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Copy file name to clipboardExpand all lines: Algebraic_foundations/doc/Algebraic_foundations/Algebraic_foundations.txt
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@@ -267,7 +267,7 @@ compute the <i>least</i> common multiple of the denominators.
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The package is part of \cgal since release 3.3. Of course the package is based
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on the former Number type support of CGAL. This goes back to Stefan Schirra and Andreas Fabri. But on the other hand the package is to a large extend influenced
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by the experience with the number type support in <span class="textsc">Exacus</span> \cgalCite{beh-eeeafcs-05},
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by the experience with the number type support in \exacus \cgalCite{beh-eeeafcs-05},
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which in the main goes back to
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Lutz Kettner, Susan Hert, Arno Eigenwillig and Michael Hemmer.
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However, the package abstracts from the pure support for
Copy file name to clipboardExpand all lines: Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Arrangement_on_surface_2.txt
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@@ -662,9 +662,9 @@ functions can be used. The program works in three
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steps, as demonstrated in \cgalFigureRef{arr_figex_3}. Note that
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here we still stick to integer coordinates, but as we work on a
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larger scale we use an unbounded integer number-type (in this
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case, the `Gmpz` type taken from the <span class="textsc">Gmp</span> library)
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case, the `Gmpz` type taken from the \gmp library)
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instead of the built-in `int` type.\cgalFootnote{As a rule of thumb, one can use a bounded integer type for representing line segments whose coordinates are bounded by \f$ \lfloor\sqrt[3]{M}\rfloor\f$, where \f$ M\f$ is the maximal representable integer value. This guarantees that no overflows occur in the computations carried out by the traits class, hence all traits-class predicates always return correct results.}
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In case the <span class="textsc">Gmp</span> library is not installed (as indicated by
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In case the \gmp library is not installed (as indicated by
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the `CGAL_USE_GMP` flag defined in `CGAL/basic.h`), we
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use `MP_Float`, a number-type included in \cgal's support
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library that is capable of storing floating-point numbers with
@@ -2064,8 +2064,7 @@ rational numbers, and this ensures the robustness and correctness of
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any computation.\cgalFootnote{Many of the example programs in the rest
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of the chapter include a header file named `arr_rational_nt.h`, which
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defines a type named `Number_type` as either `Gmpq` or
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`Quotient<MP_Float>`, depending on whether <span
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class="textsc">Gmp</span> is installed or not.}
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`Quotient<MP_Float>`, depending on whether \gmp is installed or not.}
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An instance of the `Arr_segment_traits_2<Kernel>` class
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template can be very efficient for constructing arrangements induced
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coefficients. It is highly recommended to use the
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`CORE_algebraic_number_traits` class, which is included in the
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arrangement package. It relies on the exact number types
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implemented in the <span class="textsc">Core</span> library and performs exact
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implemented in the \core library and performs exact
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computations on the number types it defines.
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</UL>
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@@ -3588,15 +3587,15 @@ and defines a simple textual input/output format.
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\section arr_secbgl Adapting to Boost Graphs
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<span class="textsc">Boost</span>\cgalFootnote{See also <span class="textsc">Boost</span>'s homepage at: <TT>www.boost.org</TT>.}
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is a collection of portable \cpp libraries that extend the Standard Template Library (<span class="textsc">Stl</span>). The <span class="textsc">Boost</span> Graph Library (<span class="textsc">bgl</span>), which one of the libraries in the collection, offers an
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\boost\cgalFootnote{See also \boost's homepage at: <TT>www.boost.org</TT>.}
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is a collection of portable \cpp libraries that extend the Standard Template Library (\stl). The \boost Graph Library (\bgl), which one of the libraries in the collection, offers an
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extensive set of generic graph algorithms parameterized through templates.
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As our arrangements are embedded as planar graphs, it is only
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natural to extend the underlying data structure with the interface that the
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<span class="textsc">bgl</span> expects, and gain the ability to perform the operations that the <span class="textsc">bgl</span> supports, such as shortest-path computation. This section describes how apply
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the graph algorithms implemented in the <span class="textsc">bgl</span> to `Arrangement_2` instances.
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\bgl expects, and gain the ability to perform the operations that the \bgl supports, such as shortest-path computation. This section describes how apply
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the graph algorithms implemented in the \bgl to `Arrangement_2` instances.
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An instance of `Arrangement_2` is adapted to a <span class="textsc">Boost</span> graph through the
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An instance of `Arrangement_2` is adapted to a \boost graph through the
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provision of a set of free functions that operate on the arrangement features
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and conform with the relevant BGL concepts. Besides the straightforward
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adaptation, which associates a vertex with each \dcel vertex and an edge
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\subsection arr_ssecbgl_primal The Primal Arrangement Representation
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Arrangement instances are adapted to <span class="textsc">Boost</span> graphs by specializing the
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Arrangement instances are adapted to \boost graphs by specializing the
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\link BGLArgtGT `boost:graph_traits` \endlink template for `Arrangement_2` instances. The
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graph-traits states the graph concepts that the arrangement class models
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(see below) and defines the types required by these concepts.
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directed as well. Moreover, as several interior-disjoint \f$ x\f$-monotone curves
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(say circular arcs) may share two common endpoints, inducing an arrangement
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with two vertices that are connected with several edges, we allow parallel
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edges in our <span class="textsc">Boost</span> graph.
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edges in our \boost graph.
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Given an `Arrangement_2` instance, we can efficiently traverse its
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vertices and halfedges. Thus, the arrangement graph is a model of the concepts
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`VertexListGraph` and `EdgeListGraph` introduced by the <span class="textsc">bgl</span>.
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`VertexListGraph` and `EdgeListGraph` introduced by the \bgl.
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At the same time, we use an iterator adapter of the circulator over the
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halfedges incident to a vertex (`Halfedge_around_vertex_circulator` - see
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Section \ref arr_sssectr_vertex), so it is possible to go over the ingoing
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It is important to notice that the vertex descriptors we use are
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`Vertex_handle` objects and <I>not</I> vertex indices. However, in order
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to gain more efficiency in most <span class="textsc">bgl</span> algorithm, it is better to have them
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to gain more efficiency in most \bgl algorithm, it is better to have them
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indexed \f$ 0, 1, \ldots, (n-1)\f$, where \f$ n\f$ is the number of vertices. We
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therefore introduce the `Arr_vertex_index_map<Arrangement>` class-template,
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which maintains a mapping of vertex handles to indices, as required by the
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<span class="textsc">bgl</span>. An instance of this class must be attached to a valid arrangement
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\bgl. An instance of this class must be attached to a valid arrangement
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vertex when it is created. It uses the notification mechanism (see
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Section \ref arr_secnotif) to automatically maintain the mapping of vertices
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to indices, even when new vertices are inserted into the arrangement or
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existing vertices are removed.
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In most algorithm provided by the <span class="textsc">bgl</span>, the output is given by
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In most algorithm provided by the \bgl, the output is given by
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<I>property maps</I>, such that each map entry corresponds to a vertex.
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For example, when we compute the shortest paths from a given source vertex
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\f$ s\f$ to all other vertices we can obtain a map of distances and a map of
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properties of type `Type`. Note however that unlike the
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`Arr_vertex_index_map` class, the vertex property-map class is not
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kept synchronized with the number of vertices in the arrangement, so it
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should not be reused in calls to <span class="textsc">bgl</span> functions in case the arrangement
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should not be reused in calls to \bgl functions in case the arrangement
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is modified in between these calls.
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\cgalFigureBegin{arr_figex_bgl,ex_bgl.png}
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In the following example we construct an arrangement of 7 line segments,
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as shown in \cgalFigureRef{arr_figex_bgl},
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then use Dijkstra's shortest-paths algorithm from the <span class="textsc">bgl</span> to compute
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then use Dijkstra's shortest-paths algorithm from the \bgl to compute
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the graph distance of all vertices from the leftmost vertex in the
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arrangement \f$ v_0\f$. Note the usage of the `boost::vector_property_map<Type, IndexMap>` and
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the `Arr_vertex_property_map` classes. The latter one, instantiated by
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`e` is directed from \f$ f_1\f$, which is its incident face, to \f$ f_2\f$, which
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is the incident face of its twin halfedge. As two arrangement faces may
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share more than a single edge on their boundary, we allow parallel
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edges in our <span class="textsc">Boost</span> graph. As is the case in the primal graph, the dual
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edges in our \boost graph. As is the case in the primal graph, the dual
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arrangement graph is also a model of the concepts `VertexListGraph`,
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`EdgeListGraph` and `BidirectionalGraph` (thus also of
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`IncidenceGraph`).
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(`Arr_segment_traits_2`
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even when the segments intersect each other, due to the small overhead
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of the latter (optimized) traits class. (For example, when the so
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called <span class="textsc">Leda</span> rational kernel is used).
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called \leda rational kernel is used).
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<LI>Prior knowledge of the combinatorial structure of the arrangement can
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be used to accelerate operations that insert \f$ x\f$-monotone curves,
Copy file name to clipboardExpand all lines: Arrangement_on_surface_2/doc/Arrangement_on_surface_2/CGAL/Arr_default_overlay_traits.h
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@@ -6,8 +6,8 @@ namespace CGAL {
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\ingroup PkgArrangementOnSurface2Overlay
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An instance of `Arr_default_overlay_traits` should be used for overlaying two arrangements
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of type `Arrangement` that store no auxiliary data with their <span class="textsc">Dcel</span> records, where the resulting overlaid arrangement stores no auxiliary
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<span class="textsc">Dcel</span> data as well. This class simply gives empty implementation for all
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of type `Arrangement` that store no auxiliary data with their \dcel records, where the resulting overlaid arrangement stores no auxiliary
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\dcel data as well. This class simply gives empty implementation for all
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traits-class functions.
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\cgalModels `OverlayTraits`
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An instance of `Arr_face_overlay_traits` should be used for overlaying two arrangements
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of types `Arr_A` and `Arr_B`, which are instantiated using the same
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geometric traits-class and with the <span class="textsc">Dcel</span> classes `Dcel_A` and
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geometric traits-class and with the \dcel classes `Dcel_A` and
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`Dcel_B` respectively, in order to store their overlay in an arrangement
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of type `Arr_R`, which is instantiated using a third <span class="textsc">Dcel</span> class
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`Dcel_R`. All three <span class="textsc">Dcel</span> classes are assumed to be instantiations of the
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of type `Arr_R`, which is instantiated using a third \dcel class
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`Dcel_R`. All three \dcel classes are assumed to be instantiations of the
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`Arr_face_extended_dcel` template with types `FaceData_A`,
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