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Initial revision

This commit is contained in:
Laurent Rineau 2002-07-15 14:23:19 +00:00
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Package Name: Visibility_complex
Author: Angelier Pierre <Pierre.Angelier@ens.fr>
Maintainer: Angelier Pierre <Pierre.Angelier@ens.fr>
Rineau Laurent <Laurent.Rineau@ens.fr>
Address: Département d'Informatique de l'Ecole Normale Supérieure
45 rue d'Ulm 75230 Paris Cedex 05 France
URL: http://www.di.ens.fr/users/angelier/visibility_complex.tar.gz
Version: 1.0 (10 July 2002)
CGAL Versions: 2.2 and greater
Supported Compilers & Platforms:
sparc_SunOS-5.8_g++-2.95
sparc_SunOS-5.7_g++-2.95.2_LEDA
i686_Linux-2.2.18_g++-2.95
i686_Linux-2.2.18_g++-2.95_LEDA
i686_Linux-2.2.18_g++-3.0.4
i686_Linux-2.2.18_g++-3.1.
Purpose:
This package implements the visibility complex data structure for planar
scenes.
Description:
Distributed with this package are the following things:
README -- this file
INSTALLATION -- file describing more details about package installation
doc_tex/ -- directory containing the package documentation in
LaTeX format
doc_ps/ -- directory containing the package documentation in
postscript format
doc_html/ -- directory containing the package documentation in
Html format
demo/ -- two small graphical (LEDA) demos illustrating the package.
see demo/README for more details.
cep_test -- shell script for running the test suite
include/ -- directory containing the Visibility_complex header files
test_suite/ -- directory containing the test suite source code,
input and output files
This package can be used to compute the visibility graph or visibility
complex of planar scenes. These scenes can consist in pairwise
non-intersecting points, segments, circles or polygons.
The package can also be used to compute the shortest path between two
points in such scenes.
Changes:
1.0 initial release
1.0.1 fix for gcc-3.x compilers
Installation:
Documentation:
Package documentation is provided in postscript and HTML format in
the subdirectory doc.
Tested items:
Testing process:
Hardware & Software requirements:
Constraints:

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#! /bin/sh
# This is a script for running the Visibility_complex test suite.
#
# This script assumes that the CGAL_MAKEFILE environment variable is set
#---------------------------------------------------------------------#
# compile_and_run <target>
#---------------------------------------------------------------------#
compile_and_run()
{
echo "Compiling $1 ... "
if eval 'make CGAL_MAKEFILE=$CGAL_MAKEFILE $1'; then
echo " compilation of $1 succeeded"
else
echo " ERROR: compilation of $1 failed"
exit 1
fi
if [ -f $1 ] ; then
OUTPUTFILE=$1.out
rm -f $OUTPUTFILE
COMMAND="./$1"
#
echo "Executing $1 ... "
echo
if eval 2>&1 $COMMAND > $OUTPUTFILE ; then
echo " execution of $1 succeeded"
else
echo " ERROR: execution of $1 failed"
exit 1
fi
else
echo " ERROR: could not execute $1"
exit 1
fi
eval "2>&1 make CGAL_MAKEFILE=$CGAL_MAKEFILE clean > /dev/null "
}
#---------------------------------------------------------------------#
# compile and run the tests
#---------------------------------------------------------------------#
cd test_suite
if [ $# -ne 0 ] ; then
for file in $* ; do
compile_and_run $file
done
else
compile_and_run visibility_complex_polygon
compile_and_run visibility_complex_segment
compile_and_run shortest_path
fi

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Two demos illustrating the Visibility_complex package.
*** visibility_graph
Reads the 50 pairwise disjoint segments from the file "segments"
and outputs the bitangents of the visibility graph.
The bitangents are in red and the segments in black.
*** shortest_path
Reads the 50 pairwise disjoint segments from the file "segments"
and outputs the shortest path between two corners of the window.
The shortest path is in red and the segments in black.

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# Created by the script create_makefile
# This is the makefile for compiling a CGAL application.
#---------------------------------------------------------------------#
# include platform specific settings
#---------------------------------------------------------------------#
# Choose the right include file from the <cgalroot>/make directory.
# CGAL_MAKEFILE = ENTER_YOUR_INCLUDE_MAKEFILE_HERE
include $(CGAL_MAKEFILE)
#---------------------------------------------------------------------#
# compiler flags
#---------------------------------------------------------------------#
CXXFLAGS = -I../include \
$(CGAL_CXXFLAGS) \
$(LONG_NAME_PROBLEM_CXXFLAGS) \
$(DEBUG_OPT)
#---------------------------------------------------------------------#
# linker flags
#---------------------------------------------------------------------#
LIBPATH = \
$(CGAL_WINDOW_LIBPATH)
LDFLAGS = \
$(LONG_NAME_PROBLEM_LDFLAGS) \
$(CGAL_WINDOW_LDFLAGS)
#---------------------------------------------------------------------#
# target entries
#---------------------------------------------------------------------#
all: \
shortest_path$(EXE_EXT) \
visibility_graph$(EXE_EXT)
shortest_path$(EXE_EXT): shortest_path$(OBJ_EXT)
$(CGAL_CXX) $(LIBPATH) $(EXE_OPT)shortest_path shortest_path$(OBJ_EXT) $(LDFLAGS)
visibility_graph$(EXE_EXT): visibility_graph$(OBJ_EXT)
$(CGAL_CXX) $(LIBPATH) $(EXE_OPT)visibility_graph visibility_graph$(OBJ_EXT) $(LDFLAGS)
clean: \
shortest_path.clean \
visibility_graph.clean
#---------------------------------------------------------------------#
# suffix rules
#---------------------------------------------------------------------#
.C$(OBJ_EXT):
$(CGAL_CXX) $(CXXFLAGS) $(OBJ_OPT) $<

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668 560 662 61
338 18 276 85
566 375 613 44
722 309 741 527
156 348 454 345
372 558 290 534
193 407 524 358
182 576 565 384
188 228 109 519
329 164 480 311
162 546 194 537
185 194 433 128
381 12 721 14
472 303 556 153
641 495 656 556
760 297 741 99
176 130 81 365
402 85 18 99
133 486 436 445
573 555 352 581
781 394 688 128
61 276 108 197
280 548 253 553
452 93 469 33
284 35 222 79
605 340 578 298
108 422 53 343
336 161 433 157
425 64 490 135
574 482 365 529
585 254 641 400
146 425 480 426
574 155 562 314
201 281 280 249
530 331 168 210
73 146 148 154
604 589 453 585
108 501 13 338
48 469 92 580
672 60 713 385
518 474 634 466
784 495 794 237
308 115 397 94
408 406 549 349
229 559 222 590
513 195 489 146
765 24 78 14
316 539 417 513
676 217 706 367
345 106 60 109

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#include <CGAL/basic.h>
#include <iostream>
#include <fstream>
#include <list>
#include <CGAL/Simple_cartesian.h>
#include <CEP/Visibility_complex/Shortest_path_segment_traits.h>
#include <CEP/Visibility_complex/Shortest_path_2.h>
#include <CGAL/IO/Window_stream.h>
#include <CGAL/IO/Ostream_iterator.h>
//
// Number type used for input
typedef int NT;
// Number type used when computing distances
typedef double SNT;
typedef CGAL::Simple_cartesian<NT> SC;
struct Rep : public SC {};
typedef CGAL::Shortest_path_segment_traits<Rep,SNT> Gt;
typedef Gt::Disk Segment;
typedef Gt::Point_2 Point;
typedef CGAL::Visibility_complex_2<Gt> VC;
typedef VC::Vertex Vertex;
typedef CGAL::Window_stream Window_stream;
int main()
{
// Reading segments from file
std::list<Segment> O;
std::ifstream ifs("segments");
std::istream_iterator<Segment> ifs_it(ifs),ifs_end;
std::copy(ifs_it,ifs_end,std::back_inserter(O));
// Open a LEDA window
Window_stream W(500,400); // physical window size
W.init(0, 800., 0.); // logical window size
CGAL::cgalize(W);
W.display();
// Two corners of the window
Point p(0,0);
Point q(1000,800);
// Otput the segments of the shortest path from p to q in reverse order.
W << CGAL::RED;
W.set_line_width(3);
shortest_path_2(O.begin(),O.end(),
p,q,
CGAL::Ostream_iterator<Segment,Window_stream>(W),
Gt());
// Output the segments in BLACK
W << CGAL::BLACK;
W.set_line_width(1);
std::copy(O.begin(),O.end(),
CGAL::Ostream_iterator<Segment,Window_stream>(W));
// Show window
std::cout << "Click in window to quit" << std::endl;
W.read_mouse();
return 0;
}

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#include <CGAL/basic.h>
#include <iostream>
#include <fstream>
#include <list>
#include <CGAL/Simple_cartesian.h>
#include <CEP/Visibility_complex/Visibility_complex_segment_traits.h>
#include <CEP/Visibility_complex/Visibility_complex_2.h>
#include <CGAL/IO/Window_stream.h>
#include <CGAL/IO/Ostream_iterator.h>
typedef int FT;
typedef CGAL::Simple_cartesian <FT> Rep;
typedef CGAL::Visibility_complex_segment_traits<Rep> Gt;
typedef CGAL::Visibility_complex_2<Gt> Visibility_complex;
typedef Visibility_complex::Vertex Vertex;
typedef Gt::Disk Segment;
typedef CGAL::Window_stream Window_stream;
int main()
{
// Reading segments from file
std::list<Segment> O;
std::ifstream ifs("segments");
std::istream_iterator<Segment> ifs_it(ifs),ifs_end;
std::copy(ifs_it,ifs_end,std::back_inserter(O));
// Computing visibility graph
Visibility_complex V(O.begin(),O.end());
// Open a LEDA window
Window_stream W(500,400); // physical window size
W.init(0, 800., 0.); // logical window size
CGAL::cgalize(W);
W.display();
// Output the bitangents in RED
W << CGAL::RED;
std::copy(V.vertices_begin(),V.vertices_end(),
CGAL::Ostream_iterator<Vertex,Window_stream>(W));
// Output the segments in BLACK
W << CGAL::BLACK;
std::copy(O.begin(),O.end(),
CGAL::Ostream_iterator<Segment,Window_stream>(W));
// Show window
std::cout << "Click in window to quit" << std::endl;
W.read_mouse();
return 0;
}

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all: to_ps to_html
to_html:
cc_manual_to_html -o html manual.tex
to_ps:
latex manual.tex
makeindex manual
index_fix manual.ind
latex manual.tex
dvips -o manual.ps manual.dvi
gzip -f manual.ps
clean:
\rm -f *.aux *.log *.dvi *.idx *.ilg *.ind *.ind.unfixed

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#include <list>
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Visibility_complex_2.h>
#include <CGAL/Visibility_complex_segment_traits.h>
typedef CGAL::Simple_cartesian<int> Rep;
typedef CGAL::Visibility_complex_segment_traits<Rep> Gt;
typedef Gt::Disk Segment;
typedef CGAL::Visibility_complex_2<Gt> Visibility_complex;
typedef Visibility_complex::Antichain Antichain;
typedef Visibility_complex::Linear_sweep_iterator Linear_sweep_iterator;
int main()
{
// Reading circles from file
std::list<Segment> D;
std::ifstream ifs("input");
std::istream_iterator<Segment> ifs_it(ifs),ifs_end;
std::copy(ifs_it,ifs_end,back_inserter(D));
// Computing the antichain
Antichain A(D.begin(),D.end());
// Sweep of the visibility complex using an iterator
Linear_sweep_iterator v(&A), vend(&A,0);
for ( ; v != vend ; ++v)
std::cout << *v << std::endl;
return 0;
}

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int main()
{
// Reading non necessarily convex polygons from file
list<Polygon> D;
// Creating a scene and inserting the polygons in D
Scene S;
copy( D.begin() , D.end() , back_inserter(S) );
// Computing the Visibility Complex
Visibility_complex V(S);
}

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% +---------------------------------------------------------------------------+
% | CGAL Reference Manual: visibility_complex.tex
% +---------------------------------------------------------------------------+
% | Visibility complex. Implemented by topological sweep using the
% | Greedy Flip Algorithm
% |
% | 21.01.2001 Pierre Angelier and Michel Pocchiola
% |
% |
\RCSdef{\vcomplexRev}{$Revision$}
\RCSdefDate{\vcomplexDate}{$Date$}
% +---------------------------------------------------------------------------+
\ccParDims
\chapter{Visibility Complexes}
\label{chapterVisibilityComplex}
\ccChapterRelease{\vcomplexRev. \ \vcomplexDate}\\
\ccChapterAuthor{Pierre Angelier and Michel Pocchiola}
\minitoc
% +----------------------------------------------------------------------------+
\section{Introduction}
\label{sectionVComplexIntroduction}
This package implements the visibility complex structure introduced by Pocchiola
and Vegter in~\cite{pv-vc-96} (see also~\cite{pv-tsvcpt-96}
and~\cite{G-ap-sstvc-01} for a more recent presentation). Although a
generalization to three dimensional environments has been proposed
in~\cite{ddp-fahrgv-99} we deal here only with planar scenes.
The visibility complex is a 2D cell complex whose 1-skeleton is isomorphic to
the visibility graph. The visibility graph can be used to compute the shortest
path between two points in a planar scene. A survey on visibility graphs and
shortest paths can be found in~\cite{m-gspno-00}.
Our implementation of the visibility complex structure follows closely the
design of the \ccc{Halfedge Data Structure} (see the introduction in
Chapter~\ref{chapterHalfedgeDS} and~\cite{k-ugpdd-99}) however this chapter can
be understood without reading the corresponding chapter.
We use the so-called Greedy Flip Algorithm (again see~\cite{pv-tsvcpt-96}) to
compute the visibility complex in optimal time complexity $O(k + n\log n)$ where
$n$ is the size of the input and $k$ the size of the output (i.e. the visibility
complex).
% +----------------------------------------------------------------------------+
\section{Definitions}
In this section we review the basic concepts defined in~\cite{G-ap-sstvc-01}
which are useful to understand the basic functionality of the package.
\paragraph{Basic definitions. }
A \emph{disk} is a bounded closed convex subset of the plane and a \emph{scene}
is a collection of pairwise disjoint disks. For the ease of the exposition we
will assume that the disks have non-empty interior and that the tangent line at
a point on the boundary of a disk is well defined. We will also suppose that our
disks are in general position that is, that there is no line tangent to three
disks. Although these definitions rule out points, segment and polygons our
implementation can handle these types of disks through a symbolic perturbation
scheme which is invisible to the user. The general position assumption is also
lifted through perturbation. More details are given in the degeneracies section
below. Note also that non convex polygons are also supported via the use of
\emph{constraints} which are defined further on.
A \emph{bitangent} is a directed line segment tangent to two disks and
intersecting these disks only at its endpoints. The disk containing the source
and target point of the bitangent are called respectively the source and target
disk. Two disks share four bitangents directed from the first towards the
second.
\begin{ccTexOnly}
%\vspace{-7mm}
\begin{center}
\parbox{0.4\textwidth}{%
\includegraphics[width=0.4\textwidth]{fig/bitangent.eps}%
}
\end{center}
%\vspace{-5mm}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<CENTER>
<img src="fig/bitangent.gif" alt="Bitangent"><P>
</CENTER>
\end{ccHtmlOnly}
The \emph{type} of a bitangent $t$ directed from a disk $A$ to a disk $B$ codes
the positions of $A$ and $B$ with respect to the supporting line of $t$. As
illustrated in the Figure above there are four possible types called \texttt{LL,
RR, LR} and \texttt{RL}. A \texttt{LR} bitangent $t$ for example, is such that
$A$ and $B$ lie respectively on the left and on the right of the supporting line
of $t$.
Consider a scene $S$. A bitangent between two disks of $S$ is \emph{free} if its
interior does not intersect a disk of $S$. A \emph{constrained scene} $S_H$ is
a scene $S$ and a subset $H$ of pairwise non crossing free bitangents. The
bitangents of $H$ are called \emph{constraints}. A bitangent is \emph{free} in
$S_H$ if it is free in $S$ and does not intersect a bitangent of $H$.
Constraints are still called free bitangents of $S_H$.
The set of free bitangents tangent to a given disk partition its boundary into
what we call \emph{arcs}. In other words an arc is a portion of the boundary of
a disk defined by two consecutively tangent bitangents.
A \emph{positive arc} belonging to a disk $d$ is an arc whose tangent lines leave
$d$ on its left. The definition of a \emph{negative arc} is obtained by
replacing the word "left" with "right". Thus for each arc we consider two
copies of it: a positive and a negative one. A \emph{signed arc} is a positive
or a negative arc.
\paragraph{Visibility graphs. }
Given a constraint scene $S_H$, the \emph{visibility graph} of $S_H$ is the
directed graph where there is:
\begin{itemize}
\item one vertex per free bitangent of $S_H$.
\item one edge per signed arc. The edge connects the two vertices
corresponding to the bitangents defining the arc. A counter-clockwise
orientation of the boundary of the disks induce an orientation of the edges
of the visibility graph.
\end{itemize}
Let $e$ be an edge. The two vertices connected by $e$ are denoted $\inf(e)$ and
$\sup(e)$ and are called the source and sink of $e$. These vertices are such
that $e$ is directed from $\inf(e)$ towards $\sup(e)$.
Each vertex of the visibility graph is incident to four edges (two on the source
disk and two on the target disk). The names of these edges are given in the
Figure below.
\begin{ccTexOnly}
\begin{center}
\psfrag{v}{$v$}
\psfrag{ccw_target_edge(v)}{\texttt{ccw\_target\_edge}($v$)}
\psfrag{cw_target_edge(v)}{\texttt{cw\_target\_edge}($v$)}
\psfrag{ccw_source_edge(v)}{\texttt{ccw\_source\_edge}($v$)}
\psfrag{cw_source_edge(v)}{\texttt{cw\_source\_edge}($v$)}
\psfrag{e}{$e$}
\psfrag{sup(e)}{$\sup(e)$}
\psfrag{inf(e)}{$\inf(e)$}
\includegraphics[height=8cm]{fig/edge-vertex.eps}%
\end{center}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<CENTER>
<img src="fig/edge-vertex.gif" alt="Edge-Vertex adjacencies"><P>
</CENTER>
\end{ccHtmlOnly}
\paragraph{Visibility complexes. }
For more sophisticated visibility queries, the visibility complex is more
appropriate than the visibility graph. It has a richer two-dimensional cell
structure whose 1-skeleton (i.e. the graph obtained by keeping only the
vertex/edges incidences) is exactly the visibility graph. The definition of the
visibility complex is more tricky and requires the reader to be familiar with
the concepts of \emph{equivalence classes}, \emph{quotient space} and the
\emph{cutting} of a surface along a curve. The rest of the definitions of this
section can be skipped if one plans to use only the functionality of the
visibility graph (shortest paths for example).
Consider a constrained scene $S_H$. The \emph{free space} of $S_H$ is the
space obtained by cutting the complement of the interior of the disks of $S$
along the constraints of $H$.
A \emph{ray} is a pair $(p,\alpha)$ consisting of a point $p$ in free space and
a direction $\alpha \in \mathbb{S}^1$ (the 1-sphere). The \emph{visibility
complex} of a constrained scene $S_H$ is the quotient space of the set of rays
with respect to the following equivalence relation $\sim$:
\begin{equation}
(p,\alpha) \sim (q,\beta) \textrm{ iff. } \alpha = \beta \textrm{ is the
direction of the line $(p,q)$ and the segment } [p,q] \textrm{ lies in the free
space of } S_H.
\end{equation}
The origins of the rays belonging to a same equivalence class define a maximal
length segment in free space or \emph{maximal segment} for short. The
visibility complex is a two dimensional cell complex. We describe its elements
in the case where the set of constraints is empty. The adjunction of constraints
is discussed later on. The 0,1 and 2-dimensional cells of the visibility complex
are respectively:
\begin{itemize}
\item \emph{Vertices}: a vertex corresponds to a maximal segment tangent to
two disks. Each vertex has its associated geometric bitangent.
\item \emph{Edges}: an edge corresponds to maximal segments tangent to one
disk and touching the same pair of disks at their extremities. \\
Let $e$ be an edge corresponding to maximal segments tangent to a disk $o$.
If $o$ lies on the left of the supporting line of these segments then $e$ is
said to be \emph{positive}. Otherwise $e$ is \emph{negative}.
\item \emph{Faces}: a face is the set of maximal segments that touch the
same pair of disks. The disk containing the source (resp. the target) of
these maximal segments is called the backward view (resp. forward view) of
the face.
\end{itemize}
\begin{ccTexOnly}
\begin{center}
\psfrag{Vertices}{Vertices}
\psfrag{Edge}{Edge}
\psfrag{Face}{Face}
\psfrag{forward view}{forward view}
\psfrag{backward view}{backward view}
\includegraphics[height=5cm,width=\linewidth]{fig/vis-complex.eps}%
\end{center}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<CENTER>
<img src="fig/vis-complex.gif" width="90%"
alt="The cells of the Visibility Complex"><P>
</CENTER>
\end{ccHtmlOnly}
The 1-skeleton of the visibility complex is isomorphic to the visibility graph.
The additional information brought by the visibility complex resides in the
faces. Each maximal segment defines an unique face.
We now describe the different incidences between vertices, edges and faces in
the visibility complex. The discussion below is valid for scene that do not
contain constraints. The adjunction of these involves a small change concerning
edges which is discussed later on.
\begin{itemize}
\item \emph{Face - Vertex incidences. } The number of vertices adjacent to a
given face $\sigma$ is at least $2$ and at most linear. We offer an access
to two vertices of particular interest denoted by $\sup(\sigma)$ and
$\inf(\sigma)$ which are intuitively the segments in the face with maximal
and minimal angle. More precisely the set of maximal segments of a face is
ordered by the following relation:
\begin{equation}
a < b \textrm{ iff. } \det(a,b) > 0.
\end{equation}
The vertices $\inf(\sigma)$ and $\sup(\sigma)$ are respectively the minimum
and maximum for this order. \emph{Warning: convex-hull vertices...}\\
Each vertex $v$ is incident to six faces.
\item \emph{Edge - Face incidences. } An edge $e$ is adjacent to three
faces. These are obtained by taking a maximal segment of the edge and moving
it slightly. See the figure below for the names of these faces.
\begin{ccTexOnly}
\begin{center}
\psfrag{dl(e)}{dl($e$)}
\psfrag{dr(e)}{dr($e$)}
\psfrag{ul(e)}{ul($e$)}
\psfrag{ur(e)}{ur($e$)}
\psfrag{e}{$e$}
\psfrag{Positive edge}{Positive edge}
\psfrag{Negative edge}{Negative edge}
\includegraphics[height=5cm,width=\linewidth]{fig/edge-face.eps}%
\end{center}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<CENTER>
<img src="fig/edge-face.gif" width="90%"
alt="Edge-Face adjacencies"><P>
</CENTER>
\end{ccHtmlOnly}
For a positive edge $e$ we set dr$(e) = $dl$(e)$ and for a negative edge
ur$(e) = $ul$(e)$.
\item \emph{Edge - Vertex incidences. } These incidences were described in
the Visibility Graph section above.
\end{itemize}
We now describe the two main differences brought by the adjunction of
constraints (a more detailed and complete discussion can be found
in~\cite{G-ap-sstvc-01}). First, the vertices corresponding to constraints are
adjacent to $8$ faces rather than $6$ before. The other difference is that the
visibility complex also contains special types of edges arousing from rays
emanating from the extremity of a constraint. Such an edge $e$ is called a
\emph{constraint edge} and is adjacent to only one face denoted by face$(e)$.
See the Figure below for an illustration.
\begin{ccTexOnly}
\begin{center}
\psfrag{face(e)}{face($e$)}
\psfrag{e}{$e$}
\psfrag{constraint}{constraint}
\includegraphics[height=5cm]{fig/constraint-edge.eps}%
\end{center}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<CENTER>
<img src="fig/constraint-edge.gif" alt="Constraint Edges"><P>
</CENTER>
\end{ccHtmlOnly}
\paragraph{Visibility complex antichains. } The antichain is an advanced feature
for users wishing to take advantage of the linear storage of the Greedy Flip
Algorithm which we have implemented to compute the visibility complex. In this
section we will only give an intuitive definition of the antichain. More details
can be found in~\cite{pv-tsvcpt-96}.
The Greedy Flip Algorithm, or \textsc{Gfa} for short, is a topological sweep
algorithm. To keep the description on the intuitive side imagine the visibility
complex embedded in $\mathbb{R}^3$. The \textsc{Gfa} sweeps the complex using a
plane. The set of faces and edges intersected by this plane is what we call the
antichain.
We have added an iterator interface to greatly ease the use of the antichain.
The small piece of code below reads a list of segments from a file and outputs
the vertices of the visibility complex in optimal time and $O(n)$ storage.
\ccIncludeExampleCode{Visibility_complex/example1.C}
The antichain is closely connected to so-called greedy pseudo-triangulations.
This is illustrated in an example below but bare in mind that this package does
not aim to provide a convivial interface to pseudo-triangulations. An extension
package concerning pseudo-triangulations should appear in the future. The
interested user is invited to contact the authors.
\paragraph{Degenerate cases.} The above discussion holds for disks which have
non empty interior and a smooth boundary. However, our implementation can handle
points, segments and convex polygons which are what we call degenerate cases.
This is achieved via a symbolic perturbation obtained by taking the Minkowski sum
of a degenerate disk with a small circle having an infinitesimal radius.
\begin{ccTexOnly}
\begin{center}
\includegraphics[height=4cm]{fig/perturbation.eps}%
\end{center}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<CENTER>
<img src="fig/perturbation.gif" alt="Perturbation"><P>
</CENTER>
\end{ccHtmlOnly}
This perturbation implies no numerical computations and is completely
invisible to the user. Each time a scene contains a degenerate disk, the user
is invited to mentally replace it by its perturbated version in order to
visualize the visibility complex or graph.
\begin{ccTexOnly}
\begin{center}
\psfrag{Perturbation}{Perturbation}
\includegraphics[height=4cm]{fig/points.eps}%
\end{center}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<CENTER>
<img src="fig/points.gif" alt="Points"><P>
</CENTER>
\end{ccHtmlOnly}
As an example, consider two points as in the figure above. They share four
bitangents which are geometrically equal. These four bitangent are nevertheless
represented by four different vertices in the visibility graph. The order of the
tangent points of these bitangents on the "boundary" of the point is determined
by the order of these tangent points if the points were replaced by small
circles. Considering the crossing predicate, the \texttt{LR} and \texttt{RL}
bitangents is the only crossing pair in the figure above. Also bear in mind
that the presence of degenerate cases may imply arcs with zero length.
Finally we explain how the general position assumption is dealt with.
\emph{A finir ...}
\paragraph{General polygonal configurations. } Recall that disks, as they are defined
above, are convex. We now describe how to treat not only non convex polygons but
also more general polygonal configurations. Consider the general setting consisting in a
family $S$ of pairwise non crossing segments which are allowed to share an
extremity. The set $S$ can for example be the edges of a non-convex polygon.
We use the following recipe to build a scene $T$ respecting our requirements:
\begin{itemize}
\item Each extremity of a segment in $S$ is inserted as a point in $T$.
Recall that points are treated symbolically as if they were small circles
(see the paragraph above concerning degenerate cases).
\item Each segment $s \in S$ directed from point $p$ to point $q$, with $p,q
\in T$ is inserted in $T$ as a constraint directed from $p$ to $q$. The type
chosen for the constraint (recall that four types are possible) is
\texttt{RL}. The reason for this choice is to avoid two crossing constraints.
\end{itemize}
The Figure below shows a polygonal configuration $S$ and the scene $T$ which
has to be given as an input to our algorithms.
\begin{ccTexOnly}
\begin{center}
\psfrag{S}{$S$}
\includegraphics[height=8cm]{fig/configuration.eps}%
\end{center}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<CENTER>
<img src="fig/configuration.gif" alt="Polygonal configurations"><P>
</CENTER>
\end{ccHtmlOnly}
The aim of the \ccc{Visibility_complex_scene} class is to make the above recipe
more transparent. In fact, one can blindly use this class to compute visibility
complexes of polygonal configurations without understanding the above
discussion. The code below illustrates the use of this class.
\ccIncludeExampleCode{Visibility_complex/example2.C}
\paragraph{Output / Partial order. }
\label{sectionVComplexInputOutput}
The vertices of the visibility complex are
accessed in our implementation via an iterator pair \texttt{vertices\_begin(),
vertices\_end()}. We now describe how these vertices are ordered. A natural idea
would be an order based on the angle of the bitangent with respect to a
horizontal direction. Unfortunately obtaining such an order would require at
least $O(k \log n)$ in time complexity, where $k$ is the size of the output and
$n$ the size of the input. Since we achieve an optimal bound of $O(k + n\log
n)$, our order is slightly more subtle than an angle-based order. It is a linear
extension of a partial order $\prec$. In other words if $a \prec b$ then $a$
appears before $b$. Moreover the partial order is compatible with angle-order:
if $a \prec b$ then the angle of $a$ with respect to a horizontal direction is
smaller than the angle of $b$. The converse is \emph{not} necessarily true.
Let $S_H$ be a constrained scene containing $n$ disks; we define the partial
order $\prec$ as the transitive closure of $2n$ total orders (two per disk). \\
Let $B_o$ be the set of free bitangents tangent to $o$ and such that $o$ lies
on the half-plane to the left of their supporting lines. The angle of a
bitangent with respect to a horizontal direction (it is a real number in
$[0,2\pi)$) defines a total order on $B_o$. See the left part of the figure below.
\begin{ccTexOnly}
\begin{center}
\psfrag{a}{\textcolor{red}{$a$}}
\psfrag{b}{\textcolor{red}{$b$}}
\psfrag{o}{$o$}
\psfrag{angle}{\textcolor{green}{angle}}
\includegraphics[width=\textwidth]{fig/order.eps}%
\end{center}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<CENTER>
<img src="fig/order.gif" alt="Partial order"><P>
</CENTER>
\end{ccHtmlOnly}
Changing the word "left" with "right" in the above paragraph defines the set
$B'_o$ and a total order on $B'_o$. We have thus defined two total orders per
disk in $S_H$. Since each bitangent is defined by two disks, it appears exactly
in two total orders. This allows us to mix the total orders on each disk to
define the partial order $\prec$. More formally $\prec$ is the transitive
closure of the $2n$ total orders defined on $B_o, B'_o$ for each $o \in S_H$.
The right part of the figure above shows an example where $a \prec b$.
% +============================================================================+
\section{Software Design}
% +============================================================================+
\begin{ccTexOnly}
\begin{figure}
\begin{center}
\parbox{0.7\textwidth}{%
\includegraphics[width=0.7\textwidth]{fig/design.eps}%
}
\end{center}
\caption{Three layer design of the visibility complex.}
\label{figureVisibilityComplexDesign}
\end{figure}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<CENTER>
<A NAME="figureVisibilityComplexDesign">
<img src="fig/design.gif"
alt="Visibility Complex Design"><BR>
Figure: Three layer design of the visibility complex.
<P>
</CENTER>
\end{ccHtmlOnly}
Figure~\ccTexHtml{\ref{figureVisibilityComplexDesign}}{}
\begin{ccHtmlOnly}
<A HREF="Chapter_main.html#figureVisibilityComplexDesign"><IMG
SRC="cc_ref_up_arrow.gif" ALT="reference arrow" WIDTH="10" HEIGHT="10"></A>
\end{ccHtmlOnly}
illustrates the responsibilities of the three layers of the software design.
The bottom layer is the geometric traits class \ccc{Traits} which defines the
geometric types (bitangents disks and arcs) and the predicates used to compute
the visibility complex. Currently four different traits classes are provided:
\begin{itemize}
\item \ccc{Visibility_complex_point_traits<R>} for use with
\ccc{CGAL::Point_2}.
\item \ccc{Visibility_complex_circle_traits<R>} for use with
\ccc{CGAL::Circle_by_radius_2}. The class \ccc{CGAL::Circle_by_radius_2}
inherits from \ccc{CGAL::Circle_2} and is implemented in the file:
\begin{quote}
\ccInclude{CGAL/Circle_by_radius_2.h}
\end{quote}
This class adds functionality to manipulate circles with their \emph{radius}
instead of their \emph{squared radius}.
\item \ccc{Visibility_complex_segment_traits<R>} for use with
\ccc{CGAL::Segment_2}.
\item \ccc{Visibility_complex_polygon_traits<R>} for use with
\ccc{CGAL::Polygon_2}.
\end{itemize}
The next layer is the \ccc{Items} class who defines, as local types, the
elements of the visibility complex. The user can enrich those classes by adding
his own data and member functions. The design used here is the same one used by
the \ccc{HalfedgeDS} in chapter~\ref{chapterHalfedgeDS}.
The two classes \ccc{Traits} and \ccc{Items} are passed as template parameters
to the class \ccc{Visibility_complex<Traits,Items>} who implements the
visibility complex structure. This former class defines the handle types which
are used to code the incidences between vertices, edges and faces.
The visibility complex structure provides three pairs of iterators to traverse
its set of vertices, edges and faces. These are ordered as a linear extension of
a certain partial order $\prec$. See the last paragraph in
section~\ref{sectionVComplexInputOutput}.
% +============================================================================+
\section{Example Programs}
% +============================================================================+
\label{sectionVComplexExamples}
\subsection{Visibility complex of circles}
The following program reads a family of pairwise disjoint circles from the file
"input" and outputs the number of vertices in the visibility complex.
\ccIncludeExampleCode{Visibility_complex/vc_circle.C}
\subsection{Extending Faces}
The following example adds a \ccc{size} member variable to count the size of the
face. We use the \ccc{Antichain} to sweep the visibility complex while using
only $O(n)$ storage. The disks are of type \ccc{CGAL::Segment\_2} and $n$ is the
number of them.
\ccIncludeExampleCode{Visibility_complex/vc_face_extension.C}
% +----------------------------------------------------------------------------+
% EOF

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#include <CGAL/Cartesian.h>
#include <CGAL/Visibility_complex_2.h>
#include <CGAL/Visibility_complex_circle_traits.h>
typedef CGAL::Cartesian<double> Rep;
typedef CGAL::Visibility_complex_circle_traits<Rep> Traits;
typedef Traits::Disk Circle;
typedef CGAL::Visibility_complex<Traits> VComplex;
int main()
{
std::list<Circle> D;
// Reading circles from file
std::ifstream ifs("input");
std::istream_iterator<Circle> ifs_it(ifs),ifs_end;
std::copy(ifs_it,ifs_end,back_inserter(D));
// Computing Visibibility Complex
VComplex V(D.begin(),D.end());
// Sice of V
int size = 0;
distance(V.vertices_begin(),V.vertices_end(),size);
std::cout << size << std::endl;
return 0;
}

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#include <CGAL/Simple_cartesian.h>
#include <CGAL/Visibility_complex_segment_traits.h>
#include <CGAL/Visibility_complex_items.h>
#include <CGAL/Visibility_complex_2.h>
typedef CGAL::Gmpz FT;
typedef CGAL::Simple_cartesian<FT> Rep;
typedef CGAL::Visibility_complex_segment_traits<Rep> Gt;
typedef Gt::Disk Segment;
// ---------------------------------------------------------------------
template <class V>
struct My_face : public CGAL::Visibility_complex_face_base<V>
{
int size;
My_face() : size(0) { }
};
struct My_items : public CGAL::Visibility_complex_items
{
template <class V>
struct Face_wrapper {
typedef My_face<V> Face;
};
};
// ---------------------------------------------------------------------
typedef CGAL::Visibility_complex_2<Gt,My_Items> Visibility_complex;
typedef Visibility_complex::Antichain Antichain;
typedef Visibility_complex::Linear_sweep_iterator Linear_sweep_iterator;
typedef Visibility_complex::Edge_handle Edge_handle;
// ---------------------------------------------------------------------
// For a positive edge the three adjacent faces of e are
// dl(e) , ur(e) , ul(e)
// Otherwise the three adjacent faces are
// dl(e) , dr(e) , ul(e)
void increment(Edge_handle e)
{
if (e->sign()) {
++e->dl()->size;
++e->ul()->size;
++e->ur()->size;
}
else {
++e->dl()->size;
++e->dr()->size;
++e->ul()->size;
}
}
// ---------------------------------------------------------------------
int main()
{
std::list<Segment> D;
// Reading segments from file
std::ifstream ifs("input");
std::istream_iterator<Segment> ifs_it(ifs),ifs_end;
std::copy(ifs_it,ifs_end,back_inserter(D));
// Computing the initia; antichain to sweep the complex
Antichain A(D.begin(),D.end());
Linear_sweep_iterator v(&A) , vend(&A,0);
for ( ; v != vend ; ++v)
{
// The two edges cw_source_edge(v) and cw_target(v) are being
// swept. We increment their three adjacent faces.
increment(v->cw_source_edge());
increment(v->cw_target_edge());
// The face inf(v) has been completely swept. We print its size
std::cout << v->inf()->size << std::endl;
}
return 0;
}

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% +------------------------------------------------------------------------+
% | Reference manual page: Arc.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +------------------------------------------------------------------------+
\begin{ccRefConcept}{Arc}
\label{pageVCArcRef}
% ------------------------------------------------------------------------------
\ccDefinition
The \ccRefName\ concept defines the minimal requirement for the
\ccc{Arc_2} type of the \ccc{VisibilityComplexTraits} concept. It
models an \emph{arc} (see the Introduction), i.e. a connected portion of a disk.
The \ccRefName\ concept is introduced for efficiency reasons. When the disks
have constant complexity, the implementation of \ccRefName\ is trivial. The
\ccc{Arc} type is used as a base type for the concept VisibilityComplexEdge
\ccIndexMainItem[c]{VisibilityComplexEdge}.
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccTypes
\ccThree{Disk_handle}{Disk_handle}{}
\ccThreeToTwo
\ccNestedType{Arc_handle}{handle to an arc. }
\ccGlue
\ccNestedType{Disk}{type of disks defining the arc. }
\ccGlue
\ccNestedType{Disk_handle}{handle to a disk.}
\ccGlue
\ccNestedType{Point_2}{the type of a point belonging to a \ccc{Disk}.}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccThree{Disk_handle}{ a.object(); }{}
\ccThreeToTwo
\ccCreation
\ccCreationVariable{a}
\ccConstructor{Arc();}{default constructor.}
\ccGlue
\ccConstructor{Arc(Disk_handle d);}
{ Creates an arc representing the complete boundary of the disk \ccc{d}. }
% ------------------------------------------------------------------------------
\ccOperations
\ccThree{Disk_handle}{ a.split (Arc_handle tmp, Point_2 p); }{}
\ccTagFullDeclarations
\ccMethod{Disk_handle object(); const}
{returns a handle to the disk containing \ccVar\ .}
\ccGlue
\ccMethod{void set_object(Disk_handle d); }
{sets \ccc{d} as the disk containing \ccVar\ .}
\ccGlue
\ccMethod{void split (Arc_handle tmp, Point_2 p);}
{splits the arc \ccVar\ at point \ccc{p}. In other words, the target (resp. source)
of \ccVar\ (resp. \ccc{tmp}) becomes \ccc{p} and the target of \ccc{tmp} becomes
the previous target of \ccVar.}
\ccGlue
\ccMethod{void join (Arc_handle y);}
{joins the two arcs \ccVar\ and \ccc{y}. That is, the target of \ccVar\ becomes
the target of \ccc{y} and the arc \ccc{y} is destroyed.}
% ------------------------------------------------------------------------------
\ccHasModels
\ccRefIdfierPage{CGAL::Arc_2<Disk>}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccSeeAlso
\ccRefConceptPage{VisibilityComplexVertex}\\
\ccRefConceptPage{VisibilityComplexTraits}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccTagDefaults
\end{ccRefConcept}
\ccRefPageEnd
% +------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Arc_2.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +------------------------------------------------------------------------+
\begin{ccRefClass}{Arc_2<Disk>}
\label{pageArc_2}
% ------------------------------------------------------------------------------
\ccDefinition
The class \ccRefName\ is a model for the \ccc{Arc} concept. \ccc{Disk} is
the disk type.
Partial specializations of \ccRefName\ are given for the types
\ccc{CGAL::Point_2<R>}, \ccc{CGAL::Segment_2<R>}, \ccc{CGAL::Circle_2<R>} and
\ccc{CGAL::Polygon_2<R>}.
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccInclude{CGAL/Arc_2.h}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccIsModel
\ccRefConceptPage{Arc}
% ------------------------------------------------------------------------------
\end{ccRefClass}
% +------------------------------------------------------------------------+
%%RefPage: end of main body, begin of footer
\ccRefPageEnd
% EOF
% +------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Bitangent.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +------------------------------------------------------------------------+
\begin{ccRefConcept}{Bitangent}
\label{pageBitangentRef}
% ------------------------------------------------------------------------------
\ccDefinition
The \ccRefName\ concept defines the minimal requirement for the local
\ccc{Bitangent} type of the \ccc{VisibilityComplexTraits} concept. The
\ccc{Bitangent} type is used as a base class for the concept
VisibilityComplexVertex \ccIndexMainItem[c]{VisibilityComplexVertex}.
A bitangent is a directed line segment tangent to two disks. The relative
positions of these two disks with respect to the supporting line of the
bitangents are coded in the \ccc{Type} information. There are four possible
types: \ccc{RR, LL, RL} and \ccc{LR}.
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccTypes
%\ccThree{aaaaaaaaBitangent_2}{aaaaaaaaDisk_handle}{}
%\ccThreeToTwo
\ccNestedType{Disk}{type of disks defining the bitangent. }
\ccGlue
\ccNestedType{Disk_handle}{handle to a \ccc{Disk}. }
\ccGlue
\ccNestedType{Arc}{type following the requirements of the \ccc{Arc} concept
\ccIndexMainItem[c]{Arc}].}
\ccGlue
\ccNestedType{Arc_handle}{handle to an \ccc{Arc}. }
\ccGlue
\ccNestedType{Type}{enumerated type to specify the type of the bitangent. The
possible values are \ccc{LL,RR,RL} or \ccc{LR} respectively
for left-left,right-right,right-left and left-right
bitangents.}{}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccThree{Disk_handle}{ b.source_object(); }{}
\ccThreeToTwo
\ccCreation
\ccCreationVariable{b}
\ccConstructor{Bitangent();}{default constructor.}
\ccGlue
\ccConstructor{Bitangent(Type t, Disk_handle source, Disk_handle target);}
{ Creates the bitangent with type t directed from source to target. }
\ccGlue
\ccConstructor{Bitangent(Type t, Arc_handle source, Arc_handle target);}
{ Creates the bitangent with type t directed from the arc source to the arc
target. If the two arcs do not have a common bitangent, the constructor is equal
to the default constructor. }
% ------------------------------------------------------------------------------
\ccOperations
\ccTagFullDeclarations
\ccMethod{Type type();}{returns the type of \ccVar.}
\ccGlue
\ccMethod{Disk_handle source_object();}{returns a handle to the source disk.}
\ccGlue
\ccMethod{Disk_handle target_object();}{returns a handle to the target disk.}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccHasModels
\ccRefIdfierPage{CGAL::Bitangent_2<Disk>}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccSeeAlso
\ccRefConceptPage{VisibilityComplexVertex}\\
\ccRefConceptPage{VisibilityComplexTraits}
% ------------------------------------------------------------------------------
% +------------------------------------------------------------------------+
\ccTagDefaults
\end{ccRefConcept}
\ccRefPageEnd
% +------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Bitangent_2.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +------------------------------------------------------------------------+
\begin{ccRefClass}{Bitangent_2<Disk>}
\label{pageBitangent_2}
% ------------------------------------------------------------------------------
\ccDefinition
The class \ccRefName\ is a model for the \ccc{Bitangent} concept. \ccc{Disk} is
the disk type.
Partial specializations of \ccRefName\ are given for the types
\ccc{CGAL::Point_2<R>}, \ccc{CGAL::Segment_2<R>}, \ccc{CGAL::Circle_2<R>} and
\ccc{CGAL::Polygon_2<R>}.
Adding a partial specialization is done by inheriting from the class
\ccc{CGAL::Bitangent_base<Disk>} and implementing the constructors described in
the \ccc{Bitangent} concept.
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccInclude{CGAL/Bitangent_2.h}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccInheritsFrom
\ccRefIdfierPage{CGAL::Bitangent_base<Disk>}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccIsModel
\ccRefConceptPage{Bitangent}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccTypes
\ccThree{const Point&;;}{source() const;MMMMMMMMMMMM}{}
\ccThreeToTwo
\ccNestedType{Point_2}{point type.}
% No longer required.
%\ccTypedef{typedef Traits::Plane_3 Plane_3;}
% {plane equation type for three argument version.}
%\ccGlue
%\ccTypedef{typedef Traits::Vector_3 Vector_3;}
% {normal vector of plane equation.}
\ccCreation
\ccCreationVariable{b}
\ccOperations
\ccTagFullDeclarations
\ccMethod{Point_2 source();}{returns the source of the bitangent.}
\ccGlue
\ccMethod{Point_2 target();}{returns the target of the bitangent.}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccSeeAlso
\ccRefIdfierPage{CGAL::Bitangent_base<Disk>}
% ------------------------------------------------------------------------------
% +----------------------------------------------------------------------------+
\ccTagDefaults
\end{ccRefClass}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Bitangent_base.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +------------------------------------------------------------------------+
\begin{ccRefClass}{Bitangent_base<Disk>}
\label{pageBitangentBaseRef}
% ------------------------------------------------------------------------------
\ccDefinition
The class \ccRefName\ defines the combinatorial layer of a bitangent. This class
is meant to simplify the implementation of new partial specializations of the
\ccc{Bitangent\_2<Disk>} class.
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccInclude{CGAL/Bitangent_2.h}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccIsModel
\ccRefConceptPage{Bitangent}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccSeeAlso
\ccRefIdfierPage{CGAL::Bitangent_2<Disk>}
% ------------------------------------------------------------------------------
\end{ccRefClass}
% +------------------------------------------------------------------------+
%%RefPage: end of main body, begin of footer
\ccRefPageEnd
% EOF
% +------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: ShortestPathTraits.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
% +----------------------------------------------------------------------------+
\begin{ccRefConcept}{ShortestPathTraits}
\label{pageShortestPathTraitsRef}
% +----------------------------------------------------------------------------+
\ccDefinition
The concept \ccRefName\ describes the set of requirements to be fulfilled by
any class used to instantiate the last parameter in the shortest\_path\_2
\ccIndexMainItem[c]{shortest_path_2} function. This function computes the shortest path between
two points in a scene.
This concept inherits all the types and requirement from the
\ccc{VisibilityComplexTraits} concept and adds a few additional primitives used
specifically for shortest path computations.
We distinguish two different number types: the one used for the input data and
the one used for the distance computations during our algorithm. The latter type
has to support the square root operation whereas the first type doesn't (i.e. it
can be of integer type).
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccTypes
\ccThree{ShortestPathTraits::Exact_NT }{}{}
\ccThreeToTwo
\ccNestedType{Exact_NT}{the number type used to compute distances. It must
support the square root operation.}
\ccGlue
\ccNestedType{Disk}{the disk type.}
\ccGlue
\ccNestedType{Arc_2}{the arc type. Model of the \ccc{Arc} concept.}
\ccGlue
\ccNestedType{Bitangent_2}{the bitangent type. Model of the \ccc{Bitangent}
concept.}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccCreation
\ccCreationVariable{traits}
\ccConstructor{ShortestPathTraits();} {default constructor.}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccOperations
\ccTagFullDeclarations
\ccMethod{Disk make_disk(Point_2 p);}{method to create a disk from a point.}
\ccGlue
\ccMethod{bool is_vertex(Disk d);}{returns true if \ccc{d} is a point. }
\ccGlue
\ccMethod{Exact_NT length(Arc_2 a, Bitangent_2 b, Bitangent_2 c);}
{returns the length of the arc \ccc{a} defined by the two bitangents \ccc{b} and
\ccc{c}.}
\ccGlue
\ccMethod{Exact_NT length(Bitangent_2 b);}{return the length of the bitangent
\ccc{b}.}
\ccTagDefaults
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccHasModels
\ccRefIdfierPage{CGAL::Shortest_path_point_traits<R,E>} \\
\ccRefIdfierPage{CGAL::Shortest_path_segment_traits<R,E>} \\
\ccRefIdfierPage{CGAL::Shortest_path_polygon_traits<R,E>}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\end{ccRefConcept}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Shortest_path_2.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\begin{ccRefFunction}{shortest_path_2}
\label{pageShortest_path_2Ref}
% ------------------------------------------------------------------------------
\ccDefinition
Function that computes the shortest path between two points $p,q$ in a planar
scene (see the basic definitions paragraph in the introduction for what we
understand by a scene). The scene is given as two pairs of iterators describing
the set of disks and constraints of the scene.
The output iterator returned by the function is an iterator on the vertices of
the shortest path. Here a vertex is a vertex of the visibility complex of the
scene plus the two points $p,q$. Recall that vertices correspond to free
bitangents.
Currently only scene containing points, segments or polygons are supported. In
other words circles are not.
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccInclude{CGAL/Shortest_path_2.h}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccFunction{
template < class DiskIterator , class ConstraintIterator ,
class OutputIterator , class Traits >
OutputIterator shortest_path_2(DiskIterator first ,
DiskIterator last ,
ConstraintIterator firstc,
ConstraintIterator lastc ,
Traits::Point_2 p , Traits::Point_2 q ,
OutputIterator result ,
Traits t);
}
{
computes the shortest path between \ccc{p} and \ccc{q} in the scene containing
the disks in the range [\ccc{first},\ccc{last}) and the constraints in the range
[\ccc{firstc},\ccc{lastc}). The past-the-end iterator for the list of vertices
of the shortest path is returned. Here vertex refers to the visibility complex
of the input scene plus the two points \ccc{p} and \ccc{q}.
}
\ccFunction{
template < class DiskIterator , class OutputIterator , class Traits >
OutputIterator shortest_path_2(DiskIterator first ,
DiskIterator last ,
Traits::Point_2 p , Traits::Point_2 q ,
OutputIterator result ,
Traits t);
}
{
Same as the previous function but with an empty list of constraints.
}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccHeading{Requirements}
\begin{enumerate}
\item \ccc{Traits} is a model for the concept ShortestPathTraits.
\ccIndexMainItem[c]{ShortestPathTraits}
\item \ccc{OutputIterator::value_type} should be the same as
\ccc{ConstraintIterator::value_type} and the same as
\ccc{Visibility_complex_2<Traits>::Vertex}.
\item \ccc{DiskIterator::value_type} should be \ccc{Traits::Disk}.
\end{enumerate}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccSeeAlso
\ccRefIdfierPage{CGAL::Shortest_path_point_traits<R,E>} \\
\ccRefIdfierPage{CGAL::Shortest_path_segment_traits<R,E>} \\
\ccRefIdfierPage{CGAL::Shortest_path_polygon_traits<R,E>}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccImplementation
The visibility graph of the input scene plus the two points is computed in time
$O(k + n \log n)$ and $O(k)$ space where $n$ and $k$ are respectively the size
of the input and output.
A Dijsktra algorithm is then applied to get the shortest path. We have
implemented the Dijsktra algorithm using a \ccc{stl::set} thus resulting in a
$O(k \log n)$ time complexity.
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccExample
The following program read a list of pairwise disjoint segments from the file
\texttt{segments} and the outputs the bitangent segments of the shortest path to
standard output in a one-per-line fashion.
\ccIncludeExampleCode{Visibility_complex_ref/shortest_path.C}
% ------------------------------------------------------------------------------
% +----------------------------------------------------------------------------+
\ccTagDefaults
\end{ccRefFunction}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Shortest_path_point_traits.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\begin{ccRefClass}{Shortest_path_point_traits<R,E>}
\label{pageShortest_path_point_traitsRef}
% ------------------------------------------------------------------------------
\ccDefinition
The class \ccRefName\ is the predefined geometric traits class provided by
\cgal\ for use with disks whith type \ccc{CGAL::Point_2<R>}.
This class is to be used as last argument of the \ccc{shortest_path_2} function
which computes the shortest path between two points in a scene.
This class is templated with a representation class \ccc{R} and a number type
\ccc{E} which supports the square root operation. The number type \ccc{E} is
used to compute distances between pairs of points.
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccInclude{CGAL/Shortest_path_point_traits.h}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccInheritsFrom
\ccRefIdfierPage{CGAL::Visibility_complex_point_traits<R>}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccIsModel
\ccRefConceptPage{ShortestPathTraits}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccSeeAlso
\ccRefIdfierPage{CGAL::Shortest_path_segment_traits<R,E>} \\
\ccRefIdfierPage{CGAL::Shortest_path_polygon_traits<R,E>}
% ------------------------------------------------------------------------------
% +----------------------------------------------------------------------------+
\ccTagDefaults
\end{ccRefClass}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Shortest_path_polygon_traits.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\begin{ccRefClass}{Shortest_path_polygon_traits<R,E>}
\label{pageShortest_path_polygon_traitsRef}
% ------------------------------------------------------------------------------
\ccDefinition
The class \ccRefName\ is the predefined geometric traits class provided by
\cgal\ for use with disks whith type \ccc{CGAL::Polygon_2<R>}.
This class is to be used as last argument of the \ccc{shortest_path_2} function
which computes the shortest path between two points in a scene.
This class is templated with a representation class \ccc{R} and a number type
\ccc{E} which supports the square root operation. The number type \ccc{E} is
used to compute distances between pairs of points.
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccInclude{CGAL/Shortest_path_polygon_traits.h}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccInheritsFrom
\ccRefIdfierPage{CGAL::Visibility_complex_polygon_traits<R>}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccIsModel
\ccRefConceptPage{ShortestPathTraits}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccSeeAlso
\ccRefIdfierPage{CGAL::Shortest_path_segment_traits<R,E>} \\
\ccRefIdfierPage{CGAL::Shortest_path_point_traits<R,E>}
% ------------------------------------------------------------------------------
% +----------------------------------------------------------------------------+
\ccTagDefaults
\end{ccRefClass}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Shortest_path_segment_traits.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\begin{ccRefClass}{Shortest_path_segment_traits<R,E>}
\label{pageShortest_path_segment_traitsRef}
% ------------------------------------------------------------------------------
\ccDefinition
The class \ccRefName\ is the predefined geometric traits class provided by
\cgal\ for use with disks whith type \ccc{CGAL::Segment_2<R>}.
This class is to be used as last argument of the \ccc{shortest_path_2} function
which computes the shortest path between two points in a scene.
This class is templated with a representation class \ccc{R} and a number type
\ccc{E} which supports the square root operation. The number type \ccc{E} is
used to compute distances between pairs of points.
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccInclude{CGAL/Shortest_path_segment_traits.h}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccInheritsFrom
\ccRefIdfierPage{CGAL::Visibility_complex_segment_traits<R>}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccIsModel
\ccRefConceptPage{ShortestPathTraits}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccSeeAlso
\ccRefIdfierPage{CGAL::Shortest_path_polygon_traits<R,E>} \\
\ccRefIdfierPage{CGAL::Shortest_path_point_traits<R,E>}
% ------------------------------------------------------------------------------
% +----------------------------------------------------------------------------+
\ccTagDefaults
\end{ccRefClass}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: VisibilityComplex.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +----------------------------------------------------------------------------+
\begin{ccRefConcept}{VisibilityComplex}
\label{pageVisibilityComplexRef}
% +----------------------------------------------------------------------------+
\ccDefinition
The class \ccRefName\ defines a vertex centered data structure capable of
maintaining the incidence between vertices, edges and faces of the visibility
complex of a collection of $n$ disks. In addition the \ccRefName\ maintains an
\ccc{VisibilityComplexAntichain} used for sweeping the visibility complex. The
\ccRefName\ offers iterators pairs for traversing the vertices,edges and faces
of the visibilty complex.
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccTypes
\ccTwo{VisibilityComplex:: Vertex_iterator }{}
\ccNestedType{Gt} {the geometric traits class.}
\ccGlue
\ccNestedType{Items} {items class, model of the \ccc{VisibilityComplex_items}
concept.}
\ccGlue
\ccNestedType{Antichain} {the antichain used for sweeping the visibility
complex, model for the
\ccc{VisibilityComplexAntichain} concept.}
\ccGlue
\ccNestedType{Vertex}{vertex type. Model of \ccc{VisibilityComplexVertex}.}
\ccGlue
\ccNestedType{Edge}{edge type. Model of \ccc{VisibilityComplexEdge}.}
\ccGlue
\ccNestedType{Face}{face type. Model of \ccc{VisibilityComplexFace}.}
\ccNestedType{Vertex_handle}{handle to vertex.}
\ccGlue
\ccNestedType{Edge_handle}{handle to edge.}
\ccGlue
\ccNestedType{Face_handle}{handle to face.}
\ccNestedType{Vertex_iterator}{iterator over all vertex.}
\ccGlue
\ccNestedType{Edge_iterator}{iterator over all edges.}
\ccGlue
\ccNestedType{Face_iterator}{iterator over all faces.}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccCreation
\ccCreationVariable{v}
\ccConstructor{VisibilityComplex();}{default constructor.}
\ccConstructor{VisibilityComplex(InputIterator first, InputIterator last,
ConstraintIt firstc,ConstraintIt lastc);}
{creates the visibility complex of the disks in the range
$[$\ccc{first,last}$)$ and constraints in the range $[$\ccc{firstc,lastc}$)$.
The value type of \ccc{ConstraintIt} is \ccc{Vertex}. }
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccOperations
\ccTagFullDeclarations
\ccMethod{Vertex_iterator vertices_begin();}{}
\ccGlue
\ccMethod{Vertex_iterator vertices_end();}{iterator over all vertices.}
\ccGlue
\ccMethod{Edge_iterator edges_begin();}{}
\ccGlue
\ccMethod{Edge_iterator edges_end();}{iterator over all edges.}
\ccGlue
\ccMethod{Face_iterator faces_begin();}{}
\ccGlue
\ccMethod{Face_iterator faces_end();}{iterator over all faces.}
\ccGlue
\ccTagDefaults
% +----------------------------------------------------------------------------+
% ------------------------------------------------------------------------------
\ccHasModels
\ccRefIdfierPage{CGAL::Visibility_complex_2<Gt,It>}
% ------------------------------------------------------------------------------
% +----------------------------------------------------------------------------+
\ccSeeAlso
\ccRefConceptPage{VisibilityComplexAntichain} \\
\ccRefConceptPage{VisibilityComplexTraits} \\
\ccRefConceptPage{VisibilityComplexVertex}\\
\ccRefConceptPage{VisibilityComplexEdge}\\
\ccRefConceptPage{VisibilityComplexFace}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\end{ccRefConcept}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: VisibilityComplexAntichain.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +----------------------------------------------------------------------------+
\begin{ccRefConcept}{VisibilityComplexAntichain}
\label{pageVisibilityComplexAntichainRef}
% +----------------------------------------------------------------------------+
\ccDefinition
The concept of a \ccRefName\ defines the sweep structure used to sweep the
visibility complex. It is a planar st-graph: a node corresponds to an edge of
visibility complex and an edge corresponds to a face of the visibility complex.
The antichain maintains a list of potential candidates which are ready to be
swept.
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccTypes
\ccTwo{VisibilityComplexAntichain:: Vertex_iterator }{}
\ccNestedType{Gt}{the geoemtric traits class. Model of a \ccc{VisibilityComplexTraits}. }
\ccGlue
\ccNestedType{Disk}{the disk type. Same as \ccc{VisibilityComplexTraits::Disk}. }
\ccGlue
\ccNestedType{Vertex}{vertex, model for \ccc{VisibilityComplexVertex}\
concept.}
\ccGlue
\ccNestedType{Edge}{edge, model for \ccc{VisibilityComplexEdge}\ concept.}
\ccGlue
\ccNestedType{Face}{face, model for \ccc{VisibilityComplexFace}\ concept.}
\ccNestedType{Vertex_handle}{handle to vertex.}
\ccGlue
\ccNestedType{Edge_handle}{handle to edge.}
\ccGlue
\ccNestedType{Face_handle}{handle to face.}
\ccNestedType{Vertex_iterator}{iterator over all vertices of the antichain.}
\ccGlue
\ccNestedType{Edge_iterator}{iterator over all edges of the antichain.}
\ccGlue
\ccNestedType{Face_iterator}{iterator over all faces of the antichain.}
\ccGlue
\ccNestedType{Minimals_iterator}{iterator over all the minimal vertices of the antichain. }
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccCreation
\ccCreationVariable{a}
\ccConstructor{VisibilityComplex_Antichain();}{default constructor.}
\ccConstructor{VisibilityComplex_Antichain(InputIterator first, InputIterator last,
ConstraintIt firstc, ConstraintIt lastc);}
{creates the antichain of the disks in the range $[$\ccc{first,last}$)$ and
constraints in the range $[$\ccc{firstc,lastc}$)$ corresponding to the filter
$I(0)$ of bitangents with positive slope. The value type of \ccc{InputIterator}
must be \ccc{Disk} and the value type of \ccc{ConstraintIt} must be
\ccc{Vertex}. }
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccOperations
\ccThree{Vertex_iterator}{a.vertices_cw_begin(); }{}
\ccThreeToTwo
\ccTagFullDeclarations
\ccMethod{Vertex_iterator vertices_begin();}{iterator over all the sinks of the
faces of the antichain. }
\ccGlue
\ccMethod{Vertex_iterator vertices_end();}{}
\ccGlue
\ccMethod{Vertex_iterator vertices_cw_begin();}{iterator over all the sources of the
faces of the antichain.}
\ccGlue
\ccMethod{Vertex_iterator vertices_cw_end();}{}
\ccGlue
\ccMethod{Edge_iterator edges_begin();}{iterator over all the edges of the
antichain.}
\ccGlue
\ccMethod{Edge_iterator edges_end();}{}
\ccGlue
\ccMethod{Face_iterator faces_begin();}{iterator over all the faces of the
antichain.}
\ccGlue
\ccMethod{Face_iterator faces_end();}{}
\ccThree{bool }{v.is_right_minimal( Vertex_handle v);}{}
\ccMethod{void set_linear_space(bool b);} {If \ccc{b} is true then the antichain
will destroy the elements which are no longer necessary after each sweep of a
vertex. This method is used to sweep the visibility complex while keeping the
storage linear (i.e. in the size of the input). The default value is true.}
\ccGlue
\ccMethod{bool linear_space(); } {returns true if the storage is linear. }
\ccMethod{void set_straight(bool b); } {if \ccc{b} is false (resp. true) then
the antichain will sweep the visibility complex using a topological (straight)
sweep. The default value is false. }
\ccGlue
\ccMethod{bool is_straight(); } { returns true if the antichain is straight. }
\ccMethod{bool is_right_minimal(Vertex_handle v);}{returns true if the vertex
$v$ is right minimal.}
\ccGlue
\ccMethod{bool is_left_minimal(Vertex_handle v);} {returns true if the vertex
$v$ is left minimal.}
\ccGlue
\ccMethod{bool is_minimal(Vertex_handle v);}{returns true if the vertex $v$ is
minimal.}
\ccThree{Minimals_iterator }{v.sweep( Vertex_handle v); }{}
\ccMethod{Minimals_iterator minimals_begin();} {}
\ccGlue
\ccMethod{Minimals_iterator minimals_end();}{ iterator over all the minimal
bitangents in the filter associated
with the antichain. If the antichain
is straight then the vertices are
ordered by increasing slope. }
\ccGlue
\ccMethod{void sweep_all_minimals();}{sweeps all the minimals vertices of the
antichain.}
\ccGlue
\ccMethod{void sweep(Vertex_handle v);}{sweeps the vertex $v$. precondition:
$v$ is minimal. If the antichain is
not straight, this operation is
amortized constant otherwise it takes
$O(\log n)$ time. The vertex is removed
from the list of minimals. }
\ccTagDefaults
% +----------------------------------------------------------------------------+
% ------------------------------------------------------------------------------
\ccHasModels
\ccRefIdfierPage{CGAL::Visibility_complex_antichain<Gt,It>}
% ------------------------------------------------------------------------------
% +----------------------------------------------------------------------------+
\ccSeeAlso
\ccRefConceptPage{VisibilityComplex} \\
\ccRefConceptPage{VisibilityComplexTraits} \\
\ccRefConceptPage{VisibilityComplexVertex}\\
\ccRefConceptPage{VisibilityComplexEdge}\\
\ccRefConceptPage{VisibilityComplexFace}
% +----------------------------------------------------------------------------+
\end{ccRefConcept}
% +----------------------------------------------------------------------------+
%%RefPage: end of main body, begin of footer
\ccRefPageEnd
% EOF
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: VisibilityComplexEdge.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +----------------------------------------------------------------------------+
\begin{ccRefConcept}{VisibilityComplexEdge}
\label{pageVCEdgeRef}
% +----------------------------------------------------------------------------+
\ccDefinition
The concept \ccRefName\ represents the edge class in the visibility complex.
It defines the minimal requirements for the \ccc{Edge} type in the
\ccc{VisibilityComplex} concept. It is also required in the
\ccc{Edge_wrapper<_Vc,_Tr>} member class template of an items class, see the
\ccc{VisibilityComplex_Items} concept. In addition to the requirement below, the
concept \ccRefName\ must inherit from the \ccc{Arc} concept
\ccIndexMainItem[c]{Arc}.
% +----------------------------------------------------------------------------+
% ------------------------------------------------------------------------------
\ccInheritsFrom
\ccRefConceptPage{Arc}\\
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccTypes
\ccThree{typedef Edge::Vertex_handle}{Vertex_handle;}{}
\ccThreeToTwo
\ccNestedType{Gt} {geometric traits class. }
\ccGlue
\ccNestedType{Vertex}{model of \ccc{Vertex}.}
\ccGlue
\ccNestedType{Edge}{model of \ccc{Edge}.}
\ccGlue
\ccNestedType{Face}{model of \ccc{Face}.}
\ccNestedType{Vertex_handle}{handle to vertex.}
\ccGlue
\ccNestedType{Edge_handle}{handle to halfedge.}
\ccGlue
\ccNestedType{Face_handle}{handle to face.}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccThree{Vertex_handle}{void v.set_inf(Vertex_handle v);}{}
\ccThreeToTwo
% ------------------------------------------------------------------------------
\ccCreation
\ccCreationVariable{e}
\ccConstructor{Edge();}{default constructor.}
\ccConstructor{Edge(bool s, Disk_handle p);}
{ creates a edge with positive sign if $s$ is true and negative otherwise.
The underlying arc is the complete boundary of the disk $p$. }
\ccConstructor{Edge(Vertex_handle v, Vertex_handle w , Disk_handle p);}
{ creates an edge on disk $p$ whose source is $v$ and sink $w$. The underlying
arc is defined by the extremities of $v$ and $w$ on $p$. }
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccOperations
\ccTagFullDeclarations
\ccMethod{Face_handle dl(); const}{}
\ccGlue
\ccMethod{Face_handle dr(); const}{}
\ccGlue
\ccMethod{Face_handle ul(); const}{}
\ccGlue
\ccMethod{Face_handle ur(); const}
{ returns a handle to one of the three faces adjacent to the edge \ccVar. For
a positive (resp. negative) edge, \ccc{dr()} and \ccc{dl()} (resp.
\ccc{ur()} and \ccc{ul()}) return the same \ccc{Face_handle}.}
\ccMethod{Face_handle face(); const}
{returns the unique face adjacent to \ccVar\ if \ccVar\ is a constraint edge.
Otherwise $0$ is returned.}
\ccMethod{void set_adjacent_faces(Face_handle f0,Face_handle f1,Face_handle f2);}
{set \ccc{f0,f1,f2} as the adjacent faces of \ccVar. For a positive edge,
\ccc{f0 = dl() = dr()}, \ccc{f1 = ur()} and \ccc{f2 = ul()}. For a negative
edge \ccc{f0 = dl()}, \ccc{f1 = dr()} and \ccc{f2 = ul() = ur()}.}
\ccMethod{Vertex_handle sup(); const}
{returns the sink vertex of \ccVar.}
\ccGlue
\ccMethod{Vertex_handle inf(); const}
{returns the source vertex of \ccVar.}
\ccGlue
\ccMethod{void set_inf(Vertex_handle v);}
{sets the source vertex of \ccVar\ to $v$ and sets the \ccc{ccw_edge}
of $v$ to \ccVar.}
\ccGlue
\ccMethod{void set_sup(Vertex_handle v);}
{sets the sink vertex of \ccVar\ to $v$ and sets the \ccc{cw_edge}
of $v$ to \ccVar.}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccHasModels
\ccRefIdfierPage{CGAL::Visibility_complex_edge_base<Vc>}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccSeeAlso
\ccRefConceptPage{Arc} \\
\ccRefConceptPage{VisibilityComplex} \\
\ccRefConceptPage{VisibilityComplexItems}\\
\ccRefConceptPage{VisibilityComplexVertex}\\
\ccRefConceptPage{VisibilityComplexFace}
% ------------------------------------------------------------------------------
% +----------------------------------------------------------------------------+
\ccTagDefaults
\end{ccRefConcept}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: VisibilityComplexFace.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +------------------------------------------------------------------------+
\begin{ccRefConcept}{VisibilityComplexFace}
\label{pageVCFaceRef}
% +----------------------------------------------------------------------------+
\ccDefinition
The concept \ccRefName\ defines the requirements for the \ccc{Face} type in the
\ccc{VisibilityComplex} concept. It is also required in the
\ccc{Face_wrapper<_Vc,_Tr>} member class template of an items class, see the
\ccc{VisibilityComplex_items} concept.
% +----------------------------------------------------------------------------+
% ------------------------------------------------------------------------------
\ccTypes
\ccThree{typedef Face::Vertex_handle}{Vertex_handle;}{}
\ccThreeToTwo
\ccNestedType{Gt} {geometric traits class. }
\ccGlue
\ccNestedType{Vertex}{model of \ccc{Vertex}.}
\ccGlue
\ccNestedType{Edge}{model of \ccc{Edge}.}
\ccGlue
\ccNestedType{Face}{model of \ccc{Face}.}
\ccNestedType{Vertex_handle}{handle to vertex.}
\ccGlue
\ccNestedType{Edge_handle}{handle to halfedge.}
\ccGlue
\ccNestedType{Face_handle}{handle to face.}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccThree{Edge_handle}{v.set_bottom_edge( Edge_handle e);}{}
\ccThreeToTwo
\ccCreation
\ccCreationVariable{f}
\ccConstructor{Face();}{default constructor.}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccOperations
\ccTagFullDeclarations
\ccMethod{Edge_handle sup(); const}
{returns the sink of face \ccVar.}
\ccGlue
\ccMethod{Edge_handle inf(); const}
{returns the source of face \ccVar.}
\ccGlue
\ccMethod{void set_sup(Vertex_handle v);}
{sets the sink of face \ccVar\ to $v$ and sets the source face of $v$ to
\ccVar.}
\ccGlue
\ccMethod{void set_inf(Vertex_handle v);}
{sets the source of face \ccVar\ to $v$ and sets the sink face of $v$ to
\ccVar.}
\ccGlue
\ccMethod{Edge_handle top_edge(); const}
{returns the last edge on the left boundary of \ccVar\ that has been swept.
If \ccVar\ belongs to the current antichain then this is the target node of the
underlying upward directed graph of the antichain. }
\ccGlue
\ccMethod{Edge_handle bottom_edge(); const}
{returns the last edge on the right boundary of \ccVar\ that has been swept.
If \ccVar\ belongs to the current antichain then this is the source node of the
underlying upward directed graph of the antichain. }
\ccGlue
\ccMethod{void set_top_edge(Edge_handle e);}
{sets $e$ as the last edge on the left boundary of \ccVar\ that has been
swept.}
\ccGlue
\ccMethod{void set_bottom_edge(Edge_handle e);}
{sets $e$ as the last edge on the right boundary of \ccVar\ that has been
swept.}
\ccThree{Disk_handle}{ v.set_backward_edge( Disk_handle d) ;}{}
\ccMethod{Disk_handle backward_object(); const}
{returns a pointer to the disk in the backward view of the face \ccVar.}
\ccGlue
\ccMethod{Disk_handle forward_object(); const}
{returns a pointer to the disk in the forward view of the face \ccVar.}
\ccMethod{void set_backward_object(Disk_handle d);}
{sets $d$ as the backward view of \ccVar.}
\ccGlue
\ccMethod{void set_forward_object(Disk_handle d);}
{sets $d$ as the forward view of \ccVar.}
\ccTagDefaults
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccHasModels
\ccRefIdfierPage{CGAL::Visibility_complex_face_base<Vc>}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccSeeAlso
\ccRefConceptPage{VisibilityComplex} \\
\ccRefConceptPage{VisibilityComplexItems}\\
\ccRefConceptPage{VisibilityComplexVertex}\\
\ccRefConceptPage{VisibilityComplexEdge}
% ------------------------------------------------------------------------------
% +----------------------------------------------------------------------------+
\ccTagDefaults
\end{ccRefConcept}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: VisibilityComplexItems.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
% +----------------------------------------------------------------------------+
\begin{ccRefConcept}{VisibilityComplexItems}
\label{pageVisibilityComplexItemsRef}
% +----------------------------------------------------------------------------+
\ccDefinition
The concept \ccRefName\ wraps the three item types -- vertex, edge, and face --
for the visibity complex. A \ccRefName\ contains three member class templates
named \ccc{Vertex_wrapper}, \ccc{Edge_wrapper}, and \ccc{Face_wrapper}, each
with one template parameter \ccc{Antichain} which is an instantiated model of
the \ccc{VisibilityComplexAntichain} concept. These three member class templates
provide local types named \ccc{Vertex}, \ccc{Edge} and \ccc{Face} respectively.
The requirements on these types are described on page~\pageref{pageVCVertexRef},
page~\pageref{pageVCEdgeRef}, and page~\pageref{pageVCFaceRef} respectively.
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccTypes
\ccTwo{VisibilityComplexItems:: Vertex_wrapper<Antichain>::Vertex }{}
\ccNestedType{Vertex_wrapper<Antichain>::Vertex}
{model of \ccc{VisibilityComplexVertex}.}
\ccGlue
\ccNestedType{Edge_wrapper<Antichain>::Edge}
{model of \ccc{VisibilityComplexEdge}.}
\ccGlue
\ccNestedType{Face_wrapper<Antichain>::Face}
{model of \ccc{VisibilityComplexFace}.}
% +----------------------------------------------------------------------------+
% ------------------------------------------------------------------------------
\ccHasModels
\ccRefIdfierPage{CGAL::Visibility_complex_items}
% ------------------------------------------------------------------------------
% +----------------------------------------------------------------------------+
\ccSeeAlso
\ccRefConceptPage{VisibilityComplexAntichain} \\
\ccRefConceptPage{VisibilityComplexVertex} \\
\ccRefConceptPage{VisibilityComplexEdge} \\
\ccRefConceptPage{VisibilityComplexFace} \\
\ccRefIdfierPage{CGAL::Visibility_complex_vertex_base<Vc>}\\
\ccRefIdfierPage{CGAL::Visibility_complex_edge_base<Vc>}\\
\ccRefIdfierPage{CGAL::Visibility_complex_face_base<Vc>}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\end{ccRefConcept}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: VisibilityComplexTraits.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
% +----------------------------------------------------------------------------+
\begin{ccRefConcept}{VisibilityComplexTraits}
\label{pageVisibilityComplexTraitsRef}
% +----------------------------------------------------------------------------+
\ccDefinition
The concept \ccRefName\ describes the set of requirements to be fulfilled by
any class used to instantiate the first template parameter of the classes
\ccc{Visibility_complex_2<Gt,It>} and
\ccc{Visibility_complex_antichain<Gt,It>}. This concept provides the
types of the geometric primitives used in the visiblity complex and some
function object types for the required predicates on those primitives.
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccTypes
\ccThree{VisibilityComplexTraits::Is_upward_directed }{}{}
\ccThreeToTwo
\ccNestedType{Point_2}{point type.}
\ccGlue
\ccNestedType{Segment_2}{segment type.}
\ccGlue
\ccNestedType{Disk}{disk type.}
\ccGlue
\ccNestedType{Bitangent_2}{model of the \ccc{Bitangent} concept. }
\ccGlue
\ccNestedType{Arc_2}{model of the \ccc{Arc} concept. }
% +----------------------------------------------------------------------------+
\ccNestedType{Orientation_object}{Must provide the operator \ccc{Orientation
operator()(Bitangent_2 a, Bitangent_2 b)}, which returns \ccc{LEFTURN,
RIGHTTURN, COLLINEAR} depending on the sign, \ccc{POSITIVE, NEGATIVE, ZERO} of
the determinant $\det(\vec{a},\vec{b})$ of the two underlying vectors of the
bitangents $a$ and $b$.}
\ccGlue
\ccNestedType{Is_upward_directed}{Must provide the operator \ccc{bool
operator()(Bitangent_2 d)}, which returns true iff. the angle of the bitangent
belongs to $[0,\pi)$. }
\ccGlue
\ccNestedType{Compare_extreme_yx}{Let \ccc{FT extreme(bool x, C c)} be a function returning
the maximal (resp. minimal) $y$-coordinate of $c$ if $x$ is false (resp. true).
The type $C$ can be either \ccc{Disk} or \ccc{Bitangent_2}. \\
The type \ccc{Compare_extreme_yx} must provide an operator
\ccc{Comparison_result operator()(bool x , C a , bool y , D b)} which returns
the comparison result between the \ccc{compare(x,a)} and \ccc{extreme(y,b)}. The
types $C$ and $D$ can be either \ccc{Disk} or \ccc{Bitangent}. }
\ccNestedType{Is_point}{Must provide the operator \ccc{bool operator()(Disk d)},
which returns true iff. $d$ is a point.}
\ccGlue
\ccNestedType{Equal_as_segments}{Must provide the operator \ccc{bool
operator()(Bitangent_2 a, Bitangent_2 b)}, which returns true iff. the
underlying segments of $a$ and $b$ are equal.}
\ccGlue
\ccNestedType{Do_intersect}{Must provide the four operators \ccc{bool
operator()(C a, D b)}-- where the types $C, D$ are either \ccc{Disk} or
\ccc{Bitangent_2}-- which return true iff. $a$ and $b$ intersect.}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccCreation
\ccCreationVariable{traits}
\ccConstructor{VisibilityComplexTraits();} {default constructor.}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccHasModels
\ccRefIdfierPage{CGAL::Visibility_complex_point_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_segment_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_polygon_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_circle_traits<R>}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\end{ccRefConcept}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: VisibilityComplexVertex.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +------------------------------------------------------------------------+
\begin{ccRefConcept}{VisibilityComplexVertex}
\label{pageVCVertexRef}
% ------------------------------------------------------------------------------
\ccDefinition
The concept \ccRefName\ represents the vertex class in the visibility complex.
It defines the requirements for the \ccc{Vertex} type in the
\ccc{VisibilityComplex} concept. It is also required in the
\ccc{Vertex_wrapper<_Vc>} member class template of an items class, see the
\ccc{VisibilityComplex_Items} concept. In addition to the requirement below, the
concept \ccRefName\ must inherit from the \ccc{Bitangent} concept defined on
page~\pageref{pageBitangentRef}.
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccInheritsFrom
\ccRefConceptPage{Bitangent}\\
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccTypes
\ccThree{typedef Vertex::Vertex_handle}{Vertex_handle;}{}
\ccThreeToTwo
\ccNestedType{Gt} {geometric traits class. }
\ccGlue
\ccNestedType{Bitangent_2} {model of \ccc{Bitangent}. Same as
\ccc{Gt::Bitangent\_2}.}
\ccNestedType{Vertex}{model of \ccc{Vertex}.}
\ccGlue
\ccNestedType{Edge}{model of \ccc{Edge}.}
\ccGlue
\ccNestedType{Face}{model of \ccc{Face}.}
\ccNestedType{Vertex_handle}{handle to vertex.}
\ccGlue
\ccNestedType{Edge_handle}{handle to halfedge.}
\ccGlue
\ccNestedType{Face_handle}{handle to face.}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccTypes
\ccThree{Vertex_handle}{v.set_ccw_edge( Edge_handle e); } {}
\ccThreeToTwo
% ------------------------------------------------------------------------------
\ccCreation
\ccCreationVariable{v}
\ccConstructor{Vertex();}{default constructor.}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccOperations
\ccTagFullDeclarations
\ccMethod{bool is_constraint(); const}
{returns true if the vertex is a constraint.}
\ccMethod{Edge_handle ccw_target_edge(); const}{}
\ccGlue
\ccMethod{Edge_handle ccw_source_edge(); const}{}
\ccGlue
\ccMethod{Edge_handle cw_source_edge(); const}{}
\ccGlue
\ccMethod{Edge_handle cw_target_edge(); const}
{returns the four adjacent edges of the vertex.}
\ccMethod{Edge_handle target_cusp_edge(); const}{}
\ccGlue
\ccMethod{Edge_handle source_cusp_edge(); const}
{If the vertex is a constraint, returns the two constraint edges with sink
\ccVar. Use this method on \ccc{\ccVar->pi()} to get the two other constraint
edges adjacent to \ccVar. If the vertex is regular, the null pointer is returned.}
\ccMethod{void set_ccw_edge(Edge_handle e);}
{sets the edge $e$ on the source or target object such that $\sup(e) = $ \ccVar.}
\ccGlue
\ccMethod{void set_cw_edge(Edge_handle e);}
{sets the edge $e$ on the source or target object such that $\sup(e) = $ \ccVar.}
\ccMethod{Face_handle sup(); const}
{returns the sink face of \ccVar.}
\ccGlue
\ccMethod{Face_handle inf(); const}
{returns the source face of \ccVar for a regular vertex and the null pointer
otherwise. To access the faces adjacent to a constraint vertex, use
the method \ccc{face()} on the two edges obtained by the methods
\ccc{target_cusp_edge()} and \ccc{source_cusp_edge()}.}
\ccGlue
\ccMethod{void set_inf(Face_handle f);}
{sets the source face of \ccVar\ to $f$.}
\ccGlue
\ccMethod{void set_sup(Face_handle f);}
{sets the sink face of \ccVar\ to $f$.}
\ccTagDefaults
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccHasModels
\ccRefIdfierPage{CGAL::Visibility_complex_vertex_base<Vc>}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccSeeAlso
\ccRefConceptPage{Bitangent} \\
\ccRefConceptPage{VisibilityComplex} \\
\ccRefConceptPage{VisibilityComplexItems}\\
\ccRefConceptPage{VisibilityComplexEdge}\\
\ccRefConceptPage{VisibilityComplexFace}
% ------------------------------------------------------------------------------
% +----------------------------------------------------------------------------+
\ccTagDefaults
\end{ccRefConcept}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Visibility_complex.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
% +----------------------------------------------------------------------------+
\begin{ccRefClass}{Visibility_complex_2<Gt,It>}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccDefinition
The class \ccRefName\ is a model of the \ccc{VisibilityComplex} concept.
\ccc{Gt} represents the geometric traits class. It is an instantiation of a
model for the VisibilityComplexTraits
\ccIndexMainItem[c]{VisibilityComplexTraits} concept. Currently four geometric
traits classes are provided. \ccc{It} is an instantiation of a model for the
VisibilityComplexItems \ccIndexMainItem[c]{VisibilityComplexItems} concept.
\ccInclude{CGAL/Visibility_complex.h}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccIsModel
\ccRefConceptPage{VisibilityComplex}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccSeeAlso
\ccRefConceptPage{VisibilityComplexItems}\\
\ccRefConceptPage{VisibilityComplexTraits}\\
\ccRefIdfierPage{CGAL::Visibility_complex_point_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_segment_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_polygon_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_circle_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_items}
% +----------------------------------------------------------------------------+
\end{ccRefClass}

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% +------------------------------------------------------------------------+
% | Reference manual page: Visibility_complex_antichain.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
% +----------------------------------------------------------------------------+
\begin{ccRefClass}{Visibility_complex_antichain<Gt,It>}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccDefinition
The class \ccRefName\ is a model of the \ccc{VisibilityComplexAntichain}
concept. \ccc{Gt} represents the geometric traits class. It is an instantiation
of a model for the VisibilityComplexTraits
\ccIndexMainItem[c]{VisibilityComplexTraits} concept. Currently four geometric
traits classes are provided. \ccc{It} is an instantiation of a model for the
VisibilityComplexItems \ccIndexMainItem[c]{VisibilityComplexItems} concept
\ccInclude{CGAL/Visibility_complex_antichain.h}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccIsModel
\ccRefConceptPage{VisibilityComplexAntichain}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccSeeAlso
\ccRefConceptPage{VisibilityComplexItems}\\
\ccRefConceptPage{VisibilityComplexTraits}\\
\ccRefIdfierPage{CGAL::Visibility_complex_point_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_segment_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_polygon_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_circle_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_items}
% +----------------------------------------------------------------------------+
\end{ccRefClass}

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% +------------------------------------------------------------------------+
% | Reference manual page: Visibility_complex_circle_traits.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\begin{ccRefClass}{Visibility_complex_circle_traits<R>}
\label{pageVisibility_complex_circle_traitsRef}
% ------------------------------------------------------------------------------
\ccDefinition
The class \ccRefName\ is the predefined geometric traits class provided by
\cgal\ for use with disks whith type \ccc{CGAL::Circle_by_radius<R>}.
It uses the kernel geometric objects and predicates. The class is templated
with a representation class \ccc{R}.
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccInclude{CGAL/Visibility_complex_circle_traits.h}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccTypes
\ccThree{typedef Bitangent_2<Disk> }{Bitangent_2; }{}
\ccTypedef{typedef R Rep;}{Representation type}
\ccGlue
\ccTypedef{typedef Point_2<R> Point_2;}{}
\ccGlue
\ccTypedef{typedef Segment_2<R> Segment_2;}{}
\ccGlue
\ccTypedef{typedef Circle_by_radius<R> Disk;}{Disk type}
\ccGlue
\ccTypedef{typedef Arc_2<Disk> Arc_2;}{}
\ccGlue
\ccTypedef{typedef Bitangent_2<Disk> Bitangent_2;}{}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccIsModel
\ccRefConceptPage{VisibilityComplexTraits}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccSeeAlso
\ccRefIdfierPage{CGAL::Arc_2<Disk>} \\
\ccRefIdfierPage{CGAL::Bitangent_2<Disk>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_point_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_segment_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_polygon_traits<R>}
% ------------------------------------------------------------------------------
% +----------------------------------------------------------------------------+
\ccTagDefaults
\end{ccRefClass}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Visibility_complex_edge_base.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
% +----------------------------------------------------------------------------+
\begin{ccRefClass}{Visibility_complex_edge_base<Vc>}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccDefinition
The class \ccRefName\ is a model of the \ccc{VisibilityComplexEdge}
concept. \ccc{Vc} is an instantiation of a \ccc{VisibilityComplexAntichain}.
\ccInclude{CGAL/Visibility_complex_edge_base.h}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccIsModel
\ccRefConceptPage{VisibilityComplexEdge}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccSeeAlso
\ccRefConceptPage{VisibilityComplexItems}\\
\ccRefIdfierPage{CGAL::Visibility_complex_items}\\
\ccRefIdfierPage{CGAL::Visibility_complex_vertex_base<Vc>}\\
\ccRefIdfierPage{CGAL::Visibility_complex_face_base<Vc>}
% +----------------------------------------------------------------------------+
\end{ccRefClass}

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% +------------------------------------------------------------------------+
% | Reference manual page: Visibility_complex_face_base.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
% +----------------------------------------------------------------------------+
\begin{ccRefClass}{Visibility_complex_face_base<Vc>}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccDefinition
The class \ccRefName\ is a model of the \ccc{VisibilityComplexFace}
concept. \ccc{Vc} is an instantiation of a \ccc{VisibilityComplexAntichain}.
\ccInclude{CGAL/Visibility_complex_face_base.h}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccIsModel
\ccRefConceptPage{VisibilityComplexFace}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccSeeAlso
\ccRefConceptPage{VisibilityComplexItems}\\
\ccRefIdfierPage{CGAL::Visibility_complex_items}\\
\ccRefIdfierPage{CGAL::Visibility_complex_vertex_base<Vc>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_edge_base<Vc>}
% +----------------------------------------------------------------------------+
\end{ccRefClass}

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% +------------------------------------------------------------------------+
% | Reference manual page: Visibility_complex_items.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
% +----------------------------------------------------------------------------+
\begin{ccRefClass}{Visibility_complex_items}
\label{pageVisibilitycomplexitemsRef}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccDefinition
The class \ccRefName\ is a model of the \ccc{VisibilityComplexItems} concept.
It uses the default types for vertices, halfedges, and faces.
\ccInclude{CGAL/Visibility_complex_items.h}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccIsModel
\ccRefConceptPage{VisibilityComplexItems}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccSeeAlso
\ccRefConceptPage{VisibilityComplexItems}\\
\ccRefIdfierPage{CGAL::Visibility_complex_vertex_base<Vc>}\\
\ccRefIdfierPage{CGAL::Visibility_complex_edge_base<Vc>}\\
\ccRefIdfierPage{CGAL::Visibility_complex_face_base<Vc>}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\end{ccRefClass}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Visibility_complex_point_traits.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\begin{ccRefClass}{Visibility_complex_point_traits<R>}
\label{pageVisibility_complex_point_traitsRef}
% ------------------------------------------------------------------------------
\ccDefinition
The class \ccRefName\ is the predefined geometric traits class provided by
\cgal\ for use with disks whith type \ccc{CGAL::Point_2<R>}.
It uses the kernel geometric objects and predicates. The class is templated
with a representation class \ccc{R}.
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccInclude{CGAL/Visibility_complex_point_traits.h}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccTypes
\ccThree{typedef Bitangent_2<Disk> }{Bitangent_2; }{}
\ccTypedef{typedef R Rep;}{Representation type}
\ccGlue
\ccTypedef{typedef Point_2<R> Point_2;}{}
\ccGlue
\ccTypedef{typedef Segment_2<R> Segment_2;}{}
\ccGlue
\ccTypedef{typedef Point_2<R> Disk;}{Disk type}
\ccGlue
\ccTypedef{typedef Arc_2<Disk> Arc_2;}{}
\ccGlue
\ccTypedef{typedef Bitangent_2<Disk> Bitangent_2;}{}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccIsModel
\ccRefConceptPage{VisibilityComplexTraits}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccSeeAlso
\ccRefIdfierPage{CGAL::Arc_2<Disk>} \\
\ccRefIdfierPage{CGAL::Bitangent_2<Disk>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_segment_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_polygon_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_circle_traits<R>}
% ------------------------------------------------------------------------------
% +----------------------------------------------------------------------------+
\ccTagDefaults
\end{ccRefClass}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Visibility_complex_polygon_traits.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\begin{ccRefClass}{Visibility_complex_polygon_traits<R>}
\label{pageVisibility_complex_polygon_traitsRef}
% ------------------------------------------------------------------------------
\ccDefinition
The class \ccRefName\ is the predefined geometric traits class provided by
\cgal\ for use with disks whith type \ccc{CGAL::Polygon_2<R>}.
It uses the kernel geometric objects and predicates. The class is templated
with a representation class \ccc{R}.
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccInclude{CGAL/Visibility_complex_polygon_traits.h}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccTypes
\ccThree{typedef Bitangent_2<Disk> }{Bitangent_2; }{}
\ccTypedef{typedef R Rep;}{Representation type}
\ccGlue
\ccTypedef{typedef Point_2<R> Point_2;}{}
\ccGlue
\ccTypedef{typedef Segment_2<R> Segment_2;}{}
\ccGlue
\ccTypedef{typedef Polygon_2<R> Disk;}{Disk type}
\ccGlue
\ccTypedef{typedef Arc_2<Disk> Arc_2;}{}
\ccGlue
\ccTypedef{typedef Bitangent_2<Disk> Bitangent_2;}{}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccIsModel
\ccRefConceptPage{VisibilityComplexTraits}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccSeeAlso
\ccRefIdfierPage{CGAL::Arc_2<Disk>} \\
\ccRefIdfierPage{CGAL::Bitangent_2<Disk>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_segment_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_polygon_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_circle_traits<R>}
% ------------------------------------------------------------------------------
% +----------------------------------------------------------------------------+
\ccTagDefaults
\end{ccRefClass}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Visibility_complex_segment_traits.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\begin{ccRefClass}{Visibility_complex_segment_traits<R>}
\label{pageVisibility_complex_segment_traitsRef}
% ------------------------------------------------------------------------------
\ccDefinition
The class \ccRefName\ is the predefined geometric traits class provided by
\cgal\ for use with disks whith type \ccc{CGAL::Segment_2<R>}.
It uses the kernel geometric objects and predicates. The class is templated
with a representation class \ccc{R}.
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccInclude{CGAL/Visibility_complex_segment_traits.h}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccTypes
\ccThree{typedef Bitangent_2<Disk> }{Bitangent_2; }{}
\ccTypedef{typedef R Rep;}{Representation type}
\ccGlue
\ccTypedef{typedef Point_2<R> Point_2;}{}
\ccGlue
\ccTypedef{typedef Segment_2<R> Segment_2;}{}
\ccGlue
\ccTypedef{typedef Segment_2<R> Disk;}{Disk type}
\ccGlue
\ccTypedef{typedef Arc_2<Disk> Arc_2;}{}
\ccGlue
\ccTypedef{typedef Bitangent_2<Disk> Bitangent_2;}{}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccIsModel
\ccRefConceptPage{VisibilityComplexTraits}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccSeeAlso
\ccRefIdfierPage{CGAL::Arc_2<Disk>} \\
\ccRefIdfierPage{CGAL::Bitangent_2<Disk>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_point_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_polygon_traits<R>} \\
\ccRefIdfierPage{CGAL::Visibility_complex_circle_traits<R>}
% ------------------------------------------------------------------------------
% +----------------------------------------------------------------------------+
\ccTagDefaults
\end{ccRefClass}
\ccRefPageEnd
% +----------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Visibility_complex_vertex_base.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\ccRefPageBegin
% +----------------------------------------------------------------------------+
\begin{ccRefClass}{Visibility_complex_vertex_base<Vc>}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccDefinition
The class \ccRefName\ is a model of the \ccc{VisibilityComplexVertex}
concept. \ccc{Vc} is an instantiation of a \ccc{VisibilityComplexAntichain}.
\ccInclude{CGAL/Visibility_complex_vertex_base.h}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccIsModel
\ccRefConceptPage{VisibilityComplexVertex}
% +----------------------------------------------------------------------------+
% +----------------------------------------------------------------------------+
\ccSeeAlso
\ccRefConceptPage{VisibilityComplexItems}\\
\ccRefIdfierPage{CGAL::Visibility_complex_items}\\
\ccRefIdfierPage{CGAL::Visibility_complex_edge_base<Vc>}\\
\ccRefIdfierPage{CGAL::Visibility_complex_face_base<Vc>}
% +----------------------------------------------------------------------------+
\end{ccRefClass}

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hds_optional.gif
hds_optional_small.gif
euler_face.gif
euler_vertex.gif
euler_center.gif
euler_loop.gif

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% +------------------------------------------------------------------------+
% | Reference manual page: intro.tex
% +------------------------------------------------------------------------+
% | Last revision: 09.07.2002
% | Author : Pierre Angelier <Pierre.Angelier@ens.fr>
% | Maintainer: Laurent Rineau <Laurent.Rineau@ens.fr>
% | Package: Visibility_complex
% +------------------------------------------------------------------------+
\clearpage
\chapter{Visibility Complexes}
\label{chapterVisibilityComplexRef}
% +------------------------------------------------------------------------+
\section*{Summary}
\subsection*{Concepts}
\ccRefConceptPage{Bitangent}\\
\ccRefConceptPage{Arc}\\
\ccRefConceptPage{VisibilityComplexTraits}\\
\ccRefConceptPage{VisibilityComplexVertex}\\
\ccRefConceptPage{VisibilityComplexEdge}\\
\ccRefConceptPage{VisibilityComplexFace}\\
\ccRefConceptPage{VisibilityComplexItems}\\
\ccRefConceptPage{VisibilityComplex} \\
\ccRefConceptPage{VisibilityComplexAntichain}
\subsection*{Classes}
\ccRefIdfierPage{CGAL::Bitangent_base<Disk>}\\
\ccRefIdfierPage{CGAL::Bitangent_2<Disk>}\\
\ccRefIdfierPage{CGAL::Arc_2<Disk>}
\ccRefIdfierPage{CGAL::Visibility_complex_antichain<Gt,It>}\\
\ccRefIdfierPage{CGAL::Visibility_complex_2<Gt,It>}\\
\ccRefIdfierPage{CGAL::Visibility_complex_items}
\ccRefIdfierPage{CGAL::Visibility_complex_vertex_base<Vc>}\\
\ccRefIdfierPage{CGAL::Visibility_complex_edge_base<Vc>}\\
\ccRefIdfierPage{CGAL::Visibility_complex_face_base<Vc>}
\ccRefIdfierPage{CGAL::Visibility_complex_point_traits<R>}\\
\ccRefIdfierPage{CGAL::Visibility_complex_segment_traits<R>}\\
\ccRefIdfierPage{CGAL::Visibility_complex_polygon_traits<R>}\\
\ccRefIdfierPage{CGAL::Visibility_complex_circle_traits<R>}
\ccRefIdfierPage{CGAL::Shortest_path_point_traits<R,E>}\\
\ccRefIdfierPage{CGAL::Shortest_path_segment_traits<R,E>}\\
\ccRefIdfierPage{CGAL::Shortest_path_polygon_traits<R,E>}
\lcHtml{\subsection*{Links to the Reference Sections}}
%% EOF %%

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% +------------------------------------------------------------------------+
% | CBP Reference Manual: main.tex
% +------------------------------------------------------------------------+
% | Automatically generated driver file for the reference manual chapter
% | of this package. Do not edit manually, you may loose your changes.
% +------------------------------------------------------------------------+
\input{Visibility_complex_ref/intro.tex}
\input{Visibility_complex_ref/Bitangent.tex}
\input{Visibility_complex_ref/Bitangent_base.tex}
\input{Visibility_complex_ref/Bitangent_2.tex}
\input{Visibility_complex_ref/Arc.tex}
\input{Visibility_complex_ref/Arc_2.tex}
\input{Visibility_complex_ref/VisibilityComplexTraits.tex}
\input{Visibility_complex_ref/Visibility_complex_point_traits.tex}
\input{Visibility_complex_ref/Visibility_complex_segment_traits.tex}
\input{Visibility_complex_ref/Visibility_complex_polygon_traits.tex}
\input{Visibility_complex_ref/Visibility_complex_circle_traits.tex}
\input{Visibility_complex_ref/VisibilityComplexVertex.tex}
\input{Visibility_complex_ref/Visibility_complex_vertex_base.tex}
\input{Visibility_complex_ref/VisibilityComplexEdge.tex}
\input{Visibility_complex_ref/Visibility_complex_edge_base.tex}
\input{Visibility_complex_ref/VisibilityComplexFace.tex}
\input{Visibility_complex_ref/Visibility_complex_face_base.tex}
\input{Visibility_complex_ref/VisibilityComplexItems.tex}
\input{Visibility_complex_ref/Visibility_complex_items.tex}
\input{Visibility_complex_ref/VisibilityComplexAntichain.tex}
\input{Visibility_complex_ref/Visibility_complex_antichain.tex}
\input{Visibility_complex_ref/VisibilityComplex.tex}
\input{Visibility_complex_ref/Visibility_complex.tex}
\input{Visibility_complex_ref/ShortestPathTraits.tex}
\input{Visibility_complex_ref/Shortest_path_point_traits.tex}
\input{Visibility_complex_ref/Shortest_path_segment_traits.tex}
\input{Visibility_complex_ref/Shortest_path_polygon_traits.tex}
\input{Visibility_complex_ref/Shortest_path_2.tex}
%% EOF

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% +------------------------------------------------------------------------+
% | Reference manual page: Bitangent_base.tex
% +------------------------------------------------------------------------+
% | 06.06.2000 Pierre Angelier
% | Package: Visibility_complex_2
% |
\RCSdef{\RCSBitangentBaseRev}{$Revision$}
\RCSdefDate{\RCSBitangentBaseDate}{$Date$}
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +------------------------------------------------------------------------+
\begin{ccRefConcept}{Bitangent_base}
\label{pageVC_Bitangent_baseRef}
The \ccRefName\ concept defines the minimal requirement for a
\emph{combinatorial} bitangent, i.e. a bitangent with no underlying geometric
segment. The only information stored by the \ccRefName\ concept is the type and
the two disks defining a bitangent. The \ccc{Bitangent_2} concept will inherit
from \ccRefName\ . The only additional functionality the \ccc{Bitangent_2} will
provide are the constructors giving the coordinates of the bitangent's
endpoints. In other words the \ccc{Bitangent_2} concept will add geometric
information.
% ------------------------------------------------------------------------------
\ccTypes
\ccThree{Bitangent_base_2}{Disk_handle}{}
\ccThreeToTwo
\ccNestedType{Disk}{type of disks defining the bitangent. }
\ccGlue
\ccNestedType{Disk_handle}{handle to \ccc{Disk}.}
\ccGlue
\ccNestedType{Type}{enumerated type to specify the type of the bitangent. The
possible values are \ccc{LL,RR,RL} or \ccc{LR} respectively
for left-left,right-right,right-left and left-right
bitangents.}{}
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccThree{Disk_handle}{ b.source_object(); }{}
\ccThreeToTwo
\ccCreation
\ccCreationVariable{b}
\ccConstructor{Bitangent_base();}{default constructor.}
\ccGlue
\ccConstructor{Bitangent_base(Type t, Disk_handle source, Disk_handle target);}
{ Creates the combinatorial bitangent with type t directed from source to target. }
\ccGlue
% ------------------------------------------------------------------------------
% ------------------------------------------------------------------------------
\ccOperations
\ccTagFullDeclarations
\ccMethod{Disk_handle source_object(); const}
{returns a pointer to the disk that \ccVar\ leaves.}
\ccGlue
\ccMethod{Disk_handle target_object(); const}
{returns a pointer to the disk that \ccVar\ enters.}
\ccGlue
\ccMethod{Type type(); const}
{returns the type of the bitangent. }
\ccTagDefaults
% ------------------------------------------------------------------------------
\ccTagDefaults
\end{ccRefConcept}
% +------------------------------------------------------------------------+
%%RefPage: end of main body, begin of footer
\ccRefPageEnd
% EOF
% +------------------------------------------------------------------------+

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#include <CGAL/basic.h>
#include <fstream>
#include <list>
#include <CGAL/Simple_cartesian.h>
#include <CEP/Visibility_complex/Shortest_path_segment_traits.h>
#include <CEP/Visibility_complex/Shortest_path_2.h>
// Number type used for input
typedef int NT;
// Number type used when computing distances
typedef double SNT;
typedef CGAL::Simple_cartesian<NT> R;
typedef CGAL::Shortest_path_segment_traits<R,SNT> Gt;
typedef Gt::Disk Segment;
typedef Gt::Point_2 Point;
int main()
{
// Reading segments from file
std::list<Segment> O;
std::ifstream ifs("segments");
std::istream_iterator<Segment> ifs_it(ifs),ifs_end;
std::copy(ifs_it,ifs_end,std::back_inserter(O));
// We want to ccompute the shortest path between p and q
Point p(0,0);
Point q(1000,800);
// Output the bitangents of the shortest path to cout
CGAL::shortest_path_2(O.begin(),O.end(),
p,q,
std::ostream_iterator<Segment>(cout , "\n"),
Gt());
return 0;
}

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currentdict
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countdictstack
save
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{
currentpoint strokepath moveto
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clip newpath PATfill
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(*** PATstroke Warning: Path is too complex, stroking
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cleartomark
restore
countdictstack exch sub dup 0 gt
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round exch round exch % tx ty cmtx a b dy.x dy.y
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grestore
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% right45
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Packages/Visibility_complex/doc_tex/fig/vis-complex.fig Normal file
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#FIG 3.2
Landscape
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Letter
100.00
Single
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1200 2
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8675 3044 8981 2516 8777 1936
0.000 1.000 0.000
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20
Packages/Visibility_complex/doc_tex/local.bib Normal file
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@techreport{G-ap-sstvc-01b
, category = "report"
, author = {P.~Angelier and M.~Pocchiola}
, title = {A sum of squares theorem for visibility complexes and applications}
, number = {G-LIENS-01-1}
, institution= {Department of Computer Science, Ecole normale supérieure Paris}
, address = {France}
, year = 2001
, month = {August}
}
@inproceedings{G-ap-sstvc-01
, category = "intc"
, author = {P. Angelier and M.~Pocchiola}
, title = {A sum of squares theorem for visibility complexes}
, booktitle = {Proc. 17th Annu. ACM Sympos. Comput. Geom.}
, year = 2001
, pages = "302--311"
, note = {}
}

6
Packages/Visibility_complex/doc_tex/main.tex Normal file
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\part{User Manual}
\input{Visibility_complex/main}
\part{Reference Manual}
\input{Visibility_complex_ref/main}

38
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\begin{thebibliography}{1}
\bibitem{G-ap-sstvc-01}
P.~Angelier and M.~Pocchiola.
\newblock A sum of squares theorem for visibility complexes.
\newblock In {\em Proc. 17th Annu. ACM Sympos. Comput. Geom.}, pages 302--311,
2001.
\bibitem{ddp-fahrgv-99}
F.~Durand, G.~Drettakis, and C.~Puech.
\newblock Fast and accurate hierarchical radiosity using global visibility.
\newblock {\em ACM Trans. Graph.}, 18(2):128--170, 1999.
\bibitem{k-ugpdd-99}
L.~Kettner.
\newblock Using generic programming for designing a data structure for
polyhedral surfaces.
\newblock {\em Comput. Geom. Theory Appl.}, 13:65--90, 1999.
\bibitem{m-gspno-00}
Joseph S.~B. Mitchell.
\newblock Geometric shortest paths and network optimization.
\newblock In J{\"o}rg-R{\"u}diger Sack and Jorge Urrutia, editors, {\em
Handbook of Computational Geometry}, pages 633--701. Elsevier Science
Publishers B.V. North-Holland, Amsterdam, 2000.
\bibitem{pv-tsvcpt-96}
M.~Pocchiola and G.~Vegter.
\newblock Topologically sweeping visibility complexes via
pseudo-triangulations.
\newblock {\em Discrete Comput. Geom.}, 16:419--453, December 1996.
\bibitem{pv-vc-96}
M.~Pocchiola and G.~Vegter.
\newblock The visibility complex.
\newblock {\em Internat. J. Comput. Geom. Appl.}, 6(3):279--308, 1996.
\end{thebibliography}

46
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% +------------------------------------------------------------------------+
% | manual.tex
% +------------------------------------------------------------------------+
% | includes the usual suspects (LaTeX style files), sets necessary
% | options, illustrates some optional options, and includes main.tex.
% +------------------------------------------------------------------------+
\documentclass{book}
\usepackage{alltt} % convinient for programs + a bit of formatting
\usepackage{cc_manual} % the style what all is about here
\usepackage{cc_manual_index} % the style file for indexing the manual
\usepackage{latex_converter} % the LaTeX to HTML converter style file
\usepackage{amssymb} % some more math symbols, needed for \mathbb
\usepackage{path} % for URL's (WWW addresses)
\usepackage{graphicx} % for PostScript figures
%\usepackage{ipe} % for IPE figures
\usepackage{makeidx} % for the index
\usepackage{minitoc} % for the local table of contents
\usepackage{color,psfrag}
\usepackage{t1enc}
% page dimensions, for example as used for the CGAL manuals
% ---------------------------------------------------------
\textwidth 15.6cm
\textheight 23 cm
\topmargin -14mm
\evensidemargin 3mm
\oddsidemargin 3mm
\sloppy
\makeindex % so the indexing commands have some effect
% New manual style including tab markers on the side margins
% ----------------------------------------------------------
\marginparsep7mm
\marginparwidth15mm
\gdef\ccNewRefManualStyle{\ccTrue} % undef this for old style manual
% Start renumbering chapters with each \part command
% ----------------------------------------------------------
\ccNumberChaptersByPart
% Tell HTML converter to use proper format for table of contents
% --------------------------------------------------------------
\ccMultiplePartsToc
% The tab markers are aligned with the top of the main text. To align
% them with the page header, the following definition of the amount
% by which the tab is shifted can be used
\setlength{\ccRefTabLift}{12.5mm}
% Optionally, package authors and a package release can be shown
% right below the chapter title. Comment out the following lines
% to get back to the default behavior.
% -----------------------------------------------------------------
\def\ccTagChapterAuthor{\ccTrue}
\def\ccTagChapterRelease{\ccTrue}
% The colunm layout can be set for all reference pages. Since
% local changes are possible, it is a good idea to enforce the
% the global layout before each reference page. This way, local
% changes of the layout do not change pages in the remainder
% of the manual.
% ------------------------------------------------------------
\newcommand{\cgalColumnLayout}{% example setting of the CGAL manuals
\ccSetThreeColumns{CGAL_Oriented_side}{}{\hspace*{8.5cm}}%
\ccPropagateThreeToTwoColumns
}
\cgalColumnLayout
\def\ccRefPageBegin{\ccParDims\cgalColumnLayout}
\def\ccRefPageEnd{\ccParDims\cgalColumnLayout}
% We define the global scope here, for example, to be 'CGAL::'.
% -------------------------------------------------------------
\ccDefGlobalScope{CGAL::}
\begin{document}
\bibliographystyle{plain}
\input{main}
\printindex %places the index in the PostScript document
\bibliography{local,geom}
\end{document}
%% EOF %%

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#ifndef CGAL_CONVEX_ARC_2
#define CGAL_CONVEX_ARC_2
#include <list>
CGAL_BEGIN_NAMESPACE
// -----------------------------------------------------------------------------
// Abstract unsigned convex arc
template < class D_ >
class Arc_base
{
public:
// -------------------------------------------------------------------------
typedef D_ Disk;
typedef const Disk* Disk_handle;
// -------------------------------------------------------------------------
public:
// -------------------------------------------------------------------------
Arc_base() : _object(0) { }
Arc_base(Disk_handle P) : _object(P) { }
// -------------------------------------------------------------------------
Disk_handle object() const { return _object; }
void set_object(Disk_handle P) { _object = P; }
// -------------------------------------------------------------------------
private:
Disk_handle _object;
};
// -----------------------------------------------------------------------------
// --------------------- Arc_2 general definition -----------------------
// ----- This definition is sufficient if arcs have constant complexity --------
// -----------------------------------------------------------------------------
template < class D_ >
struct Arc_2 : public Arc_base<D_>
{
// -------------------------------------------------------------------------
typedef D_ Disk;
typedef typename Disk::R R;
typedef typename R::FT FT;
typedef typename R::Point_2 Point_2;
typedef typename R::Segment_2 Segment_2;
typedef Arc_base<D_> Base;
typedef typename Base::Disk_handle Disk_handle;
// -------------------------------------------------------------------------
Arc_2() : Base(0) { }
Arc_2(Disk_handle P) : Base(P) { }
Arc_2(Disk_handle P,const Point_2& p, const Point_2& q)
: Base(P) { }
// -------------------------------------------------------------------------
void split (Arc_2& /*tmp*/, const Point_2& /*p*/) { }
void split_cw(Arc_2& /*tmp*/, const Point_2& /*p*/) { }
void update_begin(const Point_2& /*p*/) { }
void update_end(const Point_2& /*p*/) { }
void join (Arc_2& /*y*/) { }
// -------------------------------------------------------------------------
};
// -----------------------------------------------------------------------------
// -------------- Arc_2 Specialization for CGAL::Polygon_2 --------------
// -----------------------------------------------------------------------------
#ifdef CGAL_POLYGON_2_H
template < class R_ , class C_ >
class Arc_2 < Polygon_2<R_,C_> >
: public Arc_base< Polygon_2<R_,C_> >
{
public:
// -------------------------------------------------------------------------
typedef R_ R;
typedef typename R_::FT FT;
typedef typename R::Point_2 Point_2;
typedef Polygon_2<R_,C_> Disk;
typedef C_ Container;
typedef Arc_base<Disk> Base;
typedef typename Base::Disk_handle Disk_handle;
typedef typename Disk::Vertex_const_circulator Vertex_const_iterator;
typedef typename Disk::Vertex_const_circulator Vertex_iterator;
// -------------------------------------------------------------------------
public:
// -------------------------------------------------------------------------
Arc_2() : Base(0) { }
Arc_2(Disk_handle P)
: Base(P) , first(Vertex_iterator()) , beyond(Vertex_iterator()) { }
// -------------------------------------------------------------------------
void split (Arc_2& tmp, const Point_2& p)
{
if (first == CGAL_CIRC_NULL) {
first = object()->vertices_circulator();
while (*first != p) ++first;
beyond = first; //++beyond;
tmp.first = first; tmp.beyond = beyond;
}
else {
tmp.beyond = beyond;
tmp.first = first;
while (*tmp.first != p) ++tmp.first;
beyond = tmp.first;
++beyond;
}
}
void split_cw(Arc_2& tmp, const Point_2& p)
{
if (first == CGAL_CIRC_NULL) {
first = object()->vertices_circulator();
while (*first != p) ++first;
beyond = first; //++beyond;
tmp.first = first; tmp.beyond = beyond;
}
else {
tmp.beyond = beyond;
tmp.first = first;
while (*tmp.beyond != p) --tmp.beyond;
first = tmp.beyond;
++tmp.beyond;
}
}
void update_begin(const Point_2& p) {
first = beyond; --first;
while (*first != p) --first;
}
void update_end(const Point_2& p) {
beyond = first;
while (*beyond != p) ++beyond;
++beyond;
}
void join (Arc_2& y) { beyond = y.beyond; }
// -------------------------------------------------------------------------
// these methods are specific to this specialization
Vertex_iterator begin() const { return first; }
Vertex_iterator end() const { return beyond; }
Vertex_iterator vertices_begin() const { return first; }
Vertex_iterator vertices_end() const { return beyond; }
// -------------------------------------------------------------------------
private:
Vertex_iterator first;
Vertex_iterator beyond;
};
#endif // CGAL_POLYGON_2_H
// -----------------------------------------------------------------------------
CGAL_END_NAMESPACE
#endif

251
Packages/Visibility_complex/include/CEP/Visibility_complex/Arithmetic_filter/predicates/Circle_2_Circle_2_intersection_ftC2.h Normal file
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// ======================================================================
//
// Copyright (c) 1999,2000 The CGAL Consortium
//
// This software and related documentation is part of an INTERNAL release
// of the Computational Geometry Algorithms Library (CGAL). It is not
// intended for general use.
//
// ----------------------------------------------------------------------
//
// release :
// release_date :
//
// file : include/CGAL/Arithmetic_filter/
// package : Interval_arithmetic
// author(s) : Sylvain Pion <Sylvain.Pion@sophia.inria.fr>
//
// coordinator : INRIA Sophia-Antipolis (<Mariette.Yvinec@sophia.inria.fr>)
// ======================================================================
// This file is automatically generated by
// scripts/filtered_predicates_generator.pl
#ifndef CGAL_ARITHMETIC_FILTER_CIRCLE_2_CIRCLE_2_INTERSECTION_FTC2_H
#define CGAL_ARITHMETIC_FILTER_CIRCLE_2_CIRCLE_2_INTERSECTION_FTC2_H
CGAL_BEGIN_NAMESPACE
#ifndef CGAL_CFG_MATCHING_BUG_2
template < class CGAL_IA_CT, class CGAL_IA_ET, bool CGAL_IA_PROTECTED,
class CGAL_IA_CACHE >
#else
static
#endif
/* */
bool
circle_2_do_intersectC2(
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Dynamic, CGAL_IA_PROTECTED, CGAL_IA_CACHE> &c1x,
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Dynamic, CGAL_IA_PROTECTED, CGAL_IA_CACHE> &c1y,
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Dynamic, CGAL_IA_PROTECTED, CGAL_IA_CACHE> &R1_square,
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Dynamic, CGAL_IA_PROTECTED, CGAL_IA_CACHE> &c2x,
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Dynamic, CGAL_IA_PROTECTED, CGAL_IA_CACHE> &c2y,
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Dynamic, CGAL_IA_PROTECTED, CGAL_IA_CACHE> &R2_square)
{
try
{
Protect_FPU_rounding<CGAL_IA_PROTECTED> Protection;
return circle_2_do_intersectC2(
c1x.interval(),
c1y.interval(),
R1_square.interval(),
c2x.interval(),
c2y.interval(),
R2_square.interval());
}
catch (Interval_nt_advanced::unsafe_comparison)
{
Protect_FPU_rounding<!CGAL_IA_PROTECTED> Protection(CGAL_FE_TONEAREST);
return circle_2_do_intersectC2(
c1x.exact(),
c1y.exact(),
R1_square.exact(),
c2x.exact(),
c2y.exact(),
R2_square.exact());
}
}
#ifdef CGAL_IA_NEW_FILTERS
struct Static_Filtered_circle_2_do_intersectC2_6
{
static double _bound;
//static double ;
static unsigned number_of_failures; // ?
static unsigned number_of_updates;
static bool update_epsilon(
const Static_filter_error &c1x,
const Static_filter_error &c1y,
const Static_filter_error &R1_square,
const Static_filter_error &c2x,
const Static_filter_error &c2y,
const Static_filter_error &R2_square)
{
typedef Static_filter_error FT;
FT square_sum(R1_square + R2_square);
FT a(c2x - c1x);
FT b(c2y - c1y);
FT d_square(a*a + b*b);
if (d_square <= square_sum) return true;
FT x = d_square - square_sum;
return (x*x <= FT(4) * R1_square * R2_square);
}
// Call this function from the outside to update the context.
static void new_bound (const double b) // , const double error = 0)
{
_bound = b;
number_of_updates++;
// recompute the epsilons: "just" call it over Static_filter_error.
// That's the tricky part that might not work for everything.
(void) update_epsilon(b,b,b,b,b,b);
// TODO: We should verify that all epsilons have really been updated.
}
static bool epsilon_variant(
const Restricted_double &c1x,
const Restricted_double &c1y,
const Restricted_double &R1_square,
const Restricted_double &c2x,
const Restricted_double &c2y,
const Restricted_double &R2_square)
{
typedef Restricted_double FT;
FT square_sum(R1_square + R2_square);
FT a(c2x - c1x);
FT b(c2y - c1y);
FT d_square(a*a + b*b);
if (d_square <= square_sum) return true;
FT x = d_square - square_sum;
return (x*x <= FT(4) * R1_square * R2_square);
}
};
#ifndef CGAL_CFG_MATCHING_BUG_2
template < class CGAL_IA_CT, class CGAL_IA_ET, class CGAL_IA_CACHE >
#else
static
#endif
/* */
bool
circle_2_do_intersectC2(
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Static, true, CGAL_IA_CACHE> &c1x,
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Static, true, CGAL_IA_CACHE> &c1y,
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Static, true, CGAL_IA_CACHE> &R1_square,
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Static, true, CGAL_IA_CACHE> &c2x,
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Static, true, CGAL_IA_CACHE> &c2y,
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Static, true, CGAL_IA_CACHE> &R2_square)
{
// bool re_adjusted = false;
const double SAF_bound = Static_Filtered_circle_2_do_intersectC2_6::_bound;
// Check the bounds. All arguments must be <= SAF_bound.
if (
fabs(c1x.to_double()) > SAF_bound ||
fabs(c1y.to_double()) > SAF_bound ||
fabs(R1_square.to_double()) > SAF_bound ||
fabs(c2x.to_double()) > SAF_bound ||
fabs(c2y.to_double()) > SAF_bound ||
fabs(R2_square.to_double()) > SAF_bound)
{
// re_adjust:
// Compute the new bound.
double NEW_bound = 0.0;
NEW_bound = max(NEW_bound, fabs(c1x.to_double()));
NEW_bound = max(NEW_bound, fabs(c1y.to_double()));
NEW_bound = max(NEW_bound, fabs(R1_square.to_double()));
NEW_bound = max(NEW_bound, fabs(c2x.to_double()));
NEW_bound = max(NEW_bound, fabs(c2y.to_double()));
NEW_bound = max(NEW_bound, fabs(R2_square.to_double()));
// Re-adjust the context.
Static_Filtered_circle_2_do_intersectC2_6::new_bound(NEW_bound);
}
try
{
return Static_Filtered_circle_2_do_intersectC2_6::epsilon_variant(
c1x.dbl(),
c1y.dbl(),
R1_square.dbl(),
c2x.dbl(),
c2y.dbl(),
R2_square.dbl());
}
catch (...)
{
// if (!re_adjusted) { // It failed, we re-adjust once.
// re_adjusted = true;
// goto re_adjust;
// }
Static_Filtered_circle_2_do_intersectC2_6::number_of_failures++;
return circle_2_do_intersectC2(
c1x.exact(),
c1y.exact(),
R1_square.exact(),
c2x.exact(),
c2y.exact(),
R2_square.exact());
}
}
#ifndef CGAL_CFG_MATCHING_BUG_2
template < class CGAL_IA_CT, class CGAL_IA_ET, class CGAL_IA_CACHE >
#else
static
#endif
/* */
bool
circle_2_do_intersectC2(
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Static, false, CGAL_IA_CACHE> &c1x,
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Static, false, CGAL_IA_CACHE> &c1y,
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Static, false, CGAL_IA_CACHE> &R1_square,
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Static, false, CGAL_IA_CACHE> &c2x,
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Static, false, CGAL_IA_CACHE> &c2y,
const Filtered_exact <CGAL_IA_CT, CGAL_IA_ET, Static, false, CGAL_IA_CACHE> &R2_square)
{
CGAL_assertion_code(
const double SAF_bound = Static_Filtered_circle_2_do_intersectC2_6::_bound; )
CGAL_assertion(!(
fabs(c1x.to_double()) > SAF_bound ||
fabs(c1y.to_double()) > SAF_bound ||
fabs(R1_square.to_double()) > SAF_bound ||
fabs(c2x.to_double()) > SAF_bound ||
fabs(c2y.to_double()) > SAF_bound ||
fabs(R2_square.to_double()) > SAF_bound));
try
{
return Static_Filtered_circle_2_do_intersectC2_6::epsilon_variant(
c1x.dbl(),
c1y.dbl(),
R1_square.dbl(),
c2x.dbl(),
c2y.dbl(),
R2_square.dbl());
}
catch (...)
{
Static_Filtered_circle_2_do_intersectC2_6::number_of_failures++;
return circle_2_do_intersectC2(
c1x.exact(),
c1y.exact(),
R1_square.exact(),
c2x.exact(),
c2y.exact(),
R2_square.exact());
}
}
#endif // CGAL_IA_NEW_FILTERS
CGAL_END_NAMESPACE
#endif // CGAL_ARITHMETIC_FILTER_CIRCLE_2_CIRCLE_2_INTERSECTION_FTC2_H

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#ifndef CGAL_BITANGENT_2_H
#define CGAL_BITANGENT_2_H
#include <cmath>
#include <list>
#ifndef CGAL_CONVEX_ARC_2_H
#include <CEP/Visibility_complex/Arc_2.h>
#endif
CGAL_BEGIN_NAMESPACE
// -----------------------------------------------------------------------------
// -------------------- Base Bitangent class -----------------------------------
template< class D_ >
class Bitangent_base
{
public:
// -------------------------------------------------------------------------
typedef D_ Disk;
typedef const Disk* Disk_handle;
// -------------------------------------------------------------------------
enum Type { LL , RR , LR , RL };
class Type_util {
public:
Type operator()(bool b1,bool b2) const {
if ( b1 && b2) return LL;
if (!b1 && b2) return RL;
if (!b1 && !b2) return RR;
return LR;
}
};
public:
// Constructeurs -----------------------------------------------------------
Bitangent_base()
: _type(LL) , _source_object(0) , _target_object(0) { }
Bitangent_base(Type t, Disk_handle o1, Disk_handle o2)
: _type(t) , _source_object(o1) , _target_object(o2) { }
// ----------------------- Operators ---------------------------------------
bool operator==(const Bitangent_base& b) const{
return (b.type() == type() && source_object() == b.source_object() &&
target_object() == b.target_object());
}
bool operator!=(const Bitangent_base& b) const{ return !(*this == b); }
// -------- return the opposite oriented bitangent -------------------------
Type type() const { return _type; }
// ---- accesing the two objects defining the bitangent --------------------
Disk_handle source_object() const { return _source_object; }
Disk_handle target_object() const { return _target_object; }
// ---------- informations on the type of the bitangent --------------------
bool is_left_right() const { return (type() == LR); }
bool is_left_left() const { return (type() == LL); }
bool is_right_right() const { return (type() == RR); }
bool is_right_left() const { return (type() == RL); }
bool is_left_xx() const { return (type() == LL || type() == LR); }
bool is_xx_left() const { return (type() == LL || type() == RL); }
bool is_right_xx() const { return (type() == RR || type() == RL); }
bool is_xx_right() const { return (type() == RR || type() == LR); }
bool is_internal() const { return (type() == RL || type() == LR); }
bool is_external() const { return (type() == LL || type() == RR); }
// -------------------------------------------------------------------------
private :
Type _type;
Disk_handle _source_object , _target_object;
};
//------------------------------------------------------------------------------
//------------------ General definition of Bitangent_2 -------------------------
//------------------------------------------------------------------------------
template< class D_ >
struct Bitangent_2
: public Bitangent_base<D_>
{
//--------------------------------------------------------------------------
typedef D_ Disk;
typedef Arc_2<Disk> Arc_2;
typedef Bitangent_base<Disk> Base;
typedef typename Base::Disk_handle Disk_handle;
typedef typename Base::Type Type;
typedef typename D_::Point_2 Point_2;
typedef typename D_::Segment_2 Segment_2;
//--------------------------------------------------------------------------
Bitangent_2() : Base() { }
Bitangent_2(const Point_2& v1 , const Point_2& v2 , Type t ,
Disk_handle start, Disk_handle finish)
: Base(v1,v2,t,start,finish) { }
Bitangent_2(Type t, const Arc_2& source,const Arc_2& target)
: Base(t,source.object(),target.object()) { }
Bitangent_2(Type t , Disk_handle o1 , Disk_handle o2)
: Base(t,o1,o2) { }
//--------------------------------------------------------------------------
bool operator==(const Bitangent_2& b) const
{ return Base::operator==(b); }
bool operator!=(const Bitangent_2& b) const
{ return Base::operator!=(b); }
//--------------------------------------------------------------------------
};
//------------------------------------------------------------------------------
//------------------ Partial Specialization for CGAL::Polygon_2 ----------------
//------------------------------------------------------------------------------
#ifdef CGAL_POLYGON_2_H
template< class R_ , class C_ >
struct Bitangent_2 < Polygon_2<R_,C_> >
: public R_::Segment_2 ,
public Bitangent_base< Polygon_2<R_,C_> >
{
// -------------------------------------------------------------------------
typedef R_ R;
typedef typename R::FT FT;
typedef Polygon_2<R_,C_> Disk;
typedef Bitangent_base<Disk> Base;
typedef CGAL::Arc_2<Disk> Arc_2;
typedef CGAL::Arc_2<Disk> CCC;
typedef typename Base::Disk_handle Disk_handle;
typedef typename R_::Segment_2 Segment_2;
typedef typename R_::Point_2 Point_2;
typedef typename Base::Type Type;
// -------------------------------------------------------------------------
typedef typename CCC::Vertex_const_iterator Vertex_iterator;
typedef typename CCC::Vertex_const_iterator Vertex_const_iterator;
typedef typename Disk::Vertex_const_circulator Vertex_circulator;
typedef typename Disk::Vertex_const_circulator Vertex_const_circulator;
// Constructeurs -----------------------------------------------------------
Bitangent_2() : Base() { }
Bitangent_2(const Point_2& v1 , const Point_2& v2 , Type t ,
Disk_handle start, Disk_handle finish)
: Segment_2(v1,v2) , Base(t,start,finish) { }
Bitangent_2(Type t, const Arc_2& source, const Arc_2& target) {
if (source.begin() == source.end() || target.begin() == target.end()) {
*this = Bitangent_2(t,source.object(),target.object());
return;
}
bool t1,t2;
Vertex_const_iterator it_source = source.begin();
Vertex_const_iterator it_source_succ;
Vertex_const_iterator it_target = target.begin();
Vertex_const_iterator it_target_succ;
Point_2 p_source(*it_source);
Point_2 p_target(*it_target);
Point_2 p_target_succ,p_source_succ;
bool finished_source = false;
bool finished_target = false;
do {
if (it_source == source.end()) finished_source = true;
if (it_target == target.end()) finished_target = true;
if (finished_target && finished_source) break;
p_source = *(it_source);
p_target = *(it_target);
it_source_succ = it_source; ++it_source_succ;
if (it_source_succ != source.end())
p_source_succ = *(it_source_succ);
else p_source_succ = p_source;
it_target_succ = it_target; ++it_target_succ;
if (it_target_succ != target.end())
p_target_succ = *(it_target_succ);
else p_target_succ = p_target;
t1 = ( (t == RR || t == RL) && !leftturn (p_source,p_target,p_source_succ))
|| ( (t == LL || t == LR) && !rightturn(p_source,p_target,p_source_succ));
if ( collinear(p_source,p_target,p_source_succ) && p_source != p_source_succ)
t1 = are_ordered_along_line(p_source_succ,p_source,p_target);
t2 = ( (t == LL || t == RL) && !rightturn(p_source,p_target,p_target_succ))
|| ( (t == RR || t == LR) && !leftturn (p_source,p_target,p_target_succ));
if ( collinear(p_source,p_target,p_target_succ) && p_target != p_target_succ)
t2 = are_ordered_along_line(p_source,p_target,p_target_succ);
if (finished_target) ++it_source;
else if (finished_source) ++it_target;
else if (t1 == 1 && t2 == 0) ++it_target;
else if (t1 == 0 && t2 == 1) ++it_source;
else if (t1 == 0 && t2 == 0) {
// Le test d'orientation est different suivant que
// source_is_left et target_is_left sont de meme
// signe ou non
Point_2 tmp(p_target_succ + (p_source_succ - p_target));
if ( ( (t == LR || t == RL)
&& rightturn(p_source_succ, p_source, tmp))
|| ( (t == LL || t == RR)
&& leftturn (p_source_succ, p_source, tmp)))
++it_target;
else ++it_source;
}
} while (t1 == 0 || t2 == 0);
*this = Bitangent_2(p_source,p_target,t,source_obj,target_obj);
}
Bitangent_2(Type t , Disk_handle o1 , Disk_handle o2){
bool exists = true;
bool found = false;
bool found_start = false;
bool found_finish = false;
bool found_start_succ = false;
bool found_start_pred = false;
bool found_finish_succ = false;
bool found_finish_pred = false;
int loop_counter = o1->size() + o2->size();
Vertex_circulator start = o1->vertices_circulator();
Vertex_circulator finish = o2->vertices_circulator();
do {
Vertex_circulator start_succ = start; ++start_succ;
Vertex_circulator finish_succ = finish; ++finish_succ;
Vertex_circulator start_pred = start; --start_pred;
Vertex_circulator finish_pred = finish; --finish_pred;
if (t == LL || t == LR) {
// --------------------------------------------------------------------
found_start_succ =
(leftturn(*start,*finish,*start_succ) ||
*start == *start_succ ||
(collinear(*start,*finish,*start_succ) &&
are_ordered_along_line(*start_succ,*start,*finish)));
// --------------------------------------------------------------------
found_start_pred =
(leftturn(*start,*finish,*start_pred) ||
*start == *start_pred ||
(collinear(*start,*finish,*start_pred) &&
are_ordered_along_line(*start_pred,*start,*finish)));
// --------------------------------------------------------------------
}
else {
// --------------------------------------------------------------------
found_start_succ =
(rightturn(*start,*finish,*start_succ) ||
*start == *start_succ ||
(collinear(*start,*finish,*start_succ) &&
are_ordered_along_line(*start_succ,*start,*finish)));
// --------------------------------------------------------------------
found_start_pred =
(rightturn(*start,*finish,*start_pred) ||
*start == *start_pred ||
(collinear(*start,*finish,*start_pred) &&
are_ordered_along_line(*start_pred,*start,*finish)));
// --------------------------------------------------------------------
}
found_start = (found_start_pred && found_start_succ);
if (t == LL || t == RL) {
// --------------------------------------------------------------------
found_finish_succ =
(rightturn(*finish,*start,*finish_succ) ||
*finish == *finish_succ ||
(collinear(*finish,*start,*finish_succ) &&
are_ordered_along_line(*start,*finish,*finish_succ)));
// --------------------------------------------------------------------
found_finish_pred =
(rightturn(*finish,*start,*finish_pred) ||
*finish == *finish_pred ||
(collinear(*finish,*start,*finish_pred) &&
are_ordered_along_line(*start,*finish,*finish_pred)));
// --------------------------------------------------------------------
}
else {
// --------------------------------------------------------------------
found_finish_succ =
(leftturn(*finish,*start,*finish_succ) ||
*finish == *finish_succ ||
(collinear(*finish,*start,*finish_succ) &&
are_ordered_along_line(*start,*finish,*finish_succ)));
// --------------------------------------------------------------------
found_finish_pred =
(leftturn(*finish,*start,*finish_pred) ||
*finish == *finish_pred ||
(collinear(*finish,*start,*finish_pred) &&
are_ordered_along_line(*start,*finish,*finish_pred)));
// --------------------------------------------------------------------
}
found_finish = (found_finish_pred && found_finish_succ);
/* If found == true we have found the bitangent */
found = (found_start && found_finish);
/* Otherwise we compute one more determinant to decide on which */
/* object to advance */
if (found == false)
{
if (( (t == LR || t == RL)
&& rightturn(*start_succ,*start,*finish_succ + (*start_succ - *finish)))||
((t == LL || t == RR)
&& leftturn (*start_succ,*start,*finish_succ + (*start_succ - *finish))))
{
++finish;
}
else if (((t == LR || t == RL)
&& leftturn (*start_succ,*start,*finish_succ + (*start_succ - *finish)))||
((t == LL || t == RR)
&& rightturn(*start_succ,*start,*finish_succ + (*start_succ - *finish))))
{
++start;
}
else if (found_start == true && found_finish == false) ++finish;
else ++start;
}
--loop_counter;
if (loop_counter < -100) {exists = false; break; }
} while (found == false);
if (exists){ *this = Bitangent_2(*start,*finish,t,o1,o2); }
else { *this = Bitangent_2(); }
}
//--------------------------------------------------------------------------
bool operator==(const Bitangent_2& b) const
{ return Base::operator==(b); }
bool operator!=(const Bitangent_2& b) const
{ return Base::operator!=(b); }
//--------------------------------------------------------------------------
};
#endif // CGAL_POLYGON_2_H
//------------------------------------------------------------------------------
//------------------------------------------------------------------------------
//------------------ Partial Specialization for CGAL::Point_2 ------------------
//------------------------------------------------------------------------------
template< class R_>
struct Bitangent_2 < Point_2<R_> >
: public R_::Segment_2 ,
public Bitangent_base< Point_2<R_> >
{
// -------------------------------------------------------------------------
typedef R_ R;
typedef typename R::FT FT;
typedef Point_2<R_> Disk;
typedef Bitangent_base<Disk> Base;
typedef Arc_2<Disk> Arc_2;
typedef typename R_::Segment_2 Segment_2;
typedef typename R_::Point_2 Point_2;
typedef typename Base::Type Type;
typedef typename Base::Disk_handle Disk_handle;
// Constructeurs -----------------------------------------------------------
Bitangent_2() : Base() { }
Bitangent_2(const Point_2& v1 , const Point_2& v2 ,
Type t, Disk_handle start, Disk_handle finish)
: Segment_2(v1,v2) , Base(t,start,finish) { }
Bitangent_2(Type t, const Arc_2& source, const Arc_2& target)
: Segment_2(*source.object(),*target.object()) ,
Base(t,source.object(),target.object()) { }
Bitangent_2(Type t , Disk_handle o1 , Disk_handle o2)
: R_::Segment_2(*o1,*o2) , Base(t,o1,o2) { }
//--------------------------------------------------------------------------
bool operator==(const Bitangent_2& b) const
{ return Base::operator==(b); }
bool operator!=(const Bitangent_2& b) const
{ return Base::operator!=(b); }
//--------------------------------------------------------------------------
};
//------------------------------------------------------------------------------
//------------------------------------------------------------------------------
//---------- Partial Specialization for CGAL::Circle_by_radius -----------------
//------------------------------------------------------------------------------
#ifdef CGAL_CIRCLE_BY_RADIUS_2_H
template< class R_>
class Bitangent_2 < Circle_by_radius_2<R_> >
: public R_::Segment_2 ,
public Bitangent_base< Circle_by_radius_2<R_> >
{
public:
// -------------------------------------------------------------------------
typedef R_ R;
typedef typename R::FT FT;
typedef Circle_by_radius_2<R_> Disk;
typedef Bitangent_base<Disk> Base;
typedef Base::Disk_handle Disk_handle;
typedef Arc_2<Disk> Arc_2;
typedef typename R::Segment_2 Segment_2;
typedef typename R::Point_2 Point_2;
typedef Base::Type Type;
// -------------------------------------------------------------------------
public:
// Constructeurs -----------------------------------------------------------
Bitangent_2() : Base() { }
Bitangent_2(const Point_2& v1 , const Point_2& v2 ,
Type t , Disk_handle start, Disk_handle finish)
: Segment_2(v1,v2) , Base(t,start,finish) { }
Bitangent_2(Type t , Disk_handle o1 , Disk_handle o2) : Base(t,o1,o2)
{
compute();
}
Bitangent_2(Type t, const Arc_2& source, const Arc_2& target)
{
*this = Bitangent_2(t,source.object(),target.object());
compute();
}
//--------------------------------------------------------------------------
Point_2 source() const
{
return Point_2(source_object()->center().x() + R1*pbra,
source_object()->center().y() - R1*parb);
}
Point_2 target() const
{
return Point_2(target_object()->center().x() + R2*pbra,
target_object()->center().y() - R2*parb);
}
//--------------------------------------------------------------------------
bool operator==(const Bitangent_2& b) const
{ return Base::operator==(b); }
bool operator!=(const Bitangent_2& b) const
{ return Base::operator!=(b); }
//--------------------------------------------------------------------------
private:
void compute() {
R1 = (is_left_xx())? source_object()->radius():
- source_object()->radius();
R2 = (is_xx_left())? target_object()->radius():
- target_object()->radius();
a = target_object()->center().x() - source_object()->center().x();
b = target_object()->center().y() - source_object()->center().y();
r = R2 - R1;
FT aabb = a*a + b*b;
p = CGAL::sqrt(aabb - r*r);
pbra = (p*b - r*a) / aabb;
parb = (p*a + r*b) / aabb;
}
FT R1,R2,a,b,r,p,pbra,parb;
};
#endif // CGAL_CIRCLE_BY_RADIUS_2_H
//------------------------------------------------------------------------------
//------------------------------------------------------------------------------
//------------------ Partial Specialization for CGAL::Segment_2 ----------------
//------------------------------------------------------------------------------
#ifdef CGAL_SEGMENT_2_H
template< class R_>
class Bitangent_2 < Segment_2<R_> >
: public R_::Segment_2 ,
public Bitangent_base< Segment_2<R_> >
{
public:
// -------------------------------------------------------------------------
typedef R_ R;
typedef typename R::FT FT;
typedef typename R::Segment_2 Segment_2;
typedef typename R::Point_2 Point_2;
typedef Bitangent_base<Segment_2> Base;
typedef typename Base::Disk Disk;
typedef typename Base::Disk_handle Disk_handle;
typedef Arc_2<Disk> Arc_2;
typedef typename Base::Type Type;
// -------------------------------------------------------------------------
public:
// Constructeurs -----------------------------------------------------------
Bitangent_2() : Base() { }
Bitangent_2(const Point_2& v1 , const Point_2& v2 ,
Type t, Disk_handle start, Disk_handle finish)
: Segment_2(v1,v2) , Base(t,start,finish) { }
Bitangent_2(Type t , Disk_handle o1 , Disk_handle o2) {
if (is_bitangent(t,o1->source(),o2->source(),
o1->target(),o2->target()))
*this = Bitangent_2(o1->source(),o2->source(),t,o1,o2);
else if (is_bitangent(t,o1->source(),o2->target(),
o1->target(),o2->source()))
*this = Bitangent_2(o1->source(),o2->target(),t,o1,o2);
else if (is_bitangent(t,o1->target(),o2->source(),
o1->source(),o2->target()))
*this = Bitangent_2(o1->target(),o2->source(),t,o1,o2);
else *this = Bitangent_2(o1->target(),o2->target(),t,o1,o2);
}
Bitangent_2(Type t, const Arc_2& source, const Arc_2& target)
{ *this = Bitangent_2(t,source.object(),target.object()); }
//--------------------------------------------------------------------------
bool operator==(const Bitangent_2& b) const
{ return Base::operator==(b); }
bool operator!=(const Bitangent_2& b) const
{ return Base::operator!=(b); }
// -------------------------------------------------------------------------
private:
// b = (p1 , p2) and q1 and q2 are the two other points respectively on
// source and target object. Returns true if the bitangent is valid.
bool is_bitangent(Type t , const Point_2& p1, const Point_2& p2,
const Point_2& q1, const Point_2& q2)
{
return ((t == LL
&& (leftturn (p1,p2,q1) ||
(collinear(p1,p2,q1) && are_ordered_along_line(q1,p1,p2)))
&& (leftturn (p1,p2,q2) ||
(collinear(p1,p2,q2) && are_ordered_along_line(p1,p2,q2))))
|| (t == LR
&& (leftturn (p1,p2,q1) ||
(collinear(p1,p2,q1) && are_ordered_along_line(q1,p1,p2)))
&& (rightturn(p1,p2,q2) ||
(collinear(p1,p2,q2) && are_ordered_along_line(p1,p2,q2))))
|| (t == RR
&& (rightturn(p1,p2,q1) ||
(collinear(p1,p2,q1) && are_ordered_along_line(q1,p1,p2)))
&& (rightturn(p1,p2,q2) ||
(collinear(p1,p2,q2) && are_ordered_along_line(p1,p2,q2))))
|| (t == RL
&& (rightturn(p1,p2,q1) ||
(collinear(p1,p2,q1) && are_ordered_along_line(q1,p1,p2)))
&& (leftturn (p1,p2,q2) ||
(collinear(p1,p2,q2) && are_ordered_along_line(p1,p2,q2)))));
}
//--------------------------------------------------------------------------
};
#endif // CGAL_SEGMENT_2_H
//------------------------------------------------------------------------------
template < class D_ >
std::ostream &
operator<<(std::ostream &os, const Bitangent_2<D_> &b)
{
switch (b.type()) {
case Bitangent_2<D_>::LL : os<<"LL "; break;
case Bitangent_2<D_>::LR : os<<"LR "; break;
case Bitangent_2<D_>::RL : os<<"RL "; break;
case Bitangent_2<D_>::RR : os<<"RR "; break;
default: os<<"Unknown type ";
}
typedef Bitangent_2<D_> Bitangent_2;
os << static_cast<const typename Bitangent_2::Segment_2&>(b);
return os;
}
//------------------------------------------------------------------------------
CGAL_END_NAMESPACE
#endif

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#ifndef CGAL_BITANGENT_2_BITANGENT_2_INTERSECTION_H
#define CGAL_BITANGENT_2_BITANGENT_2_INTERSECTION_H
#include <CGAL/basic.h>
#include <CEP/Visibility_complex/Bitangent_2.h>
#include <CGAL/Segment_2_Segment_2_intersection.h>
CGAL_BEGIN_NAMESPACE
// -----------------------------------------------------------------------------
template < class D_ >
bool do_intersect( const Bitangent_2<D_>& b1, const Bitangent_2<D_>& b2 )
{
typedef typename Bitangent_2<D_>::Segment_2 Segment_2;
// -------------------------------------------------------------------------
if (b1 == b2) return true;
// -------------------------------------------------------------------------
if (b1.source_object() == b2.source_object() &&
b1.target_object() == b2.target_object())
return (b1.is_internal() && b2.is_internal());
// -------------------------------------------------------------------------
if (b1.source_object() == b2.target_object() &&
b1.target_object() == b2.source_object()) {
if (b1.is_internal() && b2.is_internal()) return true;
if (b1.is_internal() || b2.is_internal()) return false;
return (b1.type() != b2.type());
}
// -------------------------------------------------------------------------
if (b1.source() == b2.source()) {
if (b1.is_right_xx() == b2.is_right_xx()) return false;
return ((b1.is_right_xx() &&
leftturn(b1.source(),b1.target(),b2.target())) ||
(b2.is_right_xx() &&
leftturn(b2.source(),b2.target(),b1.target())));
}
// -------------------------------------------------------------------------
if (b1.target() == b2.target()) {
if (b1.is_xx_right() == b2.is_xx_right()) return false;
return ((b1.is_xx_right() &&
leftturn(b1.source(),b1.target(),b2.source())) ||
(b2.is_xx_right() &&
leftturn(b2.source(),b2.target(),b1.source())));
}
// -------------------------------------------------------------------------
if (b1.target() == b2.source()) {
if (b1.is_xx_right() != b2.is_right_xx()) return false;
return ((b1.is_xx_right() &&
leftturn(b1.source(),b1.target(),b2.target())) ||
(b1.is_xx_left() &&
rightturn(b1.source(),b1.target(),b2.target())));
}
// -------------------------------------------------------------------------
if (b1.source() == b2.target()) {
if (b1.is_right_xx() != b2.is_xx_right()) return false;
return ((b2.is_xx_right() &&
leftturn(b2.source(),b2.target(),b1.target())) ||
(b2.is_xx_left() &&
rightturn(b2.source(),b2.target(),b1.target())));
}
// -------------------------------------------------------------------------
return do_intersect(Segment_2(b1.source(),b1.target()),
Segment_2(b2.source(),b2.target()));
// -------------------------------------------------------------------------
}
// -----------------------------------------------------------------------------
CGAL_END_NAMESPACE
#endif // CGAL_BITANGENT_2_BITANGENT_2_INTERSECTION_H

24
Packages/Visibility_complex/include/CEP/Visibility_complex/Circle_2_Bitangent_2_intersection.h Normal file
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#ifndef CGAL_CIRCLE_2_BITANGENT_2_INTERSECTION_H
#define CGAL_CIRCLE_2_BITANGENT_2_INTERSECTION_H
#include <CGAL/basic.h>
#include <CGAL/Point_2.h>
#include <CGAL/Circle_2.h>
#include <CEP/Visibility_complex/Bitangent_2.h>
CGAL_BEGIN_NAMESPACE
// -----------------------------------------------------------------------------
template < class R_ , class C_ >
bool do_intersect( const Circle_2<R_>& c1, const Bitangent_2<C_>& c2 )
{
cerr << "Not implemented Circle_2 - Bitangent_2 intersection !" << endl;
return false;
}
// -----------------------------------------------------------------------------
CGAL_END_NAMESPACE
#endif // CGAL_CIRCLE_2_BITANGENT_2_INTERSECTION_H

26
Packages/Visibility_complex/include/CEP/Visibility_complex/Circle_2_Circle_2_intersection.h Normal file
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#ifndef CGAL_CIRCLE_2_CIRCLE_2_INTERSECTION_H
#define CGAL_CIRCLE_2_CIRCLE_2_INTERSECTION_H
#include <CGAL/basic.h>
#include <CGAL/Point_2.h>
#include <CGAL/Circle_2.h>
#include <CEP/Visibility_complex/predicates/Circle_2_Circle_2_intersection_ftC2.h>
CGAL_BEGIN_NAMESPACE
// -----------------------------------------------------------------------------
template < class Rep_ >
bool do_intersect( const Circle_2<Rep_>& c1, const Circle_2<Rep_>& c2 )
{
return circle_2_do_intersectC2(c1.center().x(),c1.center().y(),
c1.squared_radius(),
c2.center().x(),c2.center().y(),
c2.squared_radius());
}
// -----------------------------------------------------------------------------
CGAL_END_NAMESPACE
#endif

60
Packages/Visibility_complex/include/CEP/Visibility_complex/Circle_by_radius_2.h Normal file
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#ifndef CGAL_CIRCLE_BY_RADIUS_2_H
#define CGAL_CIRCLE_BY_RADIUS_2_H
#include <CGAL/Circle_2.h>
CGAL_BEGIN_NAMESPACE
// -----------------------------------------------------------------------------
template <class R_>
class Circle_by_radius_2 : public R_::Circle_2
{
public:
// -------------------------------------------------------------------------
typedef R_ R;
typedef typename R::RT RT;
typedef typename R::FT FT;
typedef typename R::Circle_2 Circle_2;
typedef typename R::Point_2 Point_2;
// -------------------------------------------------------------------------
Circle_by_radius_2()
: Circle_2() , _radius(0) { }
Circle_by_radius_2(const Circle_2 &t)
: Circle_2(t) , _radius(CGAL_NTS sqrt(t.squared_radius())) { }
Circle_by_radius_2(const Point_2&center,
const FT &radius,
const Orientation &orientation)
: Circle_2(center, radius*radius, orientation) , _radius(radius) { }
Circle_by_radius_2(const Point_2&center,
const FT &radius)
: Circle_2(center, radius*radius, COUNTERCLOCKWISE) , _radius(radius) { }
Circle_by_radius_2(const Point_2&p, const Point_2&q, const Point_2&r)
: Circle_2(p,q,r) ,
_radius(CGAL_NTS sqrt(squared_radius())) { }
Circle_by_radius_2(const Point_2& p, const Point_2& q,
const Orientation &orientation)
: Circle_2(p,q,orientation) ,
_radius(CGAL_NTS sqrt(squared_radius())) { }
Circle_by_radius_2(const Point_2& p,
const Point_2& q)
: Circle_2(p,q,COUNTERCLOCKWISE) ,
_radius(CGAL_NTS sqrt(squared_radius())) { }
Circle_by_radius_2(const Point_2& center,
const Orientation& orientation)
: Circle_2(center,FT(0),orientation) ,
_radius(CGAL_NTS sqrt(squared_radius())) { }
Circle_by_radius_2(const Point_2& center)
: Circle_2(center,FT(0),COUNTERCLOCKWISE) ,
_radius(CGAL_NTS sqrt(squared_radius())) { }
// -------------------------------------------------------------------------
FT radius() const { return _radius; }
private:
FT _radius;
};
// -----------------------------------------------------------------------------
CGAL_END_NAMESPACE
#endif // CGAL_CIRCLE_BY_RADIUS_2_H

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Packages/Visibility_complex/include/CEP/Visibility_complex/Polygon_2_Polygon_2_intersection.h Normal file
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#ifndef CGAL_POLYGON_2_POLYGON_2_INTERSECTION_H
#define CGAL_POLYGON_2_POLYGON_2_INTERSECTION_H
#include <CGAL/basic.h>
#include <CGAL/Segment_2.h>
#include <CGAL/Polygon_2.h>
#include <CGAL/Segment_2_Segment_2_intersection.h>
CGAL_BEGIN_NAMESPACE
// -----------------------------------------------------------------------------
template < class Traits , class Container >
bool do_intersect( const Polygon_2<Traits,Container>& P,
const Polygon_2<Traits,Container>& Q )
{
// -------------------------------------------------------------------------
// This function assumes the polygons are convex.
CGAL_precondition(P.is_convex() && Q.is_convex());
// -------------------------------------------------------------------------
// If P or Q is a point we use the CGAL bounded_side methods
if (P.size() == 1)
return (Q.bounded_side(*P.vertices_begin()) != CGAL::ON_UNBOUNDED_SIDE);
if (Q.size() == 1)
return (P.bounded_side(*Q.vertices_begin()) != CGAL::ON_UNBOUNDED_SIDE);
// -------------------------------------------------------------------------
// two booleans to avoid cycling more than twice on a Polygon.
bool a_has_cycled = false;
bool b_has_cycled = false;
// -------------------------------------------------------------------------
typedef Polygon_2<Traits,Container> Polygon_2;
typedef typename Polygon_2::Vertex_const_iterator Vertex_const_iterator;
Vertex_const_iterator a = P.vertices_begin();
Vertex_const_iterator b = Q.vertices_begin();
// -------------------------------------------------------------------------
do {
// ---------------------------------------------------------------------
if (a == P.vertices_end()) {
a = P.vertices_begin(); a_has_cycled = true;
}
if (b == Q.vertices_end()) {
b = Q.vertices_begin(); b_has_cycled = true;
}
// ---------------------------------------------------------------------
// Computations of key variables.
Vertex_const_iterator apred , bpred;
apred = (a == P.vertices_begin()) ? P.vertices_end() : a; --apred;
bpred = (b == Q.vertices_begin()) ? Q.vertices_end() : b; --bpred;
// ---------------------------------------------------------------------
// If A & B intersect, return true.
typedef typename Polygon_2::Segment_2 Segment_2;
/*
cout << "apred->Ptr() = " << long(apred->Ptr()) << endl;
cout << "bpred->Ptr() = " << long(bpred->Ptr()) << endl;
cout << "a->Ptr() = " << long(a->Ptr()) << endl;
cout << "b->Ptr() = " << long(b->Ptr()) << endl;
*/
Segment_2 s1(*apred,*a);
Segment_2 s2(*bpred,*b);
if ( do_intersect(s1,s2) ) return true;
// ---------------------------------------------------------------------
// Else Advance
// ---------------------------------------------------------------------
Orientation chi2 = orientation( *apred , *a , *b + (*apred - *bpred) );
Orientation chi1b = orientation( *bpred, *b, *a );
Orientation chi1a = orientation( *apred, *a, *b );
// ---------------------------------------------------------------------
if ( chi2 == COLLINEAR ) {
if ( chi1b == RIGHTTURN && chi1a == RIGHTTURN ) return false;
if ( chi1b == COLLINEAR && chi1a != RIGHTTURN ) ++a;
else ++b;
}
else if ( chi2 == LEFTTURN ) {
if ( chi1a == LEFTTURN ) ++a;
else ++b;
}
else {
if ( chi1b == LEFTTURN ) ++b;
else ++a;
}
// ---------------------------------------------------------------------
} while ( (a != P.vertices_end() || !a_has_cycled) &&
(b != Q.vertices_end() || !b_has_cycled) );
// -------------------------------------------------------------------------
return ((P.bounded_side(*Q.vertices_begin()) != CGAL::ON_UNBOUNDED_SIDE) ||
(Q.bounded_side(*P.vertices_begin()) != CGAL::ON_UNBOUNDED_SIDE));
// -------------------------------------------------------------------------
//return false;
// -------------------------------------------------------------------------
}
// -----------------------------------------------------------------------------
CGAL_END_NAMESPACE
#endif

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