9 +444,12 @@ incident to a given edge. They are bidirectional and non mutable.
\ccMethod{Tds operator=(const Tds & tds1);}
{Assignment. All the vertices and cells are duplicated.}
The previous first two methods are equivalent.
\ccMethod{void swap(Tds & tds1);}
{Swaps \ccVar\ and \ccc{tds1}. Should be preferred to \ccVar=\ccc{tds1}
or \ccVar(\ccc{tds1}) when tds1 is deleted after that.}
or \ccVar(\ccc{tds1}) when tds1 is deleted after that. Indeed, there is no
copy of cells and vertices, thus this method runs in constant time.}
\ccMethod{Vertex* copy_tds(const Tds & tds1, Vertex* v = NULL);}
{\ccc{tds1} is copied into \ccVar. The vertex of \ccVar\
@ -288,7 +458,8 @@ corresponding to \ccc{v} is returned.
\ccc{tds1}.}}
\ccFunction{void \ccTilde\ccVar();}
{Destructor. All vertices and cells are deleted.}
{Destructor. All vertices and cells are deleted, and \ccVar\ itself is
deleted.}
\ccAccessFunctions
@ -424,11 +595,12 @@ operation is exactly reciprocal to \ccc{\ccVar.insert_in_cell(v)}.
\textit{not yet implemented}}
\ccMethod{void clear();}
{Deletes all cells and vertices.}
{Deletes all cells and vertices. \ccVar\ is reset as a triangulation
data structure constructed by the default constructor.}
In addition to these requirements, in order to be used as a triangulation
data structure by the class
\ccc{CGAL_Delaunay_triangulation_3<Traits,Tds>}, the triangulation
\ccc{Delaunay_triangulation_3<Traits,Tds>}, the triangulation
must offer the following method:
\ccMethod{void star_region( set<void*, less<void*> > & region,
@ -542,8 +714,6 @@ each cell given by their index in the preceding list of cells.
\ccCreationVariable{v}
The vertex stores a point and a pointer to an incident cell.
The vertex class of a 3D- triangulation data structure must define
the types and operations listed in this section. Some of these
requirements are of geometric nature, they are \textit{optional}
@ -725,8 +895,8 @@ a precise indication on the kind of invalidity encountered.}
\end{ccClass}
\begin{ccClassTemplate}{CGAL_Triangulation_data_structure_3<Vb,Cb>}
\section{The Triangulation Data Structure Class\\
\begin{ccClassTemplate}{Triangulation_data_structure_3<Vb,Cb>}
\section{A model of Triangulation Data Structure:\\
\protect \ccClassTemplateName}
\label{TDS3-sec-class}
@ -736,26 +906,24 @@ geometric functionalities. It is templated by base classes for
vertices and cells described in Section~\ref{TDS3-sec-concept-Base}.
\ccInheritsFrom
\ccc{CGAL_Triangulation_utils_3}
\ccc{Triangulation_utils_3}
\\This class defines basic computations on indices of vertices and
neighbors of cells (see Section~\ref{Triangulation3-sec-class-Utils} of
neighbors of cells (Details are presented in
Section~\ref{Triangulation3-sec-class-Utils} of
Chapter~\ref{chapter-Triangulation3}).
\ccInclude{CGAL/Triangulation_data_structure_3.h}
The class \ccc{CGAL_Triangulation_ds_vertex_3} is a model for the vertex
The class \ccc{Triangulation_ds_vertex_3} is a model for the vertex
concept described in Section~\ref{TDS3-sec-concept-Tds_Vertex}.
\ccInclude{CGAL/Triangulation_ds_vertex_3.h}
The class \ccc{CGAL_Triangulation_ds_cell_3} is a model for the
The class \ccc{Triangulation_ds_cell_3} is a model for the
cell concept described in Section~\ref{TDS3-sec-concept-Tds_Cell}.
\ccInclude{CGAL/Triangulation_ds_cell_3.h}
\ccInheritsFrom
\ccc{CGAL_Triangulation_utils_3}
\end{ccClassTemplate}
\section{Concepts for the Base Vertices and Cells}
@ -943,29 +1111,33 @@ a precise indication on the kind of invalidity encountered.}
\end{ccClass}
\section{The Base Classes for Vertices and Cells}
\section{Models of Base Vertices and Cells}
\label{TDS3-sec-class-Base}
\begin{ccClassTemplate}{CGAL_Triangulation_vertex_base_3<Traits>}
\begin{ccClassTemplate}{Triangulation_vertex_base_3<Traits>}
\subsection{The Base Class \protect \ccClassTemplateName}
\label{TDS3-sec-class-Base_Vertex}
This class is proposed by \cgal\ as a base vertex class.
It is templated by a geometric traits class. Using the same
geometric traits class \ccc{Traits} as the one used for
\ccc{CGAL_Triangulation_3<Traits,Tds>} is strongly advised. In this way, the
point type defined by \ccClassName\ is the same as the point type
defined by the geometric traits class.
This class is proposed by \cgal\ as a base vertex class. It is
templated by a geometric traits class \ccc{Traits} that provides the
geometric types. The user who uses the geometric layer (see
Section~\ref{TDS3-sec-design} and
Chapter~\ref{chapter-Triangulation3}) is strongly advised to use the
same geometric traits class \ccc{Traits} as the one used for
\ccc{Triangulation_3<Traits,Tds>}. In this way, the point type
defined by \ccClassName\ is the same as the point type defined by the
geometric traits class. The default geometric traits class proposed by
\cgal\ is presented in Section~\ref{Triangulation3-sec-Traits}.
This base class can be used directly or can serve as a base to derive
other base classes with some additionnal attribute (a color for example)
tuned for a specific application.
This base class can be used directly or can serve as a base to derive
other base classes with some additionnal attributes (a color for
example) tuned for a specific application.
\ccInclude{CGAL/Triangulation_vertex_base_3.h}
\end{ccClassTemplate}
\begin{ccClassTemplate}{CGAL_Triangulation_cell_base_3<Traits>}
\begin{ccClassTemplate}{Triangulation_cell_base_3<Traits>}
\subsection{The Base Class \protect \ccClassTemplateName}
This class is proposed by \cgal\ as a base cell class, and is similar
@ -975,4 +1147,3 @@ Section~\ref{TDS3-sec-class-Base_Vertex}).
\ccInclude{CGAL/Triangulation_cell_base_3.h}
\end{ccClassTemplate}

353
Packages/Triangulation_3/doc_tex/basic/Triangulation_3/Triangulation3.tex
View File

@ -46,7 +46,7 @@ fourth vertex an auxiliary vertex called the \ccc{infinite vertex}. In
that way, each facet is incident to exactly two cells and special cases
at the boundary of the convex hull are simple to deal with.
The class \ccc{CGAL_Triangulation_3<Traits,Tds>} of \cgal\ implements this
The class \ccc{Triangulation_3<Traits,Tds>} of \cgal\ implements this
point of view and therefore considers the triangulation of the set
of points as a set of finite and infinite tetrahedra. Notice that the
infinite vertex has no significant coordinates and that no
@ -58,23 +58,45 @@ access to its four incident vertices and to its four adjacent
cells. Each vertex gives access to one of its incident cells.
The four vertices of a cell are indexed with 0, 1, 2 and 3 in positive
orientation. The neighbors of a cell are also indexed with 0, 1, 2, 3
in such a way that the neighbor indexed by $i$ is opposite to the vertex
with the same index.
orientation, the orientation being defined by the orientation of the
underlying space $\R^3$. The neighbors of a cell are also indexed with
0, 1, 2, 3 in such a way that the neighbor indexed by $i$ is opposite
to the vertex with the same index. See
Figure~\ref{Triangulation3-fig-orient}.
Edges ($1$-faces) and facets ($2$-faces) are not explicitely represented:
a facet is given by a cell and an index (the facet
\begin{ccTexOnly}
\begin{figure}[htbp]
\begin{center}
\includegraphics{orient.eps}
\end{center}
\caption{Orientation of a cell (3-dimensional case)
\label{Triangulation3-fig-orient}}
\end{figure}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<img border=0 src="./orient.gif" align=center alt="Orientation of a cell
(3-dimensional case)">
\end{ccHtmlOnly}
As in the underlying combinatorial triangulation (see
Chapter~\ref{chapter-TDS3}), edges ($1$-faces) and facets ($2$-faces)
are not explicitely
represented: a facet is given by a cell and an index (the facet
\ccc{i} of a cell \ccc{c} is the facet of \ccc{c} that is opposite to
the vertex of index \ccc{i}) and an edge is given by a cell and two
indices (the edge \ccc{(i,j)} of a cell \ccc{c} is the edge
whose endpoints are the vertices of indices \ccc{i} and \ccc{j} of
\ccc{c}).
indices (the edge \ccc{(i,j)} of a cell \ccc{c} is the edge whose
endpoints are the vertices of indices \ccc{i} and \ccc{j} of
\ccc{c}). See Figure~\ref{TDS3-fig-repres}.
\subsection{Degenerate Dimensions}
\label{Triangulation3-sec-degen_dim}
The class \ccc{CGAL_Triangulation_3<Traits,Tds>} can deal with degenerate
dimensions:
The class \ccc{Triangulation_3<Traits,Tds>} can deal with degenerate
dimensions. A triangulation of a set of points in $\R^d$ is a partition of the
whole space $\R^d$ into cells having $d+1$ vertices: some of them
are infinite, they are obtained by linking the additional infinite
vertex to each facet of the convex hull of the points.
\begin{itemize}
\item {} \emph{dimension 2:} when a triangulation only contains
coplanar points (which is the case when there are only three points),
@ -121,21 +143,6 @@ When all the points are collinear, this condition becomes:
{\bf (c-1D)} For any two adjacent edges $(u,v)$ and $(v,w)$, $u$ and
$w$ lie on opposite sides of the common vertex $v$.
\begin{ccTexOnly}
\begin{figure}[htbp]
\begin{center}
\includegraphics{orient.eps}
\end{center}
\caption{Orientation of a cell (3-dimensional case)
\label{Triangulation3-fig-orient}}
\end{figure}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<img border=0 src="./orient.gif" align=center alt="Orientation of a cell
(3-dimensional case)">
\end{ccHtmlOnly}
The \ccc{is_valid()} method provided by \cgal\ checks the local
validity of a given triangulation. This does not always
ensure global validity \cite{mnssssu-cgpvg-96,dlpt-ccpps-98} but it is
@ -144,7 +151,7 @@ sufficient for practical cases.
\section{Software Design}
\label{Triangulation3-sec-design}
The class \ccc{CGAL_Triangulation_3<Traits,Tds>} is designed to be used as
The class \ccc{Triangulation_3<Traits,Tds>} is designed to be used as
a layer upon a 3D-triangulation data structure as presented in
Section~\ref{TDS3-sec-design} of Chapter~\ref{chapter-TDS3}.
It provides high level geometric operations such as location of a point
@ -163,22 +170,114 @@ described in Chapter~\ref{chapter-TDS3}.
Delaunay triangulations as well as hierarchical Delaunay triangulations
\cite{d-iirdt-98} are also implemented in the package:
\ccc{CGAL_Delaunay_triangulation_3<Traits,Tds>} inherits from
\ccc{CGAL_Triangulation_3<Traits,Tds>} and
\ccc{CGAL_Delaunay_hierarchic_triangulation_3<Traits,Tds>} inherits from
\ccc{CGAL_Delaunay_triangulation_3<Traits,Tds>}.
\ccc{Delaunay_triangulation_3<Traits,Tds>} inherits from
\ccc{Triangulation_3<Traits,Tds>} and
\ccc{Delaunay_hierarchic_triangulation_3<Traits,Tds>} inherits from
\ccc{Delaunay_triangulation_3<Traits,Tds>}. \textit{(hierarchical
Delaunay triangulations are not yet implemented).}
\ccc{CGAL_Triangulation_3<Traits,Tds>} derives from
\ccc{CGAL_Triangulation_utils_3<Traits,Tds>}, which defines tools on
\ccc{Triangulation_3<Traits,Tds>} derives from
\ccc{Triangulation_utils_3<Traits,Tds>}, which defines tools on
the indices of vertices in cells (see
Section~\ref{Triangulation3-sec-class-Utils}).
\section{Examples}
\label{Triangulation3-sec-examples}
This example shows the incremental construction of a 3D triangulation,
the location of a point, and how to manipulate elementary operations
on indices in a cell.
\begin{verbatim}
#include <CGAL/basic.h>
#include <cstring>
#include <iostream>
#include <fstream>
#include <strstream.h>
#include <list>
#include <vector>
#include <CGAL/triangulation_assertions.h>
#include <CGAL/Cartesian.h>
#include <CGAL/Triangulation_cell_base_3.h>
#include <CGAL/Triangulation_vertex_base_3.h>
#include <CGAL/Triangulation_data_structure_3.h>
#include <CGAL/Triangulation_geom_traits_3.h>
#include <CGAL/Triangulation_3.h>
typedef CGAL::Cartesian<double> Rep;
typedef CGAL::Triangulation_geom_traits_3<Rep> Gt;
typedef CGAL::Triangulation_vertex_base_3<Gt> Vb;
typedef CGAL::Triangulation_cell_base_3<Gt> Cb;
typedef CGAL::Triangulation_data_structure_3<Vb,Cb> TDS;
typedef CGAL::Triangulation_3<Gt,TDS> Triangulation;
typedef typename Triangulation::Cell_handle Cell_handle;
typedef typename Triangulation::Vertex_handle Vertex_handle;
typedef typename Triangulation::Locate_type Locate_type;
typedef Gt::Point Point;
int main(int argc, char* argv[])
{
Triangulation T;
// insertion from a list :
std::list<Point> L;
L.push_front(Point(0,0,0));
L.push_front(Point(1,0,0));
L.push_front(Point(0,1,0));
int n = T.insert(L.begin(), L.end());
// insertion from a vector :
std::vector<Point> V(3);
V[0] = Point(0,0,1);
V[1] = Point(1,1,1);
V[2] = Point(2,2,2);
n = n + T.insert(V.begin(), V.end());
// 6 points have been inserted :
assert( n == 6 );
// checking validity of T :
assert( T.is_valid(false) );
Locate_type lt;
int li, lj;
Point p(0,0,0);
Cell_handle c = T.locate(p, lt, li, lj);
// p is the vertex of c of index li :
assert( lt == Triangulation::VERTEX );
assert( c->vertex(li)->point() == p );
Vertex_handle v = c->vertex( (li+1)&3 );
// v is another vertex of c
Cell_handle nc = c->neighbor(li);
// nc = neighbor of c opposite to the vertex associated with p
// nc must have vertex v :
int nli;
assert( nc->has_vertex( v, nli ) );
// nli is the index of v in nc
std::ofstream oFileT("output",ios::out);
// writing file output;
oFileT << T;
return 0;
}
\end{verbatim}
\clearpage
\begin{ccClass}{CGAL_Triangulation_utils_3}
\begin{ccClass}{Triangulation_utils_3}
\section{A Class of Tools \protect \ccClassTemplateName}
\label{Triangulation3-sec-class-Utils}
@ -223,7 +322,7 @@ Figure~\ref{Triangulation3-fig-utils}).
\end{ccClass}
\begin{ccClassTemplate}{CGAL_Triangulation_3<Traits,Tds>}
\begin{ccClassTemplate}{Triangulation_3<Traits,Tds>}
\section{The Triangulation Class \protect \ccClassTemplateName}
\ccDefinition
@ -237,7 +336,7 @@ and~\ref{Triangulation3-sec-class-Traits}.
\ccInclude{CGAL/Triangulation_3.h}
\ccInheritsFrom
\ccc{CGAL_Triangulation_utils_3}
\ccc{Triangulation_utils_3}
(see Section~\ref{Triangulation3-sec-class-Utils})
\ccTypes
@ -278,7 +377,8 @@ A handle is a type which supports the two dereference operators
bidirectionnel and non mutable. Circulators and iterators are
assignable to the corresponding handle types. Whenever a handle appears
in the parameter list of a function, an appropriate iterator or
circulator can be used as well \textit{not yet implemented}. The edges and facets of the
circulator can be used as well \textit{(not yet implemented)}. The
edges and facets of the
triangulation can also be visited through iterators and circulators
which are bidirectionnel and non mutable.
@ -314,7 +414,7 @@ OUTSIDE_AFFINE_HULL};}
{Introduces a triangulation \ccVar\ having only one vertex which is the
infinite vertex.}
\ccConstructor{\ccClassName(const CGAL_Triangulation_3<Traits,Tds> & tr);}
\ccConstructor{\ccClassName(const Triangulation_3<Traits,Tds> & tr);}
{Copy constructor. All the vertices and faces are duplicated.
\ccVar\ and \ccc{tr}\ refer to different triangulations. After the copy,
if \ccc{tr} is modified, \ccVar\ is not modified.}
@ -325,19 +425,22 @@ deleted.}
\ccHeading{Assignment}
\ccMethod{CGAL_Triangulation_3<Traits,Tds>
operator=(const CGAL_Triangulation_3<Traits,Tds> & tr);}
\ccMethod{Triangulation_3<Traits,Tds>
operator=(const Triangulation_3<Traits,Tds> & tr);}
{The triangulation is duplicated, and modifying one after the
copy does not modify the other.}
\ccMethod{copy_triangulation
(const CGAL_Triangulation_3<Traits,Tds> & tr);}
\ccMethod{void copy_triangulation
(const Triangulation_3<Traits,Tds> & tr);}
{The triangulation is duplicated.}
\ccMethod{void swap(CGAL_Triangulation_3<Traits,Tds> & tr);}
The previous first two methods are equivalent.
\ccMethod{void swap(Triangulation_3<Traits,Tds> & tr);}
{The triangulations \ccc{tr} and \ccVar\ are swapped.
\ccc{\ccVar.swap(tr)} should be preferred to \ccVar\ = \ccc{tr} or to
\ccc{t(tr)} if \ccc{tr} is deleted after that.}
\ccc{t(tr)} if \ccc{tr} is deleted after that. Indeed, there is no
copy of cells and vertices, thus this method runs in constant time.}
\ccAccessFunctions
\ccMethod{const Traits & geom_traits() const;}
@ -553,69 +656,69 @@ triangulation, \ccc{lt} is set to \ccc{OUTSIDE_AFFINE_HULL} and
{Same as the previous method, but \ccc{start} is used as a starting
place for the location.}
\ccMethod{CGAL_Bounded_side
\ccMethod{Bounded_side
side_of_cell(const Point & p,
Cell_handle c,
Locate_type & lt, int & li, int & lj) const;}
{Returns on which side of the oriented boundary of \ccc{c} lies
the point \ccc{p}. More precisely, it returns:\\
- \ccc{CGAL_ON_BOUNDED_SIDE} if \ccc{p} is inside the cell (for an infinite
- \ccc{ON_BOUNDED_SIDE} if \ccc{p} is inside the cell (for an infinite
cell this means that \ccc{p} lies strictly in the half space limited by
its finite facet) \\
- \ccc{CGAL_ON_BOUNDARY} if p on the boundary of the cell (for an infinite
- \ccc{ON_BOUNDARY} if p on the boundary of the cell (for an infinite
cell this means that \ccc{p} lies on the *finite* facet). Then
\ccc{lt} together with \ccc{li} and \ccc{lj} give the precise location
on the boundary (see the descriptions of \ccc{locate} methods)\\
- \ccc{CGAL_ON_UNBOUNDED_SIDE} if \ccc{p} lies outside the cell (for an
- \ccc{ON_UNBOUNDED_SIDE} if \ccc{p} lies outside the cell (for an
infinite cell this means that \ccc{p} is not in the preceding two
cases).
\ccPrecond{\ccc{\ccVar.dimension()} $=3$}}
\ccMethod{CGAL_Bounded_side
\ccMethod{Bounded_side
side_of_facet(const Point & p,
const Facet & f,
Locate_type & lt, int & li, int & lj) const;}
{Returns on which side of the oriented boundary of \ccc{f} lies
the point \ccc{p}:\\
- \ccc{CGAL_ON_BOUNDED_SIDE} if \ccc{p} is inside the facet (for an
- \ccc{ON_BOUNDED_SIDE} if \ccc{p} is inside the facet (for an
infinite facet this means that \ccc{p} lies strictly in the half plane
limited by its finite edge) \\
- \ccc{CGAL_ON_BOUNDARY} if \ccc{p} is on the boundary of the facet
- \ccc{ON_BOUNDARY} if \ccc{p} is on the boundary of the facet
(for an infinite facet this means that \ccc{p} lies on the finite
edge). \ccc{lt}, \ccc{li} and \ccc{lj} give the precise location of
\ccc{p} on the boundary of the facet. \ccc{li} and \ccc{lj} refer to
indices in the cell \ccc{c} storing \ccc{f}\\
- \ccc{CGAL_ON_UNBOUNDED_SIDE} if \ccc{p} lies outside the facet (for
- \ccc{ON_UNBOUNDED_SIDE} if \ccc{p} lies outside the facet (for
an infinite facet this means that \ccc{p} is not in the preceding two
cases)
\ccPrecond{\ccc{\ccVar.dimension()} $=2$ and \ccc{p} lies in the
plane containing the triangulation. \ccc{f.first} $=3$ (in dimension~2
there is only one facet per cell).}}
\ccGlue
\ccMethod{CGAL_Bounded_side
\ccMethod{Bounded_side
side_of_facet(const Point & p,
Cell_handle c,
Locate_type & lt, int & li, int & lj) const;}
{Same as the previous method for the facet \ccc{(c,3)}.}
\ccMethod{CGAL_Bounded_side
\ccMethod{Bounded_side
side_of_edge(const Point & p,
const Edge & e,
Locate_type & lt, int & li) const;}
{Returns on which side of the oriented boundary of \ccc{e} lies
the point \ccc{p}:\\
- \ccc{CGAL_ON_BOUNDED_SIDE} if \ccc{p} is inside the edge (for an
- \ccc{ON_BOUNDED_SIDE} if \ccc{p} is inside the edge (for an
infinite edge this means that \ccc{p} lies in the half line defined by
the vertex)\\
- \ccc{CGAL_ON_BOUNDARY} if \ccc{p} equals one of the vertices,
- \ccc{ON_BOUNDARY} if \ccc{p} equals one of the vertices,
\ccc{li} give the index of the vertex in the cell storing \ccc{e}\\
- \ccc{CGAL_ON_UNBOUNDED_SIDE} if \ccc{p} lies outside the edge (for
- \ccc{ON_UNBOUNDED_SIDE} if \ccc{p} lies outside the edge (for
an infinite edge this means that \ccc{p} lies on the other half line)\\
\ccPrecond{\ccc{\ccVar.dimension()} $=1$ and \ccc{p} is collinear
with the points of the triangulation. \ccc{e.first} $=0$ and
\ccc{e.second} $=1$ (in dimension~2 there is only one edge per cell).}}
\ccGlue
\ccMethod{CGAL_Bounded_side
\ccMethod{Bounded_side
side_of_edge(const Point & p,
Cell_handle c,
Locate_type & lt, int & li) const;}
@ -691,8 +794,8 @@ inside cell \ccc{c}.}}
{Inserts point \ccc{p} in facet \ccc{f}. In dimension~3, the 2
neighboring cells are split into 3 tetrahedra; in dimension~2, the facet
is split into 3 triangles.
\ccPrecond{\ccc{\ccVar.dimension()} $\geq 2$ and \ccc{p} lies inside
face \ccc{f}.}}
\ccPrecond{\ccc{\ccVar.dimension()} $\geq 2$ and \ccc{p} lies strictly
inside face \ccc{f}.}}
\ccGlue
\ccMethod{Vertex_handle insert_in_facet(const Point & p,
Cell_handle c, int i);}
@ -717,7 +820,7 @@ dimension~1.}}
\ccMethod{Vertex_handle insert_outside_convex_hull(const Point & p,
Cell_handle c,
int li, int lj=0);}
{\ccc{c}, together with \ccc{li} and possibly \ccc{lj}, give a
{The cell \ccc{c}, together with \ccc{li} and possibly \ccc{lj}, give a
separator (facet, edge or vertex, depending on the dimension) for
\ccc{p} from the triangulation (see the description of method
\ccc{locate()} for more details on the way the separator is represented).\\
@ -792,7 +895,7 @@ and \ccc{Vertex}. They are all invalidated by any change in the
triangulation.
\ccMethod{Vertex_iterator finite_vertices_begin() const;}
{Starts at an arbitrary finite vertex. Then \ccc{$++$} and \ccc{$--$} will
{Starts at an arbitrary finite vertex. Then \ccc{++} and \ccc{--} will
iterate on finite vertices. Returns \ccc{vertices_end()} when
\ccc{\ccVar.number_of_vertices()} $<1$.}
\ccGlue
@ -805,7 +908,7 @@ ones). Returns \ccc{vertices_end()} when
{Past-the-end iterator}
\ccMethod{Edge_iterator finite_edges_begin() const;}
{Starts at an arbitrary finite edge. Then \ccc{$++$} and \ccc{$--$} will
{Starts at an arbitrary finite edge. Then \ccc{++} and \ccc{--} will
iterate on finite edges. Returns \ccc{edges_end()} when
\ccc{\ccVar.dimension()} $<1$.}
\ccGlue
@ -817,7 +920,7 @@ ones). Returns \ccc{edges_end()} when \ccc{\ccVar.dimension()} $<1$.}
{Past-the-end iterator}
\ccMethod{Facet_iterator finite_facets_begin() const;}
{Starts at an arbitrary finite facet. Then \ccc{$++$} and \ccc{$--$} will
{Starts at an arbitrary finite facet. Then \ccc{++} and \ccc{--} will
iterate on finite facets. Returns \ccc{facets_end()} when
\ccc{\ccVar.dimension()} $<2$.}
\ccGlue
@ -830,7 +933,7 @@ ones). Returns \ccc{facets_end()} when
{Past-the-end iterator}
\ccMethod{Cell_iterator finite_cells_begin() const;}
{Starts at an arbitrary finite cell. Then \ccc{$++$} and \ccc{$--$} will
{Starts at an arbitrary finite cell. Then \ccc{++} and \ccc{--} will
iterate on finite cells. Returns \ccc{cells_end()} when
\ccc{\ccVar.dimension()} $<3$.}
\ccGlue
@ -907,7 +1010,7 @@ debugging.
\ccMethod{bool
is_valid(bool verbose = false) const;}
{Checks the combinatorial validity of the triangulation and also the
{Checks the combinatorial validity of the triangulation. Checks also the
validity of its geometric embedding (see
Section~\ref{Triangulation3-sec-Valid}). When \ccc{verbose} is set to true,
messages describing the first invalidity encountered are printed.\\
@ -936,7 +1039,7 @@ corresponding input operators of the vertex and the cell
classes. Assigns the resulting triangulation to \ccc{t}.}
\ccFunction{ostream& operator<<
(ostream& os, const CGAL_Triangulation_3<Traits, Tds> &t);}
(ostream& os, const Triangulation_3<Traits, Tds> &t);}
{Writes the triangulation \ccc{t} into \ccc{os}.}
The information in the \ccc{iostream} is: the dimension, the number of
@ -949,7 +1052,7 @@ of maximal dimension.)
\end{ccClassTemplate}
\begin{ccClassTemplate}{CGAL_Triangulation_vertex_3<Traits,Tds>}
\begin{ccClassTemplate}{Triangulation_vertex_3<Traits,Tds>}
\section{The Vertex Class of a Triangulation \protect \ccClassTemplateName}
\label{Triangulation3-sec-class-Vertex}
@ -973,9 +1076,9 @@ defined in the class \ccc{Triangulation_3<Traits,Tds>}:
\ccGlue
\ccTypedef{typedef typename Tds::Cell Ctds;}{cell of the triangulation data structure}
\ccGlue
\ccTypedef{typedef CGAL_Triangulation_vertex_handle_3<Traits,Tds> Vertex_handle;}{handle to a vertex}
\ccTypedef{typedef Triangulation_vertex_handle_3<Traits,Tds> Vertex_handle;}{handle to a vertex}
\ccGlue
\ccTypedef{typedef CGAL_Triangulation_cell_handle_3<Traits,Tds> Cell_handle;}{handle to a cell}
\ccTypedef{typedef Triangulation_cell_handle_3<Traits,Tds> Cell_handle;}{handle to a cell}
\begin{ccAdvanced}
\ccCreation
@ -1034,7 +1137,7 @@ of maximal dimension (Section~\ref{Triangulation3-sec-degen_dim}).}
\end{ccClassTemplate}
\begin{ccClassTemplate}{CGAL_Triangulation_cell_3<Traits,Tds>}
\begin{ccClassTemplate}{Triangulation_cell_3<Traits,Tds>}
\section{The Cell Class of a Triangulation \protect \ccClassTemplateName}
\label{Triangulation3-sec-class-Cell}
@ -1056,7 +1159,7 @@ dimension: (Section~\ref{Triangulation3-sec-degen_dim}).
\ccTypes
The class \ccClassTemplateName\ defines the same types as the
\ccc{CGAL_Triangulation_vertex_3<Traits,Tds>} class.
\ccc{Triangulation_vertex_3<Traits,Tds>} class.
\begin{ccAdvanced}
\ccCreation
@ -1168,7 +1271,7 @@ computes the index \ccc{i} of the neighbor.}
\label{Triangulation3-sec-Traits}
The first template parameter of the triangulation class
\ccc{CGAL_Triangulation_3<Traits,Tds>} of \cgal\ is the geometric traits class.
\ccc{Triangulation_3<Traits,Tds>} of \cgal\ is the geometric traits class.
The first subsection of this section describes the requirements
that the geometric traits class must fulfill. The second subsection
@ -1180,12 +1283,12 @@ presents a predefined geometric traits class available in \cgal.
\protect \ccClassName}
\label{Triangulation3-sec-concept-Traits}
\subsubsection{To be used by \protect \ccc{CGAL_Triangulation_3<Traits, Tds>}}
\subsubsection{To be used by \protect \ccc{Triangulation_3<Traits, Tds>}}
\ccCreationVariable{traits}
The geometric traits class \ccClassName\ of the triangulation
class \ccc{CGAL_Triangulation_3<Traits, Tds>} must define the geometric
class \ccc{Triangulation_3<Traits, Tds>} must define the geometric
objects (points, segments, triangles and tetrahedra) forming the
triangulation together with a few geometric predicates on these objects:
equality, coordinates comparison, orientation in the space, orientation
@ -1225,36 +1328,36 @@ can be provided.
\ccMethod{bool equal(const Point & p, const Point & q) const;}
{Equality test.}
\ccMethod{CGAL_Comparison_result compare_x(const Point & p, const Point
\ccMethod{Comparison_result compare_x(const Point & p, const Point
& q) const;}
{Comparison of \ccc{x}-coordinates. Returns \ccc{CGAL_LARGER}
(resp. \ccc{CGAL_EQUAL}, \ccc{CGAL_SMALLER}) when the \ccc{x}
{Comparison of \ccc{x}-coordinates. Returns \ccc{LARGER}
(resp. \ccc{EQUAL}, \ccc{SMALLER}) when the \ccc{x}
coordinate of \ccc{p} is larger than (resp. equal to, smaller than)
the \ccc{x} coordinate of \ccc{q}.}
\ccGlue
\ccMethod{CGAL_Comparison_result compare_y(const Point & p, const Point
\ccMethod{Comparison_result compare_y(const Point & p, const Point
& q) const;}
{Comparison of \ccc{y}-coordinates.}
\ccGlue
\ccMethod{CGAL_Comparison_result compare_z(const Point & p, const Point
\ccMethod{Comparison_result compare_z(const Point & p, const Point
& q) const;}
{Comparison of \ccc{z}-coordinates.}
\ccMethod{CGAL_Orientation orientation(const Point& p0,
\ccMethod{Orientation orientation(const Point& p0,
const Point& p1,
const Point& p2,
const Point& p3) const;}
{Orientation test in three dimensions.}
\ccMethod{CGAL_Orientation orientation_in_plane
\ccMethod{Orientation orientation_in_plane
(const Point & q,
const Point & r,
const Point & s,
const Point & test) const;}
{When \ccc{test, q, r, s} are coplanar, tests whether \ccc{test} is on the
same side of \ccc{(q, r)} as \ccc{s}. Returns \ccc{CGAL_COLLINEAR} if
\ccc{test, q, r} are collinear, \ccc{CGAL_POSITIVE} if \ccc{(q, r, test)}
and \ccc{(q, r, s)} have the same orientation, \ccc{CGAL_NEGATIVE} if
same side of \ccc{(q, r)} as \ccc{s}. Returns \ccc{COLLINEAR} if
\ccc{test, q, r} are collinear, \ccc{POSITIVE} if \ccc{(q, r, test)}
and \ccc{(q, r, s)} have the same orientation, \ccc{NEGATIVE} if
\ccc{(q, r, test)} and \ccc{(q, r, s)} have opposite orientations.
\ccPrecond{\ccc{test,q,r,s} are coplanar and \ccc{q,r,s} are not collinear.}}
@ -1264,7 +1367,7 @@ and \ccc{(q, r, s)} have the same orientation, \ccc{CGAL_NEGATIVE} if
{Collinearity test.}
\subsubsection{To be used by \protect
\ccc{CGAL_Delaunay_triangulation_3<Traits, Tds>}}
\ccc{Delaunay_triangulation_3<Traits, Tds>}}
In addition to the requirements described before, the geometric traits
class of a
@ -1272,7 +1375,7 @@ Delaunay triangulation must fulfill the following requirements:
\ccHeading{Predicates}
\ccMethod{CGAL_Oriented_side
\ccMethod{Oriented_side
side_of_oriented_sphere(const Point & p,
const Point & q,
const Point & r,
@ -1281,23 +1384,23 @@ Delaunay triangulation must fulfill the following requirements:
{Computes the side of point \ccc{test} with respect to the oriented
sphere circumscribing \ccc{p,q,r,s}.}
\ccMethod{CGAL_Oriented_side
\ccMethod{Oriented_side
side_of_oriented_circle(const Point & p,
const Point & q,
const Point & r,
const Point & test) const;}
{computes the side of \ccc{test} with respect to the oriented circle
{Computes the side of \ccc{test} with respect to the oriented circle
circumscribing \ccc{p,q,r}.
\ccPrecond{\ccc{p,q,r,test} are coplanar.}}
\subsubsection{To be used by \protect
\ccc{CGAL_Delaunay_hierarchic_triangulation_3<Traits, Tds>}}
\ccc{Delaunay_hierarchic_triangulation_3<Traits, Tds>}}
\textit{Not yet implemented}
\end{ccClass}
\begin{ccClassTemplate}{CGAL_Triangulation_geom_traits_3<R>}
\begin{ccClassTemplate}{Triangulation_geom_traits_3<R>}
\subsection{The Class \protect \ccClassTemplateName}
\label{Triangulation3-sec-class-Traits}
@ -1314,13 +1417,13 @@ Section~\ref{Triangulation3-sec-concept-Traits}.
\ccTypes
\ccTypedef{typedef CGAL_Point_3<R> Point;}{}
\ccTypedef{typedef Point_3<R> Point;}{}
\ccGlue
\ccTypedef{typedef CGAL_Segment_3<R> Segment;}{}
\ccTypedef{typedef Segment_3<R> Segment;}{}
\ccGlue
\ccTypedef{typedef CGAL_Triangle_3<R> Triangle;}{}
\ccTypedef{typedef Triangle_3<R> Triangle;}{}
\ccGlue
\ccTypedef{typedef CGAL_Tetrahedron_3<R> Tetrahedron;}{}
\ccTypedef{typedef Tetrahedron_3<R> Tetrahedron;}{}
\end{ccClassTemplate}
@ -1328,32 +1431,32 @@ Section~\ref{Triangulation3-sec-concept-Traits}.
\label{Triangulation3-sec-tds}
The second template parameter of the basic triangulation class
\ccc{CGAL_Triangulation_3<Traits,Tds>} is a triangulation data structure
\ccc{Triangulation_3<Traits,Tds>} is a triangulation data structure
class. This class can be seen as a container for the cells and
vertices maintaining incidence and adjacency relations.
The concept for the triangulation data structure is described in
Section~\ref{TDS3-sec-concept} of Chapter~\ref{chapter-TDS3}. Its optional
arguments related to geometry are compulsory for this use as a
template parameter of \ccc{CGAL_Triangulation_3<Traits,Tds>}.
template parameter of \ccc{Triangulation_3<Traits,Tds>}.
A model of this triangulation data structure is
\ccc{CGAL_Triangulation_data_structuree_3} presented in
\ccc{Triangulation_data_structuree_3} presented in
Section~\ref{TDS3-sec-class}.
\begin{ccClassTemplate}{CGAL_Delaunay_triangulation_3<Traits,Tds>}
\begin{ccClassTemplate}{Delaunay_triangulation_3<Traits,Tds>}
\section{The Delaunay Triangulation Class \protect
\ccClassTemplateName}
\ccDefinition
\ccInclude{CGAL/Delaunay_triangulation_3.h}
\ccInheritsFrom{\ccc{CGAL_Triangulation_3<Traits,Tds>}}
\ccInheritsFrom{\ccc{Triangulation_3<Traits,Tds>}}
\ccCreationVariable{dt}
\ccTypes
Inherits the types of \ccc{CGAL_Triangulation_3<Traits,Tds>}.
Inherits the types of \ccc{Triangulation_3<Traits,Tds>}.
\ccCreation
@ -1364,13 +1467,17 @@ Inherits the types of \ccc{CGAL_Triangulation_3<Traits,Tds>}.
{Creates an empty Delaunay triangulation with traits class
\ccc{traits}.}
\ccConstructor{\ccClassName(const CGAL_Delaunay_triangulation_3<Traits,Tds> & dt1)}
\ccConstructor{\ccClassName(const Delaunay_triangulation_3<Traits,Tds> & dt1)}
{Copy constructor.}
\ccModifiers
\ccHeading{Insertion}
The following methods, which already exist in triangulations, are
overloaded to ensure the empty sphere property of Delaunay
triangulations.
\ccMethod{Vertex_handle insert(const Point & p );}
{Inserts point \ccc{p} in the triangulation and returns the corresponding
vertex. Similar to the insertion in a triangulation, but insures in
@ -1408,45 +1515,45 @@ number of inserted points.
\ccHeading{Queries}
\ccMethod{CGAL_Bounded_side
\ccMethod{Bounded_side
side_of_sphere(Cell_handle c, const Point & p) const;}
{Returns on which side of the circumscribed sphere of \ccc{c} lies
the point \ccc{p}. More precisely, it returns:\\
- \ccc{CGAL_ON_BOUNDED_SIDE} if \ccc{p} is inside the sphere (for an infinite
- \ccc{ON_BOUNDED_SIDE} if \ccc{p} is inside the sphere (for an infinite
cell this means that \ccc{p} lies strictly either in the half space
limited by its finite facet, or in the interior of the disk
circumscribing the *finite* facet) \\
- \ccc{CGAL_ON_BOUNDARY} if p on the boundary of the sphere (for an infinite
- \ccc{ON_BOUNDARY} if p on the boundary of the sphere (for an infinite
cell this means that \ccc{p} lies on the circle circumscribing
the *finite* facet)\\
- \ccc{CGAL_ON_UNBOUNDED_SIDE} if \ccc{p} lies outside the sphere (for an
- \ccc{ON_UNBOUNDED_SIDE} if \ccc{p} lies outside the sphere (for an
infinite cell this means that \ccc{p} is not in the preceding two
cases).
\ccPrecond{\ccc{\ccVar.dimension()} $=3$.}}
\ccMethod{CGAL_Bounded_side
\ccMethod{Bounded_side
side_of_circle(const Facet & f, const Point & p) const;}
{Returns on which side of the circumscribed circle of \ccc{f} lies
the point \ccc{p}. More precisely, it returns:\\
- in dimension~3:\\
-- for a finite facet, \ccc{CGAL_ON_BOUNDARY} if \ccc{p} lies
on the circle, \ccc{CGAL_ON_UNBOUNDED_SIDE} when it lies in the exterior of
the disk, \ccc{CGAL_ON_BOUNDED_SIDE} when it lies in its interior.\\
-- for a finite facet, \ccc{ON_BOUNDARY} if \ccc{p} lies
on the circle, \ccc{ON_UNBOUNDED_SIDE} when it lies in the exterior of
the disk, \ccc{ON_BOUNDED_SIDE} when it lies in its interior.\\
-- for an infinite facet, it considers the plane defined by the finite
facet of the same cell, and does the same as in dimension~2 in this
plane.\\
- in dimension~2:\\
-- for a finite facet, \ccc{CGAL_ON_BOUNDARY} if \ccc{p} lies
on the circle, \ccc{CGAL_ON_UNBOUNDED_SIDE} when it lies in the exterior of
the disk, \ccc{CGAL_ON_BOUNDED_SIDE} when it lies in its interior.\\
-- for an infinite facet, \ccc{CGAL_ON_BOUNDARY} if the
-- for a finite facet, \ccc{ON_BOUNDARY} if \ccc{p} lies
on the circle, \ccc{ON_UNBOUNDED_SIDE} when it lies in the exterior of
the disk, \ccc{ON_BOUNDED_SIDE} when it lies in its interior.\\
-- for an infinite facet, \ccc{ON_BOUNDARY} if the
point lies on the finite edge of \ccc{f} (endpoints included),
\ccc{CGAL_ON_BOUNDED_SIDE} for a point in the open half-plane defined
by \ccc{f}, \ccc{CGAL_ON_UNBOUNDED_SIDE} elsewhere.
\ccc{ON_BOUNDED_SIDE} for a point in the open half-plane defined
by \ccc{f}, \ccc{ON_UNBOUNDED_SIDE} elsewhere.
\ccPrecond{\ccc{\ccVar.dimension()} $\geq 2$.}}
\ccMethod{CGAL_Bounded_side
\ccMethod{Bounded_side
side_of_circle(Cell_handle c, int i, const Point & p);}
{Same as the previous method.}
{Same as the previous method for facet \ccc{i} of cell \ccc{c}.}
\begin{ccAdvanced}
\ccHeading{Checking}
@ -1464,7 +1571,7 @@ This method is mainly a debugging help for the users of advanced features.
\end{ccClassTemplate}
\begin{ccClassTemplate}{CGAL_Delaunay_hierarchic_triangulation_3<Traits,Tds>}
\begin{ccClassTemplate}{Delaunay_hierarchic_triangulation_3<Traits,Tds>}
\section{The Hierarchic Delaunay Triangulation Class \protect
\ccClassTemplateName}
@ -1491,13 +1598,13 @@ To work around this problem, two functions have been defined to transform
\ccc{handles} into usual $C^{++}$ pointers for debugging purposes.
\ccFunction{template <class Traits, class Tds>
CGAL_Triangulation_vertex_3<Traits,Tds> *
CGAL_debug(const CGAL_Triangulation_vertex_handle_3<Traits,Tds> v);}
Triangulation_vertex_3<Traits,Tds> *
debug(const Triangulation_vertex_handle_3<Traits,Tds> v);}
{}
\ccFunction{template <class Traits, class Tds>
CGAL_Triangulation_cell_3<Traits,Tds> *
CGAL_debug(const CGAL_Triangulation_cell_handle_3<Traits,Tds>
Triangulation_cell_3<Traits,Tds> *
debug(const Triangulation_cell_handle_3<Traits,Tds>
c);}
{}