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Shai Hirsch 2001-05-22 12:13:15 +00:00
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% +------------------------------------------------------------------------+
% | Reference manual page: Pmwx.tex (Planar map with intersections)
% +------------------------------------------------------------------------+
% |
% | Package: arr (Planar_map_with_intersections_2)
% |
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +------------------------------------------------------------------------+
% +========================================================================+
% Planar_map_with_intersections_2
% +========================================================================+
\begin{ccRefClass}{Planar_map_with_intersections_2<Planar_map>}
\ccDefinition An object \ccStyle{pmwx} of the class
\ccClassTemplateName\ represents the planar
subdivision induced by a set of curves that are possibly intersecting.
This class extends the class \ccStyle{Planar_map_2<Dcel,Traits>}
which handles $x$-monotone pairwise interior disjoint
curves. %The class have the following internal typedefs:
An object of the class \ccClassTemplateName\ is also a planar map and
therefore the types, constants, and traversal iterators and operations remain
the same.
\ccInclude{CGAL/Pm_with_intersections.h}
\ccInheritsFrom
\ccStyle{Planar_map_2<Dcel,Traits>}
The modifying functions \ccStyle{insert}
override the inherited functions
%\comment{iddo: need to be virtual in planar map}.
\ccTypes
\ccNestedType{Pmwx_change_notification}{}
\ccCreation
\ccCreationVariable{pmwx}
\ccConstructor{Planar_map_with_intersections_2<Planar_map>();}
{creates an empty planar map with the default point location strategy.}
\ccConstructor{Planar_map_with_intersections_2<Planar_map>(Pm_point_location_base<Planar_map> *pl);}
{creates an empty planar map with \ccc{*pl} as the point location strategy.}
\ccConstructor{Planar_map_with_intersections_2<Planar_map>(const Traits& tr, Pm_point_location_base<Planar_map> *pl);}
{creates an empty planar map with \ccc{tr} as the traits class and \ccc{*pl} as the point location strategy.}
\ccOperations
\ccMethod{Halfedge_handle insert(const X_curve& cv, Pmwx_change_notification *en = NULL);}
{inserts the $x$-monotone curve \ccStyle{cv} into the planar map.
Returns the last inserted halfedge whose target points to the
target point of \ccStyle{cv}.
\ccPrecond{\ccc{cv} is not equivalent to a point. }
}
\ccMethod{Halfedge_handle insert_from_vertex(const X_curve& cv, Vertex_handle src,
Pmwx_change_notification *en = NULL);}
{inserts the $x$-monotone curve \ccStyle{cv}, for
which the source vertex, \ccStyle{src}, is already in the
arrangement.
Returns the last inserted halfedge whose target points to the
target point of \ccStyle{cv}.
This function is more efficient than the \ccc{insert} function,
since it does not invoke a point location query for the
endpoint \ccc{src}. Therefore,
whenever possible, it is recommended to use this one rather
than \ccc{insert}.
\ccPrecond{\ccc{src} holds a point which is the same as \ccc{cv's}
source.}
\ccPrecond{\ccc{cv} is not equivalent to a point. }
}
\begin{ccAdvanced}
\ccMethod{Halfedge_handle non_intersecting_insert_from_vertex(const X_curve& cv, Vertex_handle src, bool
source);}
{invokes the planar map implementation of
\ccc{insert_from_vertex} function. Since this function does not
check for intersections it is more efficient than \ccc{insert_from_vertex} but
it should not be called when \ccc{cv} could have intersections with other
curves of the planar map.
\ccPrecond{\ccc{cv} has no intersection in the interior with any curve of the map. }
\ccPrecond{\ccc{src} holds a point which is the same as \ccc{cv}'s source.}
}
\ccMethod{Halfedge_handle non_intersecting_insert(const X_curve&
cv);}
{invokes the planar map implementation of
\ccc{insert} function. Since this function does not
check for intersections it is more efficient than \ccc{insert} but
it should not be called when \ccc{cv} could have intersections with other
curves of the planar map.
\ccPrecond{\ccc{cv} has no intersection in the interior with any curve of the map. }
}
\end{ccAdvanced}
% The following operations have the same functionality as their
% counterparts in the planar map.
%
% \ccMethod{Vertex_iterator vertices_begin();}{}
% \ccMethod{Vertex_iterator vertices_end();}{}
% \ccMethod{Size number_of_vertices();}{}
% \ccMethod{Halfedge_iterator halfedges_begin();}{}
% \ccMethod{Halfedge_iterator halfedges_end();}{}
% \ccMethod{Size number_of_halfedges();}{}
% \ccMethod{Face_iterator faces_begin();}{}
% \ccMethod{Face_iterator faces_end();}{}
% \ccMethod{Size number_of_faces();}{}
%
% \ccMethod{Halfedge_handle split_edge(Halfedge_handle e, const X_curve& c1,
% const X_curve& c2);}{}
% \ccMethod{Halfedge_handle locate(const Point& p, Locate_type& lt);}{}
% \ccMethod{Halfedge_handle vertical_ray_shoot(const Point& p, Locate_type& lt, bool up);}{}
%
% \ccMethod{Face_handle unbounded_face();}{}
%
% \ccMethod{bool is_valid(bool verbose=false);}{}
\end{ccRefClass}
% +------------------------------------------------------------------------+
%%RefPage: end of main body, begin of footer
\ccRefPageEnd
% EOF
% +------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Pmwx_change_notification.tex
% +------------------------------------------------------------------------+
% |
% | Package: arr (Planar_map_with_intersections_2)
% |
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +------------------------------------------------------------------------+
% +========================================================================+
% Pmwx_change_notification
% +========================================================================+
\begin{ccRefConcept}{Planar_Map_with_Intersections_Change_Notification}
\ccDefinition
An instance of \ccRefName\ can be passed to the modification
methods of a planar map with intersections object. An insertion
of an intersecting curve to a planar map may add several
halfedges and modify several features of the map. The relevant
function of the change notification instance is called after
each such modification.
\ccTypes
\ccNestedType{Planar_map}{the planar map in which the object is being used.}
\ccNestedType{Traits}{the traits class of \ccc{Planar_map}.}
\ccCreationVariable{pmwx_cn}
\ccMethod{virtual void add_edge(
const typename Traits::X_curve& cv,
typename Planar_map::Halfedge_handle e,
bool original_direction, bool overlap);}
{called each time a pair of twin halfedges is added to the planar map.
\ccc{cv} is the curve of the added halfedges.
\ccc{e} is on of the new halfedges.
\ccc{original_direction} is true iff \ccc{cv} is directed the same as the curve that we insert to the map.
\ccc{overlap} is true iff the added halfedges overlap previously inserted halfedges in the map.}
\ccMethod{virtual void split_edge(
typename Planar_map::Halfedge_handle orig_edge,
typename Planar_map::Halfedge_handle new_edge,
const typename Traits::X_curve& c1,
const typename Traits::X_curve& c2);}
{called each time a pair of twin halfedges is split into two
pairs. The split operation creates a new pair of twin halfedges
(\ccc{new_edge} and its twin) and modifies the original
pair (\ccc{orig_edge} an its twin).
\ccc{c1} and \ccc{c2} are the curves after the split.
}
\ccMethod{virtual void split_face(
typename Planar_map::Face_handle orig_face,
typename Planar_map::Face_handle new_face);}
{called each time a face of the planar map is being split by
the inserted curve into two faces: \ccc{orig_face} is the
original face and \ccc{new_face} is the new face.
}
\ccMethod{virtual void add_hole(
typename Planar_map::Face_handle in_face,
typename Planar_map::Halfedge_handle new_hole);}
{called each time a hole is added to a face \ccc{in_face}.
\ccc{new_hole} is a halfedge on the new hole.
}
\ccMethod{virtual bool have_support_curve();}
{returns true if \ccc{edge_support_curve} function can return the
supporting curve (used by the arrangement class).
}
\ccMethod{virtual const typename Traits::X_curve&
edge_support_curve(typename Planar_map::Halfedge_handle edge);}
{returns the supporting curve of a \ccc{edge}, if it exists
(used by the arrangement class).
}
\ccHasModels
The package does not supply specific implementations of
\ccc{Planar Map with Intersections Change Notification}. The arrangement class has an
internal implementation that is used for the updating of its
hierarchy structure.
\end{ccRefConcept}
% +------------------------------------------------------------------------+
%%RefPage: end of main body, begin of footer
\ccRefPageEnd
% EOF
% +------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Pmwx_2_traits.tex (Arrangement)
% +------------------------------------------------------------------------+
% |
% | Package: arr (Planar_map_with_intersections_2)
% |
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +------------------------------------------------------------------------+
% +========================================================================+
% Arrangement_2_traits
% +========================================================================+
\begin{ccRefConcept}{Planar_Map_with_Intersections_Traits}
\ccDefinition
The \ccRefName\ concept is a refinement of the \ccc{Planar Map Traits}
concept \lcTex{(\ccRefPage{Planar_map_traits})}.
The requirements of the latter
are requirements for the former.
The additional requirements are the following types and methods.
%The \ccStyle{Point} can be implemented for non-linear curves as the
%intersection of 2 curves.
\ccTypes
\ccNestedType{Point}{same as the planar map traits' \ccc{Point}.}
\ccNestedType{X_curve}{same as the planar map traits' \ccc{X_curve}.}
\ccCreationVariable{pmwx_traits}
\ccMethod{Point point_reflect_in_x_and_y(const Point& pt);}{returns the
\ccc{Point}
that is the the reflection of \ccc{pt} in the $x$ and $y$ axes. E.g.,
the point $(2,2)$ will be reflected as $(-2,-2)$. }
\ccMethod{X_curve curve_reflect_in_x_and_y(const X_curve& cv);}{returns the \ccc{X_curve}
that is the the reflection of \ccc{cv} in the $x$ and $y$ axes. E.g., the line segment $((2,2),(3,3))$ will be reflected
as $((-2,-2),(-3,-3))$. }
\ccMethod{void curve_split(const X_curve& cv, X_curve& c1, X_curve& c2,
const Point& split_pt);}{splits $cv$ at \ccc{split_pt}
and assigns the resulting two curves to \ccc{c1} and \ccc{c2}.
\ccPrecond{\ccc{split_pt} is on \ccc{cv}.}
\ccPrecond{\ccc{split_pt} is not an endpoint of \ccc{cv}.}
}
\ccMethod{bool do_intersect_to_right(const X_curve& c1, const X_curve& c2,
const Point& pt);}{returns $true$ if \ccc{c1}
and \ccc{c2} intersect to the right of the point \ccc{pt}.
Intersection to the right of \ccc{pt} is defined as
an intersection which is lexicographically strictly to the right of \ccc{pt}
(i.e., an intersection above or to the right of
\ccc{pt} but {\em not} on \ccc{pt}).}
\ccMethod{bool nearest_intersection_to_right(const X_curve& c1,
const X_curve& c2,
const Point& pt,
Point& p1, Point& p2);}
{finds the
nearest intersection point to the right of \ccc{pt}. Nearest is defined as the
one lexicographically closest to \ccc{pt} not including \ccc{pt} itself.
If the intersection of \ccc{c1} and \ccc{c2} to the
right of \ccc{pt} is an \ccc{X_curve} (i.e., there is an overlapping subcurve
to the right of \ccc{pt}),
then \ccc{p1} will hold the source point of the intersection and
\ccc{p2} will hold its target point. Otherwise \ccc{p1} and \ccc{p2} are
identical. In the case of an overlap we regard the intersection
to the right as the open curve between \ccc{p1} and \ccc{p2}. This means that
if the overlapping subcurve contains \ccc{pt} either \ccc{p1} or \ccc{p2} will
be equal to \ccc{pt}, this is the only case in which this can happen.
If \ccc{c1} and \ccc{c2} do not intersect to the right of \ccc{pt}
the function returns \ccc{false}.
%\ccPrecond{\ccc{do_intersect_to_right(c1,c2,pt) == true}}
}
%The intersection function is defined in such a way to enable dealing with
%degenerate cases when two curves inserted into the arrangement overlap.
\ccMethod{bool curves_overlap(const X_curve& cv1, const X_curve& cv2);}
{returns \ccc{true} if \ccc{cv1} and \ccc{cv2} overlap
in a one-dimensional subcurve (i.e., not in a finite number
of points), otherwise returns
\ccc{false}.}
\ccHasModels
The package does not supply specific implementations of the planar
map with intersections traits concept. However, since the
\ccc{Arrangement Traits} concept is a refinment of the \ccRefName\
concept, all of the models of the former are models of the latter.
\ccc{Arr_segment_exact_traits<R>}\\
\ccc{Arr_leda_segment_exact_traits}\\
\ccc{Arr_circles_real_traits<NT>}\\
\ccc{Arr_polyline_traits<R, Container>}
\end{ccRefConcept}
% +------------------------------------------------------------------------+
%%RefPage: end of main body, begin of footer
\ccRefPageEnd
% EOF
% +------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual chapter: intro.tex (Pmwx)
% +------------------------------------------------------------------------+
% |
% | Package: arr (Arrangement_2)
% |
% +------------------------------------------------------------------------+
%+---------------------------------------------------------------------------80
%| update log
%|
%| 04 May 2001 - Eyal Flato
%| Created from arr documentation
%|
%+---------------------------------------------------------------------------80
% +========================================================================+
% Introduction
% +========================================================================+
\clearpage
\section{Planar Map with Intersections Reference Pages}
\subsection*{Introduction}
\ccRefLabel{pmwx_ref_intro}
The planar map with intersections class extends the planar map
class. The planar map with intersections adds insertion functions
that handle intersections and overlapping among the curves of the
planar map. The basic types of the planar map (vertex, halfedge,
face, Dcel, etc.) are kept. \ccc{Planar_map_with_intersections_2}
is templated with the planar map class it derives from.
\subsection*{Planar Map with Intersections Traits Class}
%-----------------------------------------------
Since the planar map with intersections has an additional
functionality of intersecting curves it needs additional
functionality of the traits class. We describe the planar map with
intersections traits concept following in this chapter.
Nevertheless, we do not have specific implementation of this
concept, and we use the arrangement traits instead (see
Chapter~\ref{I1_ChapterArrangement}).
\begin{ccAdvanced}
\subsection*{Change Notification}
%-----------------------------------------------
An insertion of an intersecting curve to a planar map may add
several halfedges and modify several features of the map (i.e.,
split halfedges, split faces). The relevant function of the change
notification instance, that is passed to the modification methods
of a planar map with intersections, is called after each such
modification. The change notification class is useful in many
cases. For example, one may add a color (or other extra data) to
any halfedge of a planar map. An insertion of a new curve can
split halfedges that were previously in the map. After such split
the color of the newly created halfedges should be updated
according to the original color of the split halfedge. One can do
this by implementing the \ccc{split_edge} function of the change
notification class. This function will be called after each split
of an halfedge in the map.
\subsubsection{Example of Change Notification}
\label{ssec:example2}
The following example demonstrates the usage of the change
notification concept during the construction of a
planar map out of three segments --- $(0,1)-(1,0)$, $(0,0)-(1,1)$
and $(0,1)-(1,1)$.
During the insertion we use \ccc{My_notification} instance to
output the internal process of the construction of the planar map.
We also count how many edges are in the map by incrementing a
counter each time an edge is added (\ccc{add_edge})or
split (\ccc{split_edge}).
\ccIncludeExampleCode{Pm_with_intersections/example2.C}
The output of the program looks like this:
\begin{verbatim}
inserting 0 1 1 0
add_edge
add_hole
inserting 0 0 1 1
split_edge
add_edge
add_edge
inserting 0 1 1 1
add_edge
split_face
Total number of edges 5
\end{verbatim}
\end{ccAdvanced}
\begin{ccTexOnly}
\subsection*{Concepts}
\ccRefConceptPage{Planar_Map_with_Intersections_Traits}\\
\ccRefConceptPage{Planar_Map_with_Intersections_Change_Notification}\\
\subsection*{Classes}
\ccRefIdfierPage{CGAL::Planar_map_with_intersections_2<Planar_map>}%\\
\end{ccTexOnly}

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% +------------------------------------------------------------------------+
% | Reference manual chapter: pmwx_ref.tex (Planar_map_with_intersections_2)
% +------------------------------------------------------------------------+
% |
% | Package: arr (Planar_map_with_intersections_2)
% |
% +------------------------------------------------------------------------+
\input{Pm_with_intersections_2_ref/intro.tex}
\input{Pm_with_intersections_2_ref/Pmwx.tex}
\input{Pm_with_intersections_2_ref/Pmwx_traits.tex}
\input{Pm_with_intersections_2_ref/Pmwx_change_notification.tex}

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\input{pmwx}
% we temporarily include the reference pages in the regular manual
% later this input will be in the reference wrapper
\input{Pm_with_intersections_2_ref/pmwx_ref}

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%+-------------------------------------------------------
%| CGAL Manual : pmwx.tex
%|
%| Soon to be split into User and Reference Manuals
%+--------------------------------------------------------
%| update log
%|
%| 04 May 2001 - Eyal Flato
%| Created from arr documentation
%|
%--------------------------------------------------------
%\documentclass[12pt]{book}
%\usepackage{graphics, amssymb,epsfig}
%\usepackage{cprog,cc_manual}
%
%\def\Ipe#1{\def\IPEfile{#1}\input{#1}}
%
%\setlength{\evensidemargin}{.0in}
%\setlength{\oddsidemargin}{-0.3in}
%\setlength{\textwidth}{6.8in}
%\setlength{\textheight}{8in}
%
%\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}}
%\newcommand{\normal}[1]{\eta_{#1}}
%\newtheorem{theorem}{Theorem}[section] % section
%\newtheorem{remark}[theorem]{Remark}
%\newtheorem{lemma}[theorem]{Lemma}
%\newenvironment{dfn}{{\vspace*{1ex} \noindent \bf Definition }}{\vspace*{1ex}}
%\newcommand{\bigdef}[2]{\index{#1}\begin{dfn} {\rm #2} \end{dfn}}
%\newenvironment{proof}{{\em Proof:}}{\hfill{\hfill\rule{2mm}{2mm}}}
%
%\newcommand{\comment}[1]{{\sf * #1 *}}
%\newcommand{\ncomment}[1]{\noindent {\sf * #1 * }}
%
%\newtheorem{defn}[theorem]{Definition}
%\newcommand{\intsupplanes}{P}
%\def\C{{\cal C}}
%\def\G{{\cal G}}
%\def\F{{\cal F}}
%\def\I{{\cal I}}
%\def\U{{\cal U}}
%\def\M{{\cal M}}
%\def\eps{{\varepsilon}}
%\def\bd{{\partial}}
%\def\dm{{\cal D}}
%
% Title
%\title{CGAL Arrangement Specifications}
%
%\date{ \today }
%
%\begin{document}
%
%\tableofcontents
%
%\maketitle
\chapter{2D Planar Maps with Intersections} \label{I1_ChapterPmwx}
\section{Introduction}
\paragraph{2D Planar Map with Intersections:} Given a collection $C$ of (possibly
intersecting) $x$-monotone curves in the plane, we construct a
collection $C'$ as follows: We decompose each curve in $C$ into
maximal connected pieces not intersecting any other curve in $C$.
This way we obtain the collection $C'$ of $x$-monotone, pairwise
interior disjoint curves.
Constructing the {\it planar map with intersection} of the curves
in $C$ gives the {\it planar map}(see
Chapter~\ref{I1_ChapterPlanarMap}) induced by the curves in $C'$.
Planar map with intersections extends the functionality of planar
map by enabling simple insertion of intersecting $x$-monotone
curves. The planar map with intersections class has different
insertion functions but it uses the same data structures of the
planar map. Therefore, any functionality of the planar map is also
supported here (e.g., traversal of planar map features, point
location queries). However, The planar map with intersections
needs additional intersection functions in the geometric traits
class. Note that if one needs to build a planar map of
$x$-monotone, pairwise interior disjoint curves, then it would be
more efficient (in running time) and less demanding (in traits
class functionality) to use the planar map class.
\paragraph{Degeneracies} Like the planar map class (see
Chapter~\ref{I1_ChapterPlanarMap}), the planar map with intersections class can deal with
$x$-degenerate input (including vertical segments). However, while in the
planar map the input curves were assumed to be non
intersecting in their interiors, there is no such assumption when using
planar map with intersections. Furthermore, overlapping curves are
supported. If two curves overlap the traits intersection function returns
the two endpoints of the common part.
%******************************************************************************
\section{Example Programs}
%---------------------------------------------------
\subsection{Simple Example of a Segment Planar Map with Intersections}
\label{ssec:example1}
The following example demonstrates the construction of an
$X$-shaped planar subdivision out of two intersecting segments.
We output the coordinates of the halfedges of the constructed
subdivision.
\ccIncludeExampleCode{Pm_with_intersections/example1.C}
The output of the program looks like this:
\begin{verbatim}
Inserting the segments:
0 0 1 1
0 1 1 0
Edges of the planar map:
0 0 --- 0.5 0.5
0.5 0.5 --- 1 1
0 1 --- 0.5 0.5
1 0 --- 0.5 0.5
\end{verbatim}
% EOF ------------------------------------------------------------------------80

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% +------------------------------------------------------------------------+
% | Reference manual page: Pmwx.tex (Planar map with intersections)
% +------------------------------------------------------------------------+
% |
% | Package: arr (Planar_map_with_intersections_2)
% |
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +------------------------------------------------------------------------+
% +========================================================================+
% Planar_map_with_intersections_2
% +========================================================================+
\begin{ccRefClass}{Planar_map_with_intersections_2<Planar_map>}
\ccDefinition An object \ccStyle{pmwx} of the class
\ccClassTemplateName\ represents the planar
subdivision induced by a set of curves that are possibly intersecting.
This class extends the class \ccStyle{Planar_map_2<Dcel,Traits>}
which handles $x$-monotone pairwise interior disjoint
curves. %The class have the following internal typedefs:
An object of the class \ccClassTemplateName\ is also a planar map and
therefore the types, constants, and traversal iterators and operations remain
the same.
\ccInclude{CGAL/Pm_with_intersections.h}
\ccInheritsFrom
\ccStyle{Planar_map_2<Dcel,Traits>}
The modifying functions \ccStyle{insert}
override the inherited functions
%\comment{iddo: need to be virtual in planar map}.
\ccTypes
\ccNestedType{Pmwx_change_notification}{}
\ccCreation
\ccCreationVariable{pmwx}
\ccConstructor{Planar_map_with_intersections_2<Planar_map>();}
{creates an empty planar map with the default point location strategy.}
\ccConstructor{Planar_map_with_intersections_2<Planar_map>(Pm_point_location_base<Planar_map> *pl);}
{creates an empty planar map with \ccc{*pl} as the point location strategy.}
\ccConstructor{Planar_map_with_intersections_2<Planar_map>(const Traits& tr, Pm_point_location_base<Planar_map> *pl);}
{creates an empty planar map with \ccc{tr} as the traits class and \ccc{*pl} as the point location strategy.}
\ccOperations
\ccMethod{Halfedge_handle insert(const X_curve& cv, Pmwx_change_notification *en = NULL);}
{inserts the $x$-monotone curve \ccStyle{cv} into the planar map.
Returns the last inserted halfedge whose target points to the
target point of \ccStyle{cv}.
\ccPrecond{\ccc{cv} is not equivalent to a point. }
}
\ccMethod{Halfedge_handle insert_from_vertex(const X_curve& cv, Vertex_handle src,
Pmwx_change_notification *en = NULL);}
{inserts the $x$-monotone curve \ccStyle{cv}, for
which the source vertex, \ccStyle{src}, is already in the
arrangement.
Returns the last inserted halfedge whose target points to the
target point of \ccStyle{cv}.
This function is more efficient than the \ccc{insert} function,
since it does not invoke a point location query for the
endpoint \ccc{src}. Therefore,
whenever possible, it is recommended to use this one rather
than \ccc{insert}.
\ccPrecond{\ccc{src} holds a point which is the same as \ccc{cv's}
source.}
\ccPrecond{\ccc{cv} is not equivalent to a point. }
}
\begin{ccAdvanced}
\ccMethod{Halfedge_handle non_intersecting_insert_from_vertex(const X_curve& cv, Vertex_handle src, bool
source);}
{invokes the planar map implementation of
\ccc{insert_from_vertex} function. Since this function does not
check for intersections it is more efficient than \ccc{insert_from_vertex} but
it should not be called when \ccc{cv} could have intersections with other
curves of the planar map.
\ccPrecond{\ccc{cv} has no intersection in the interior with any curve of the map. }
\ccPrecond{\ccc{src} holds a point which is the same as \ccc{cv}'s source.}
}
\ccMethod{Halfedge_handle non_intersecting_insert(const X_curve&
cv);}
{invokes the planar map implementation of
\ccc{insert} function. Since this function does not
check for intersections it is more efficient than \ccc{insert} but
it should not be called when \ccc{cv} could have intersections with other
curves of the planar map.
\ccPrecond{\ccc{cv} has no intersection in the interior with any curve of the map. }
}
\end{ccAdvanced}
% The following operations have the same functionality as their
% counterparts in the planar map.
%
% \ccMethod{Vertex_iterator vertices_begin();}{}
% \ccMethod{Vertex_iterator vertices_end();}{}
% \ccMethod{Size number_of_vertices();}{}
% \ccMethod{Halfedge_iterator halfedges_begin();}{}
% \ccMethod{Halfedge_iterator halfedges_end();}{}
% \ccMethod{Size number_of_halfedges();}{}
% \ccMethod{Face_iterator faces_begin();}{}
% \ccMethod{Face_iterator faces_end();}{}
% \ccMethod{Size number_of_faces();}{}
%
% \ccMethod{Halfedge_handle split_edge(Halfedge_handle e, const X_curve& c1,
% const X_curve& c2);}{}
% \ccMethod{Halfedge_handle locate(const Point& p, Locate_type& lt);}{}
% \ccMethod{Halfedge_handle vertical_ray_shoot(const Point& p, Locate_type& lt, bool up);}{}
%
% \ccMethod{Face_handle unbounded_face();}{}
%
% \ccMethod{bool is_valid(bool verbose=false);}{}
\end{ccRefClass}
% +------------------------------------------------------------------------+
%%RefPage: end of main body, begin of footer
\ccRefPageEnd
% EOF
% +------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Pmwx_change_notification.tex
% +------------------------------------------------------------------------+
% |
% | Package: arr (Planar_map_with_intersections_2)
% |
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +------------------------------------------------------------------------+
% +========================================================================+
% Pmwx_change_notification
% +========================================================================+
\begin{ccRefConcept}{Planar_Map_with_Intersections_Change_Notification}
\ccDefinition
An instance of \ccRefName\ can be passed to the modification
methods of a planar map with intersections object. An insertion
of an intersecting curve to a planar map may add several
halfedges and modify several features of the map. The relevant
function of the change notification instance is called after
each such modification.
\ccTypes
\ccNestedType{Planar_map}{the planar map in which the object is being used.}
\ccNestedType{Traits}{the traits class of \ccc{Planar_map}.}
\ccCreationVariable{pmwx_cn}
\ccMethod{virtual void add_edge(
const typename Traits::X_curve& cv,
typename Planar_map::Halfedge_handle e,
bool original_direction, bool overlap);}
{called each time a pair of twin halfedges is added to the planar map.
\ccc{cv} is the curve of the added halfedges.
\ccc{e} is on of the new halfedges.
\ccc{original_direction} is true iff \ccc{cv} is directed the same as the curve that we insert to the map.
\ccc{overlap} is true iff the added halfedges overlap previously inserted halfedges in the map.}
\ccMethod{virtual void split_edge(
typename Planar_map::Halfedge_handle orig_edge,
typename Planar_map::Halfedge_handle new_edge,
const typename Traits::X_curve& c1,
const typename Traits::X_curve& c2);}
{called each time a pair of twin halfedges is split into two
pairs. The split operation creates a new pair of twin halfedges
(\ccc{new_edge} and its twin) and modifies the original
pair (\ccc{orig_edge} an its twin).
\ccc{c1} and \ccc{c2} are the curves after the split.
}
\ccMethod{virtual void split_face(
typename Planar_map::Face_handle orig_face,
typename Planar_map::Face_handle new_face);}
{called each time a face of the planar map is being split by
the inserted curve into two faces: \ccc{orig_face} is the
original face and \ccc{new_face} is the new face.
}
\ccMethod{virtual void add_hole(
typename Planar_map::Face_handle in_face,
typename Planar_map::Halfedge_handle new_hole);}
{called each time a hole is added to a face \ccc{in_face}.
\ccc{new_hole} is a halfedge on the new hole.
}
\ccMethod{virtual bool have_support_curve();}
{returns true if \ccc{edge_support_curve} function can return the
supporting curve (used by the arrangement class).
}
\ccMethod{virtual const typename Traits::X_curve&
edge_support_curve(typename Planar_map::Halfedge_handle edge);}
{returns the supporting curve of a \ccc{edge}, if it exists
(used by the arrangement class).
}
\ccHasModels
The package does not supply specific implementations of
\ccc{Planar Map with Intersections Change Notification}. The arrangement class has an
internal implementation that is used for the updating of its
hierarchy structure.
\end{ccRefConcept}
% +------------------------------------------------------------------------+
%%RefPage: end of main body, begin of footer
\ccRefPageEnd
% EOF
% +------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual page: Pmwx_2_traits.tex (Arrangement)
% +------------------------------------------------------------------------+
% |
% | Package: arr (Planar_map_with_intersections_2)
% |
% +------------------------------------------------------------------------+
\ccRefPageBegin
%%RefPage: end of header, begin of main body
% +------------------------------------------------------------------------+
% +========================================================================+
% Arrangement_2_traits
% +========================================================================+
\begin{ccRefConcept}{Planar_Map_with_Intersections_Traits}
\ccDefinition
The \ccRefName\ concept is a refinement of the \ccc{Planar Map Traits}
concept \lcTex{(\ccRefPage{Planar_map_traits})}.
The requirements of the latter
are requirements for the former.
The additional requirements are the following types and methods.
%The \ccStyle{Point} can be implemented for non-linear curves as the
%intersection of 2 curves.
\ccTypes
\ccNestedType{Point}{same as the planar map traits' \ccc{Point}.}
\ccNestedType{X_curve}{same as the planar map traits' \ccc{X_curve}.}
\ccCreationVariable{pmwx_traits}
\ccMethod{Point point_reflect_in_x_and_y(const Point& pt);}{returns the
\ccc{Point}
that is the the reflection of \ccc{pt} in the $x$ and $y$ axes. E.g.,
the point $(2,2)$ will be reflected as $(-2,-2)$. }
\ccMethod{X_curve curve_reflect_in_x_and_y(const X_curve& cv);}{returns the \ccc{X_curve}
that is the the reflection of \ccc{cv} in the $x$ and $y$ axes. E.g., the line segment $((2,2),(3,3))$ will be reflected
as $((-2,-2),(-3,-3))$. }
\ccMethod{void curve_split(const X_curve& cv, X_curve& c1, X_curve& c2,
const Point& split_pt);}{splits $cv$ at \ccc{split_pt}
and assigns the resulting two curves to \ccc{c1} and \ccc{c2}.
\ccPrecond{\ccc{split_pt} is on \ccc{cv}.}
\ccPrecond{\ccc{split_pt} is not an endpoint of \ccc{cv}.}
}
\ccMethod{bool do_intersect_to_right(const X_curve& c1, const X_curve& c2,
const Point& pt);}{returns $true$ if \ccc{c1}
and \ccc{c2} intersect to the right of the point \ccc{pt}.
Intersection to the right of \ccc{pt} is defined as
an intersection which is lexicographically strictly to the right of \ccc{pt}
(i.e., an intersection above or to the right of
\ccc{pt} but {\em not} on \ccc{pt}).}
\ccMethod{bool nearest_intersection_to_right(const X_curve& c1,
const X_curve& c2,
const Point& pt,
Point& p1, Point& p2);}
{finds the
nearest intersection point to the right of \ccc{pt}. Nearest is defined as the
one lexicographically closest to \ccc{pt} not including \ccc{pt} itself.
If the intersection of \ccc{c1} and \ccc{c2} to the
right of \ccc{pt} is an \ccc{X_curve} (i.e., there is an overlapping subcurve
to the right of \ccc{pt}),
then \ccc{p1} will hold the source point of the intersection and
\ccc{p2} will hold its target point. Otherwise \ccc{p1} and \ccc{p2} are
identical. In the case of an overlap we regard the intersection
to the right as the open curve between \ccc{p1} and \ccc{p2}. This means that
if the overlapping subcurve contains \ccc{pt} either \ccc{p1} or \ccc{p2} will
be equal to \ccc{pt}, this is the only case in which this can happen.
If \ccc{c1} and \ccc{c2} do not intersect to the right of \ccc{pt}
the function returns \ccc{false}.
%\ccPrecond{\ccc{do_intersect_to_right(c1,c2,pt) == true}}
}
%The intersection function is defined in such a way to enable dealing with
%degenerate cases when two curves inserted into the arrangement overlap.
\ccMethod{bool curves_overlap(const X_curve& cv1, const X_curve& cv2);}
{returns \ccc{true} if \ccc{cv1} and \ccc{cv2} overlap
in a one-dimensional subcurve (i.e., not in a finite number
of points), otherwise returns
\ccc{false}.}
\ccHasModels
The package does not supply specific implementations of the planar
map with intersections traits concept. However, since the
\ccc{Arrangement Traits} concept is a refinment of the \ccRefName\
concept, all of the models of the former are models of the latter.
\ccc{Arr_segment_exact_traits<R>}\\
\ccc{Arr_leda_segment_exact_traits}\\
\ccc{Arr_circles_real_traits<NT>}\\
\ccc{Arr_polyline_traits<R, Container>}
\end{ccRefConcept}
% +------------------------------------------------------------------------+
%%RefPage: end of main body, begin of footer
\ccRefPageEnd
% EOF
% +------------------------------------------------------------------------+

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% +------------------------------------------------------------------------+
% | Reference manual chapter: intro.tex (Pmwx)
% +------------------------------------------------------------------------+
% |
% | Package: arr (Arrangement_2)
% |
% +------------------------------------------------------------------------+
%+---------------------------------------------------------------------------80
%| update log
%|
%| 04 May 2001 - Eyal Flato
%| Created from arr documentation
%|
%+---------------------------------------------------------------------------80
% +========================================================================+
% Introduction
% +========================================================================+
\clearpage
\section{Planar Map with Intersections Reference Pages}
\subsection*{Introduction}
\ccRefLabel{pmwx_ref_intro}
The planar map with intersections class extends the planar map
class. The planar map with intersections adds insertion functions
that handle intersections and overlapping among the curves of the
planar map. The basic types of the planar map (vertex, halfedge,
face, Dcel, etc.) are kept. \ccc{Planar_map_with_intersections_2}
is templated with the planar map class it derives from.
\subsection*{Planar Map with Intersections Traits Class}
%-----------------------------------------------
Since the planar map with intersections has an additional
functionality of intersecting curves it needs additional
functionality of the traits class. We describe the planar map with
intersections traits concept following in this chapter.
Nevertheless, we do not have specific implementation of this
concept, and we use the arrangement traits instead (see
Chapter~\ref{I1_ChapterArrangement}).
\begin{ccAdvanced}
\subsection*{Change Notification}
%-----------------------------------------------
An insertion of an intersecting curve to a planar map may add
several halfedges and modify several features of the map (i.e.,
split halfedges, split faces). The relevant function of the change
notification instance, that is passed to the modification methods
of a planar map with intersections, is called after each such
modification. The change notification class is useful in many
cases. For example, one may add a color (or other extra data) to
any halfedge of a planar map. An insertion of a new curve can
split halfedges that were previously in the map. After such split
the color of the newly created halfedges should be updated
according to the original color of the split halfedge. One can do
this by implementing the \ccc{split_edge} function of the change
notification class. This function will be called after each split
of an halfedge in the map.
\subsubsection{Example of Change Notification}
\label{ssec:example2}
The following example demonstrates the usage of the change
notification concept during the construction of a
planar map out of three segments --- $(0,1)-(1,0)$, $(0,0)-(1,1)$
and $(0,1)-(1,1)$.
During the insertion we use \ccc{My_notification} instance to
output the internal process of the construction of the planar map.
We also count how many edges are in the map by incrementing a
counter each time an edge is added (\ccc{add_edge})or
split (\ccc{split_edge}).
\ccIncludeExampleCode{Pm_with_intersections/example2.C}
The output of the program looks like this:
\begin{verbatim}
inserting 0 1 1 0
add_edge
add_hole
inserting 0 0 1 1
split_edge
add_edge
add_edge
inserting 0 1 1 1
add_edge
split_face
Total number of edges 5
\end{verbatim}
\end{ccAdvanced}
\begin{ccTexOnly}
\subsection*{Concepts}
\ccRefConceptPage{Planar_Map_with_Intersections_Traits}\\
\ccRefConceptPage{Planar_Map_with_Intersections_Change_Notification}\\
\subsection*{Classes}
\ccRefIdfierPage{CGAL::Planar_map_with_intersections_2<Planar_map>}%\\
\end{ccTexOnly}

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% +------------------------------------------------------------------------+
% | Reference manual chapter: pmwx_ref.tex (Planar_map_with_intersections_2)
% +------------------------------------------------------------------------+
% |
% | Package: arr (Planar_map_with_intersections_2)
% |
% +------------------------------------------------------------------------+
\input{Pm_with_intersections_2_ref/intro.tex}
\input{Pm_with_intersections_2_ref/Pmwx.tex}
\input{Pm_with_intersections_2_ref/Pmwx_traits.tex}
\input{Pm_with_intersections_2_ref/Pmwx_change_notification.tex}

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\input{pmwx}
% we temporarily include the reference pages in the regular manual
% later this input will be in the reference wrapper
\input{Pm_with_intersections_2_ref/pmwx_ref}

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%+-------------------------------------------------------
%| CGAL Manual : pmwx.tex
%|
%| Soon to be split into User and Reference Manuals
%+--------------------------------------------------------
%| update log
%|
%| 04 May 2001 - Eyal Flato
%| Created from arr documentation
%|
%--------------------------------------------------------
%\documentclass[12pt]{book}
%\usepackage{graphics, amssymb,epsfig}
%\usepackage{cprog,cc_manual}
%
%\def\Ipe#1{\def\IPEfile{#1}\input{#1}}
%
%\setlength{\evensidemargin}{.0in}
%\setlength{\oddsidemargin}{-0.3in}
%\setlength{\textwidth}{6.8in}
%\setlength{\textheight}{8in}
%
%\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}}
%\newcommand{\normal}[1]{\eta_{#1}}
%\newtheorem{theorem}{Theorem}[section] % section
%\newtheorem{remark}[theorem]{Remark}
%\newtheorem{lemma}[theorem]{Lemma}
%\newenvironment{dfn}{{\vspace*{1ex} \noindent \bf Definition }}{\vspace*{1ex}}
%\newcommand{\bigdef}[2]{\index{#1}\begin{dfn} {\rm #2} \end{dfn}}
%\newenvironment{proof}{{\em Proof:}}{\hfill{\hfill\rule{2mm}{2mm}}}
%
%\newcommand{\comment}[1]{{\sf * #1 *}}
%\newcommand{\ncomment}[1]{\noindent {\sf * #1 * }}
%
%\newtheorem{defn}[theorem]{Definition}
%\newcommand{\intsupplanes}{P}
%\def\C{{\cal C}}
%\def\G{{\cal G}}
%\def\F{{\cal F}}
%\def\I{{\cal I}}
%\def\U{{\cal U}}
%\def\M{{\cal M}}
%\def\eps{{\varepsilon}}
%\def\bd{{\partial}}
%\def\dm{{\cal D}}
%
% Title
%\title{CGAL Arrangement Specifications}
%
%\date{ \today }
%
%\begin{document}
%
%\tableofcontents
%
%\maketitle
\chapter{2D Planar Maps with Intersections} \label{I1_ChapterPmwx}
\section{Introduction}
\paragraph{2D Planar Map with Intersections:} Given a collection $C$ of (possibly
intersecting) $x$-monotone curves in the plane, we construct a
collection $C'$ as follows: We decompose each curve in $C$ into
maximal connected pieces not intersecting any other curve in $C$.
This way we obtain the collection $C'$ of $x$-monotone, pairwise
interior disjoint curves.
Constructing the {\it planar map with intersection} of the curves
in $C$ gives the {\it planar map}(see
Chapter~\ref{I1_ChapterPlanarMap}) induced by the curves in $C'$.
Planar map with intersections extends the functionality of planar
map by enabling simple insertion of intersecting $x$-monotone
curves. The planar map with intersections class has different
insertion functions but it uses the same data structures of the
planar map. Therefore, any functionality of the planar map is also
supported here (e.g., traversal of planar map features, point
location queries). However, The planar map with intersections
needs additional intersection functions in the geometric traits
class. Note that if one needs to build a planar map of
$x$-monotone, pairwise interior disjoint curves, then it would be
more efficient (in running time) and less demanding (in traits
class functionality) to use the planar map class.
\paragraph{Degeneracies} Like the planar map class (see
Chapter~\ref{I1_ChapterPlanarMap}), the planar map with intersections class can deal with
$x$-degenerate input (including vertical segments). However, while in the
planar map the input curves were assumed to be non
intersecting in their interiors, there is no such assumption when using
planar map with intersections. Furthermore, overlapping curves are
supported. If two curves overlap the traits intersection function returns
the two endpoints of the common part.
%******************************************************************************
\section{Example Programs}
%---------------------------------------------------
\subsection{Simple Example of a Segment Planar Map with Intersections}
\label{ssec:example1}
The following example demonstrates the construction of an
$X$-shaped planar subdivision out of two intersecting segments.
We output the coordinates of the halfedges of the constructed
subdivision.
\ccIncludeExampleCode{Pm_with_intersections/example1.C}
The output of the program looks like this:
\begin{verbatim}
Inserting the segments:
0 0 1 1
0 1 1 0
Edges of the planar map:
0 0 --- 0.5 0.5
0.5 0.5 --- 1 1
0 1 --- 0.5 0.5
1 0 --- 0.5 0.5
\end{verbatim}
% EOF ------------------------------------------------------------------------80