mirror of https://github.com/CGAL/cgal
add documentation for lloyd_optimize_mesh_2
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Jane Tournois
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@ -19,7 +19,7 @@ specific application.
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be the same as the one used for the Delaunay mesh.
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\tparam Fb is the base class from which `Delaunay_mesh_face_base_2`
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derives. It must be a model of the `TriangulationFaceBase_2` concept.
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derives. It must be a model of the `ConstrainedTriangulationFaceBase_2` concept.
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\cgalModels `DelaunayMeshFaceBase_2`
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@ -0,0 +1,30 @@
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namespace CGAL {
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/*!
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\ingroup PkgMesh2
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The class `Delaunay_mesh_vertex_base_2` is a model for the concept
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`DelaunayMeshVertexBase_2`.
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This class can be used directly or it can serve as a base to derive other
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classes with some additional attributes tuned to a
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specific application.
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\tparam Traits is the geometric traits class. It must
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be the same as the one used for the Delaunay mesh.
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\tparam Vb is the base class from which `Delaunay_mesh_vertex_base_2`
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derives. It must be a model of the `TriangulationVertexBase_2` concept.
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\cgalModels `DelaunayMeshVertexBase_2`
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*/
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template< typename Traits, typename Vb >
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class Delaunay_mesh_vertex_base_2 : Vb {
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public:
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/// @}
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}; /* end Delaunay_mesh_vertex_base_2 */
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} /* end namespace CGAL */
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@ -38,7 +38,7 @@ components including at least one seed, or the union of the bounded
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connected components that do no contain any seed. Note that the unbounded
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component of the plane is never meshed.
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\sa `lloyd_optimize_mesh_2()`
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\sa `CGAL::lloyd_optimize_mesh_2`
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*/
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template< typename CDT, typename Criteria >
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@ -257,7 +257,7 @@ bool step_by_step_refine_mesh();
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}; /* end Delaunay_mesher_2 */
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/*!
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\ingroup PkgMesh2
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\ingroup PkgMesh2Functions
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Refines the default domain defined by a constrained Delaunay
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triangulation without seeds into a mesh satisfying the criteria
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@ -279,7 +279,7 @@ void refine_Delaunay_mesh_2 (CDT &t, const Criteria& criteria = Criteria());
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/*!
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\ingroup PkgMesh2
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\ingroup PkgMesh2Functions
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Refines the default domain defined by a constrained
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Delaunay triangulation into a mesh
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@ -1,7 +1,7 @@
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namespace CGAL {
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/*!
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\ingroup PkgMesh2
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\ingroup PkgMesh2Functions
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@ -19,7 +19,7 @@ template<class CDT> void make_conforming_Delaunay_2 (CDT &t);
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namespace CGAL {
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/*!
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\ingroup PkgMesh2
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\ingroup PkgMesh2Functions
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@ -1,7 +1,7 @@
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namespace CGAL {
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/*!
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\ingroup PkgMesh_2Functions
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\ingroup PkgMesh2Functions
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The function `lloyd_optimize_mesh_2()` is a mesh optimization process
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based on the minimization of a global energy function.
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@ -18,13 +18,15 @@ positions that bring to zero the energy gradient
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and the Delaunay triangulation is updated.
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Vertices on the mesh boundaries are not moved.
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\pre `time_limit` \f$ \geq\f$ 0 and 0 \f$ \leq\f$ `convergence` \f$ \leq\f$ 1 and 0 \f$ \leq\f$ `freeze_bound` \f$ \leq\f$ 1
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\tparam CDT is required to be or derive from
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`CGAL::Constrained_Delaunay_triangulation_2`.
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\tparam CDT is required to be or derive from `CGAL::Constrained_Delaunay_triangulation_2`,
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with vertex base and face base of its underlying `TriangulationDataStructure_2`
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respectively implementing the concepts `DelaunayMeshFaceBase_2` and `DelaunayMeshVertexBase_2`.
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The argument `cdt`, passed by reference, provides the initial mesh
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and is modified by the algorithm to represent the final optimized mesh.
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\tparam PointIterator must be an iterator with value type `CGAL::Kernel::Point_2`
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The function has several optional parameters which are named parameters
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(we use the Boost.Parameter library).
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Therefore, when calling the function, the parameters can be provided in any order
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@ -36,24 +38,41 @@ provided that the names of the parameters are used
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- <b>`parameters::time_limit`</b>
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is used to set up, in seconds,
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a CPU time limit after which the optimization process is stopped. This time is
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measured using `Timer`.
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measured using `CGAL::Timer`.
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The default value is 0 and means that there is no time limit.
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\pre `time_limit` \f$ \geq\f$ 0
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- <b>`parameters::%max_iteration_number`</b> sets a limit on the
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number of performed iterations. The default value of 0 means that there is
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no limit on the number of performed iterations.
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\pre `max_iteration_number`\f$ \geq\f$ 0
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- <b>`parameters::%convergence`</b> is a stopping criterion based on convergence:
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the optimization process is stopped, when at the last iteration,
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the displacement of any vertex is less than a given fraction of the
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length of the shortest edge incident to that vertex.
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The parameter `convergence` gives the threshold ratio.
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\pre 0 \f$ \leq\f$ `convergence` \f$ \leq\f$ 1
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- <b>`parameters::freeze_bound`</b> is designed to reduce running time of each
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optimization iteration. Any vertex that has a displacement less than a given
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fraction of the length of its shortest incident edge, is frozen (i.e.\ is
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not relocated). The parameter `freeze_bound` gives the threshold ratio.
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The default value is 0.001. If it is set to 0, freezing of vertices is disabled.
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\pre 0 \f$ \leq\f$ `freeze_bound` \f$ \leq\f$ 1
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- <b>`parameters::seeds_begin`</b> and <b>`parameters::seeds_end`</b>
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are begin and end input iterators to iterate on seed points.
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The sequence [`parameters::seeds_begin`, `parameters::seeds_end`)
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defines the domain in which the mesh was generated, and should be optimized.
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- <b>`parameters::mark`</b>. If `mark` is set to true, the mesh domain
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is the union of the bounded connected components including at least one seed.
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If `mark` is false, the domain is the union of the bounded components including
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no seed. Note that the unbounded component of the plane is never optimized.
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The default value is false.
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\return
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@ -83,17 +102,19 @@ lloyd_optimize_mesh_2(cdt,
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\endcode
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\sa `CGAL::Mesh_optimization_return_code`
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\sa `CGAL::refine_mesh_2()`
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\sa `CGAL::refine_Delaunay_mesh_2()`
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*/
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template<typename CDT>
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template<typename CDT, typename PointIterator>
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Mesh_optimization_return_code
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lloyd_optimize_mesh_2(CDT& cdt,
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double parameters::time_limit=0,
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std::size_t parameters::max_iteration_number=0,
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double parameters::convergence=0.02,
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double parameters::freeze_bound = 0.01,
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bool parameters::do_freeze=true);
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double parameters::time_limit=0,
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std::size_t parameters::max_iteration_number=0,
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double parameters::convergence=0.001,
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double parameters::freeze_bound = 0.001,
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PointIterator parameters::seeds_begin = InputIterator(),
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PointIterator parameters::seeds_end = InputIterator(),
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bool parameters::mark = false);
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} /* namespace CGAL */
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@ -0,0 +1,40 @@
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/*!
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\ingroup PkgMesh2Concepts
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\cgalConcept
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The concept `DelaunayMeshVertexBase_2` refines the concept
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`TriangulationVertexBase_2`. It adds two functions giving access
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to a `double` marker, that is useful for the mesh optimizers to keep
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the mesh density everywhere while modifying the mesh.
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\cgalRefines `TriangulationVertexBase_2`
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\cgalHasModel `CGAL::Delaunay_mesh_vertex_base_2<Traits, Vb>`
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*/
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class DelaunayMeshVertexBase_2 {
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public:
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/// \name Access Functions
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/// @{
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/*!
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returns the size requested locally around the vertex
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*/
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const double& sizing_info() const;
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/*!
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sets the size requested locally around the vertex
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*/
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void set_sizing_info(const double& s);
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/// @}
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}; /* end DelaunayMeshVertexBase_2 */
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@ -51,9 +51,16 @@ to be Delaunay or Gabriel edges.
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Constrained Delaunay triangulations can be refined into
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conforming triangulations by the two following global functions:
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\link make_conforming_Delaunay_2() `template<class CDT> void make_conforming_Delaunay_2 (CDT& t)`\endlink and
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\link make_conforming_Gabriel_2() `template<class CDT> void make_conforming_Gabriel_2 (CDT& t)`\endlink.
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\code{.cpp}
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template<class CDT>
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void make_conforming_Delaunay_2 (CDT& t)
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\endcode
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and
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\code{.cpp}
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template<class CDT>
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void make_conforming_Gabriel_2 (CDT& t)
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\endcode
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.
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In both cases, the template parameter `CDT` must be instantiated by a
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constrained Delaunay triangulation class (see Chapter \ref
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@ -193,9 +200,12 @@ angle bound greater than \f$ 20.7\f$ degrees, but there is no such guarantee.
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\subsection secMesh_2_building_meshes Building Meshes
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Meshes are obtained from
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constrained Delaunay triangulations by calling the global function
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constrained Delaunay triangulations by calling the global function :
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\link refine_Delaunay_mesh_2() `template<class CDT, class Criteria> void refine_Delaunay_mesh_2 (CDT &t, const Criteria& criteria)`\endlink.
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\code{.cpp}
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template<class CDT, class Criteria>
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void refine_Delaunay_mesh_2 (CDT &t, const Criteria& criteria)
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\endcode
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The template parameter `CDT` must be instantiated by a constrained
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Delaunay triangulation class. In order to override the domain,
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@ -234,6 +244,28 @@ Once the mesh is constructed, one can determine which faces of the
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triangulation are in the mesh domain using the `is_in_domain()` member
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function of the face type (see the concept `DelaunayMeshFaceBase_2`).
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\subsection secMesh_2_optimization Optimization of Meshes with Lloyd
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The package also provides a global function that runs Lloyd optimization iterations on the
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mesh generated by Delaunay refinement.
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\code{.cpp}
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template< class CDT >
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Mesh_optimization_return_code lloyd_optimize_mesh_2(CDT& cdt);
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\endcode
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Note that this global function has more parameters (see details in reference pages) to tune the optimization process.
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This optimization process alternates relocating vertices to the center of mass
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of their Voronoi cells, and updating the Delaunay connectivity of the triangulation.
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The center of mass is computed with respect to a sizing function that was designed to
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preserve the local density of points in the mesh generated by Delaunay refinement.
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\section Mesh_2Examples Examples
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\subsection Mesh_2ExampleUsingtheGlobalFunction Example Using the Global Function
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The following example inserts several segments into a constrained
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@ -268,6 +300,21 @@ faces is used to count them.
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\cgalExample{Mesh_2/mesh_with_seeds.cpp}
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\subsection Mesh_2ExampleLloyd Example Using the Lloyd optimizer
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This example uses the global function `lloyd_optimize_mesh_2()`.
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The mesh is generated using the function `refine_mesh()` of `CGAL::Delaunay_mesher_2`,
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and is then optimized using `lloyd_optimize_mesh_2()`. The optimization will stop
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after 10 (set by `max_iteration_number`) iterations of alternating vertex relocations
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and Delaunay connectivity updates. More termination conditions can be used and are detailed
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in the Reference Manual.
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\cgalExample{Mesh_2/mesh_optimization.cpp}
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*/
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} /* namespace CGAL */
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@ -2,6 +2,15 @@
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/// \defgroup PkgMesh2Concepts Concepts
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/// \ingroup PkgMesh2
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/// \defgroup PkgMesh2Functions Mesh Generation Functions
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/// \ingroup PkgMesh2
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/// The main function to generate a mesh is `refine_Delaunay_mesh_2()`.
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/// The function `lloyd_optimize_mesh_2()` allows to optimize an existing mesh.
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/// \defgroup PkgMesh2Enum Enumerations
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/// \ingroup PkgMesh2
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/*!
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\addtogroup PkgMesh2
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\cgalPkgDescriptionBegin{2D Conforming Triangulations and Meshes,PkgMesh2Summary}
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@ -32,11 +41,13 @@ The package can handle intersecting input constraints and set no restriction on
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- `DelaunayMeshTraits_2`
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- `MeshingCriteria_2`
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- `DelaunayMeshFaceBase_2`
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- `DelaunayMeshVertexBase_2`
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## Classes##
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- `CGAL::Triangulation_conformer_2<CDT>`
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- `CGAL::Delaunay_mesher_2<CDT, Criteria>`
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- `CGAL::Delaunay_mesh_face_base_2<Traits, Fb>`
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- `CGAL::Delaunay_mesh_vertex_base_2<Traits, Vb>`
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- `CGAL::Delaunay_mesh_criteria_2<CDT>`
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- `CGAL::Delaunay_mesh_size_criteria_2<CDT>`
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- `CGAL::Mesh_2::Face_badness`
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