CGAL 6.2 - 2D Conforming Triangulations and Meshes

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Functions
Mesh Generation Functions
Reference Manual

##

The main function to generate a mesh is `refine_Delaunay_mesh_2()`.

The function `lloyd_optimize_mesh_2()` allows to optimize an existing mesh.

## Functions
template<typename CDT , typename NamedParameters = CGAL::parameters::Default_named_parameters>
void | CGAL::refine_Delaunay_mesh_2 (CDT &t, const NamedParameters &np)
| refines the domain defined by a constrained Delaunay triangulation into a mesh satisfying the criteria defined by the traits `criteria`.
template<class CDT , class Criteria >
void | CGAL::refine_Delaunay_mesh_2 (CDT &t, const Criteria &criteria=Criteria())
template<class CDT , class Criteria , class InputIterator >
void | CGAL::refine_Delaunay_mesh_2 (CDT &t, InputIterator begin, InputIterator end, const Criteria &criteria=Criteria(), bool mark=false)
template<class CDT >
void | CGAL::make_conforming_Delaunay_2 (CDT &t)
| Refines the constrained Delaunay triangulation `t` into a conforming Delaunay triangulation.
template<class CDT >
void | CGAL::make_conforming_Gabriel_2 (CDT &t)
| Refines the constrained Delaunay triangulation `t` into a conforming Gabriel triangulation.
template<typename CDT , typename NamedParameters = CGAL::parameters::Default_named_parameters>
Mesh_optimization_return_code | CGAL::lloyd_optimize_mesh_2 (CDT &cdt, const NamedParameters &np=parameters::default_values())
| The function `lloyd_optimize_mesh_2()` is a mesh optimization process based on the minimization of a global energy function.

## Function Documentation

## ◆ lloyd_optimize_mesh_2()

template<typename CDT , typename NamedParameters = CGAL::parameters::Default_named_parameters>

Mesh_optimization_return_code CGAL::lloyd_optimize_mesh_2 | ( | CDT & | cdt,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

`#include <CGAL/lloyd_optimize_mesh_2.h>`

The function `lloyd_optimize_mesh_2()` is a mesh optimization process based on the minimization of a global energy function.

In `lloyd_optimize_mesh_2()`, the minimized global energy may be interpreted as the \( L^1\)-norm of the error achieved when the function \( x^2\) is interpolated on the mesh domain using a piecewise linear function which is linear in each cell of the Voronoi diagram of the mesh vertices.

The optimizer `lloyd_optimize_mesh_2()` works in iterative steps. At each iteration, mesh vertices are moved into positions that bring to zero the energy gradient and the Delaunay triangulation is updated. Vertices on the mesh boundaries are not moved.

Template Parameters

  CDT | is required to be or derive from `CGAL::Constrained_Delaunay_triangulation_2`, with vertex base and face base of its underlying `TriangulationDataStructure_2` being models of `DelaunayMeshVertexBase_2` and `DelaunayMeshFaceBase_2`, respectively.
  NamedParameters | a sequence of Named Parameters

Parameters

  cdt | the initial mesh that will be modified by the algorithm to represent the final optimized mesh.
  np | an optional sequence of Named Parameters among the ones listed below:

Optional Named Parameters

  2D points used to define the domain to mesh. If `seeds_are_in_domain==true`, the mesh domain is the union of the bounded connected components including at least one seed. If `seeds_are_in_domain==false`, the domain is the union of the bounded components including no seed. Type: a class model of `ConstRange` whose iterator is a model of `InputIterator` with `CDT::Point_2` as value type. Default: No seed. |
  specified if seeds indicate bounded connected components inside or outside of the domain. Type: `bool` Default: false |
  limit on the number of performed iterations. 0 means that there is no limit on the number of performed iterations. Precondition: `max_iteration_number >=0` Type: `int` Default: 0 |
  CPU time limit (in seconds) after which the optimization process is stopped. This time is measured using `CGAL::Real_timer`. 0 means that there is no time limit. Type: `double` Precondition: `time_limit` \( \geq\) 0 Default: 0 |
  designed to reduce running time of each optimization iteration. Any vertex that has a displacement less than a given fraction of the length of its shortest incident edge, is frozen (i.e. is not relocated). The parameter `freeze_bound` gives the threshold ratio. If it is set to 0, freezing of vertices is disabled. Precondition: `0<= freeze_bound <=1` Type: `double` Default: 0.001 |
  threshold ratio of stopping criterion based on convergence: the optimization process is stopped when at the last iteration the displacement of any vertex is less than a given fraction of the length of the shortest edge incident to that vertex. Precondition: `0 <=convergence <= 1` Type: `double` Default: 0.001 |

Returns
  an enum value providing some information about the outcome of the algorithm.

See also
  `CGAL::Mesh_optimization_return_code`
  `CGAL::refine_Delaunay_mesh_2()`

Examples
  Mesh_2/mesh_optimization.cpp.

## ◆ make_conforming_Delaunay_2()

template<class CDT >

void CGAL::make_conforming_Delaunay_2 | ( | CDT & | t | ) |

`#include <CGAL/Triangulation_conformer_2.h>`

Refines the constrained Delaunay triangulation `t` into a conforming Delaunay triangulation.

After a call to this function, all edges of `t` are Delaunay edges.

Template Parameters

  CDT | must be a 2D constrained Delaunay triangulation and its geometric traits class must be a model of `ConformingDelaunayTriangulationTraits_2`.

Examples
  Mesh_2/conforming.cpp.

## ◆ make_conforming_Gabriel_2()

template<class CDT >

void CGAL::make_conforming_Gabriel_2 | ( | CDT & | t | ) |

`#include <CGAL/Triangulation_conformer_2.h>`

Refines the constrained Delaunay triangulation `t` into a conforming Gabriel triangulation.

After a call to this function, all constrained edges of `t` have the Gabriel property: the circle that has \( e\) as diameter does not contain any vertex from the triangulation.

Template Parameters

  CDT | must be a 2D constrained Delaunay triangulation and its geometric traits class must be a model of `ConformingDelaunayTriangulationTraits_2`.

Examples
  Mesh_2/conforming.cpp.

## ◆ refine_Delaunay_mesh_2() [1/3]

template<class CDT , class Criteria >

void CGAL::refine_Delaunay_mesh_2 | ( | CDT & | t,
|  | const Criteria & | criteria = `Criteria()`
| ) |  |

`#include <CGAL/Delaunay_mesher_2.h>`

Deprecated:
  This function is deprecated since CGAL 5.6

refines the default domain defined by a constrained Delaunay triangulation without seeds into a mesh satisfying the criteria defined by the traits `criteria`. The domain of the mesh covers all the connected components of the plane defined by the constrained edges of `t`, except for the unbounded component.

Template Parameters

  CDT | must be a 2D constrained Delaunay triangulation and its geometric traits class must be a model of `DelaunayMeshTraits_2`. The face of the constrained Delaunay triangulation must be a model of the concept `DelaunayMeshFaceBase_2`.
  Criteria | must be a model of the concept `MeshingCriteria_2`. `Criteria::Face_handle` must be the same as `CDT::Face_handle`.

## ◆ refine_Delaunay_mesh_2() [2/3]

template<typename CDT , typename NamedParameters = CGAL::parameters::Default_named_parameters>

void CGAL::refine_Delaunay_mesh_2 | ( | CDT & | t,
|  | const NamedParameters & | np
| ) |  |

`#include <CGAL/Delaunay_mesher_2.h>`

refines the domain defined by a constrained Delaunay triangulation into a mesh satisfying the criteria defined by the traits `criteria`.

Note that the unbounded component of the plane is never meshed.

Template Parameters

  CDT | must be a 2D constrained Delaunay triangulation and its geometric traits class must be a model of `DelaunayMeshTraits_2`. The face of the constrained Delaunay triangulation must be a model of the concept `DelaunayMeshFaceBase_2`.
  NamedParameters | a sequence of Named Parameters

Parameters

  t | the constrained Delaunay triangulation that will be refined
  np | an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

  2D points used to define the domain to mesh. If `seeds_are_in_domain==true`, the mesh domain is the union of the bounded connected components including at least one seed. If `seeds_are_in_domain==false`, the domain is the union of the bounded components including no seed. Type: a class model of `ConstRange` with iterator being a model of `InputIterator` with `CDT::Point` as value type. Default: No seed. |
  The meshing criteria Type: a class model of a model of the concept `MeshingCriteria_2` with `Criteria::Face_handle` being the same as `CDT::Face_handle`. Default: `CGAL::Delaunay_mesh_size_criteria_2` |
  If `true` the domain is considered as already initialized (see `DelaunayMeshFaceBase_2::is_in_domain()`). If `false` and no seeds are provided, the domain of the mesh covers all the connected components of the plane defined by the constrained edges of `t`, except for the unbounded component. Type: `bool` Default: false |
  specified if seeds indicates bounded connected components inside or outside of the domain. Type: `bool` Default: false |

Examples
  Mesh_2/mesh_global.cpp, Mesh_2/mesh_marked_domain.cpp, and Mesh_2/mesh_with_seeds.cpp.

## ◆ refine_Delaunay_mesh_2() [3/3]

template<class CDT , class Criteria , class InputIterator >

void CGAL::refine_Delaunay_mesh_2 | ( | CDT & | t,
|  | InputIterator | begin,
|  | InputIterator | end,
|  | const Criteria & | criteria = `Criteria()`,
|  | bool | mark = `false`
| ) |  |

`#include <CGAL/Delaunay_mesher_2.h>`

Deprecated:
  This function is deprecated since CGAL 5.6

refines the default domain defined by a constrained Delaunay triangulation into a mesh satisfying the criteria defined by the traits `criteria`.The sequence `[begin, end)` gives a set of seeds points, that defines the domain to be meshed as follows. The constrained edges of `t` partition the plane into connected components. If `mark==true`, the mesh domain is the union of the bounded connected components including at least one seed. If `mark==false`, the domain is the union of the bounded components including no seed. Note that the unbounded component of the plane is never meshed.

Template Parameters

  CDT | must be 2D constrained Delaunay triangulation and its geometric traits class must be a model of `DelaunayMeshTraits_2`. The face of the constrained Delaunay triangulation must be a model of the concept `DelaunayMeshFaceBase_2`.
  Criteria | must be a model of the concept `MeshingCriteria_2`. `Criteria::Face_handle` must be the same as `CDT::Face_handle`.
  InputIterator | must be an input iterator with value type `CDT::Geom_traits::Point_2`.