CGALWrappers Namespace Reference

Namespaces
namespace	internal

Classes
struct	AdditionalData
struct	AdditionalData< 2 >
struct	FaceInfo2

Typedefs
using	ConcurrencyTag = CGAL::Sequential_tag
using	K = CGAL::Exact_predicates_exact_constructions_kernel_with_sqrt
using	K_exact = CGAL::Exact_predicates_exact_constructions_kernel
using	CGALPolygon = CGAL::Polygon_2< K >
using	Polygon_with_holes_2 = CGAL::Polygon_with_holes_2< K >
using	CGALTriangle2 = K::Triangle_2
using	CGALTriangle3 = K::Triangle_3
using	CGALTriangle3_exact = K_exact::Triangle_3
using	CGALPoint2 = K::Point_2
using	CGALPoint3 = K::Point_3
using	CGALPoint3_exact = K_exact::Point_3
using	CGALSegment2 = K::Segment_2
using	Surface_mesh = CGAL::Surface_mesh< K_exact::Point_3 >
using	CGALSegment3 = K::Segment_3
using	CGALSegment3_exact = K_exact::Segment_3
using	CGALTetra = K::Tetrahedron_3
using	CGALTetra_exact = K_exact::Tetrahedron_3
using	Triangulation2 = CGAL::Triangulation_2< K >
using	Triangulation3 = CGAL::Triangulation_3< K >
using	Triangulation3_exact = CGAL::Triangulation_3< K_exact >
using	Vb = CGAL::Triangulation_vertex_base_2< K >
using	Fbb = CGAL::Triangulation_face_base_with_info_2< FaceInfo2, K >
using	CFb = CGAL::Constrained_triangulation_face_base_2< K, Fbb >
using	Fb = CGAL::Delaunay_mesh_face_base_2< K, CFb >
using	Tds = CGAL::Triangulation_data_structure_2< Vb, Fb >
using	Itag = CGAL::Exact_predicates_tag
using	CDT = CGAL::Constrained_Delaunay_triangulation_2< K, Tds, Itag >
using	Criteria = CGAL::Delaunay_mesh_size_criteria_2< CDT >
using	Vertex_handle = CDT::Vertex_handle
using	Face_handle = CDT::Face_handle

Enumerations
enum class	FacetTopology { facet_vertices_on_surface = CGAL::FACET_VERTICES_ON_SURFACE , facet_vertices_on_same_surface_patch , facet_vertices_on_same_surface_patch_with_adjacency_check }
enum class	BooleanOperation { compute_corefinement = 1 << 0 , compute_difference = 1 << 1 , compute_intersection = 1 << 2 , compute_union = 1 << 3 }

Functions
template<int structdim0, int structdim1, int spacedim>
std::vector< std::array< Point< spacedim >, structdim1+1 > >	compute_intersection_of_cells (const typename Triangulation< structdim0, spacedim >::cell_iterator &cell0, const typename Triangulation< structdim1, spacedim >::cell_iterator &cell1, const Mapping< structdim0, spacedim > &mapping0, const Mapping< structdim1, spacedim > &mapping1, const double tol=1e-9)
template<int structdim0, int structdim1, int spacedim>
std::vector< std::array< Point< spacedim >, structdim1+1 > >	compute_intersection_of_cells (const ArrayView< const Point< spacedim > > &vertices0, const ArrayView< const Point< spacedim > > &vertices1, const double tol=1e-9)
template<typename CGALPointType , int dim, int spacedim>
void	dealii_cell_to_cgal_surface_mesh (const typename ::Triangulation< dim, spacedim >::cell_iterator &cell, const ::Mapping< dim, spacedim > &mapping, CGAL::Surface_mesh< CGALPointType > &mesh)
template<typename CGALPointType , int dim, int spacedim>
void	dealii_tria_to_cgal_surface_mesh (const ::Triangulation< dim, spacedim > &triangulation, CGAL::Surface_mesh< CGALPointType > &mesh)
template<typename C3t3 >
void	cgal_surface_mesh_to_cgal_triangulation (CGAL::Surface_mesh< typename C3t3::Point::Point > &surface_mesh, C3t3 &triangulation, const AdditionalData< 3 > &data=AdditionalData< 3 >{})
template<typename CGALPointType >
void	compute_boolean_operation (const CGAL::Surface_mesh< CGALPointType > &surface_mesh_1, const CGAL::Surface_mesh< CGALPointType > &surface_mesh_2, const BooleanOperation &boolean_operation, CGAL::Surface_mesh< CGALPointType > &output_surface_mesh)
template<typename CGALTriangulationType >
::Quadrature< CGALTriangulationType::Point::Ambient_dimension::value >	compute_quadrature (const CGALTriangulationType &tria, const unsigned int degree)
template<int dim0, int dim1, int spacedim>
::Quadrature< spacedim >	compute_quadrature_on_boolean_operation (const typename ::Triangulation< dim0, spacedim >::cell_iterator &cell0, const typename ::Triangulation< dim1, spacedim >::cell_iterator &cell1, const unsigned int degree, const BooleanOperation &bool_op, const Mapping< dim0, spacedim > &mapping0=(ReferenceCells::get_hypercube< dim0 >() .template get_default_linear_mapping< dim0, spacedim >()), const Mapping< dim1, spacedim > &mapping1=(ReferenceCells::get_hypercube< dim1 >() .template get_default_linear_mapping< dim1, spacedim >()))
template<int dim0, int dim1, int spacedim>
::Quadrature< spacedim >	compute_quadrature_on_intersection (const typename ::Triangulation< dim0, spacedim >::cell_iterator &cell0, const typename ::Triangulation< dim1, spacedim >::cell_iterator &cell1, const unsigned int degree, const Mapping< dim0, spacedim > &mapping0, const Mapping< dim1, spacedim > &mapping1)
template<typename CGALPointType >
void	remesh_surface (CGAL::Surface_mesh< CGALPointType > &surface_mesh, const AdditionalData< 3 > &data=AdditionalData< 3 >{})
template<class T , class... Types>
const T *	get_if_ (const std::variant< Types... > *v)
template<class T , class... Types>
const T *	get_if_ (const boost::variant< Types... > *v)

Detailed Description

Interface to the Computational Geometry Algorithm Library (CGAL).

CGAL is a software project that provides easy access to efficient and reliable geometric algorithms. The library offers data structures and algorithms like triangulations, Voronoi diagrams, Boolean operations on polygons and polyhedra, point set processing, arrangements of curves, surface and volume mesh generation, geometry processing, alpha shapes, convex hull algorithms, shape reconstruction, AABB and KD trees...

You can learn more about the CGAL library at https://www.cgal.org/

Typedef Documentation

ConcurrencyTag

using CGALWrappers::ConcurrencyTag = typedef CGAL::Sequential_tag

Definition at line 82 of file utilities.h.

K

using CGALWrappers::K = typedef CGAL::Exact_predicates_exact_constructions_kernel_with_sqrt

Definition at line 61 of file intersections.cc.

K_exact

using CGALWrappers::K_exact = typedef CGAL::Exact_predicates_exact_constructions_kernel

Definition at line 62 of file intersections.cc.

CGALPolygon

using CGALWrappers::CGALPolygon = typedef CGAL::Polygon_2<K>

Definition at line 63 of file intersections.cc.

Polygon_with_holes_2

using CGALWrappers::Polygon_with_holes_2 = typedef CGAL::Polygon_with_holes_2<K>

Definition at line 64 of file intersections.cc.

CGALTriangle2

using CGALWrappers::CGALTriangle2 = typedef K::Triangle_2

Definition at line 65 of file intersections.cc.

CGALTriangle3

using CGALWrappers::CGALTriangle3 = typedef K::Triangle_3

Definition at line 66 of file intersections.cc.

CGALTriangle3_exact

using CGALWrappers::CGALTriangle3_exact = typedef K_exact::Triangle_3

Definition at line 67 of file intersections.cc.

CGALPoint2

using CGALWrappers::CGALPoint2 = typedef K::Point_2

Definition at line 68 of file intersections.cc.

CGALPoint3

using CGALWrappers::CGALPoint3 = typedef K::Point_3

Definition at line 69 of file intersections.cc.

CGALPoint3_exact

using CGALWrappers::CGALPoint3_exact = typedef K_exact::Point_3

Definition at line 70 of file intersections.cc.

CGALSegment2

using CGALWrappers::CGALSegment2 = typedef K::Segment_2

Definition at line 71 of file intersections.cc.

Surface_mesh

using CGALWrappers::Surface_mesh = typedef CGAL::Surface_mesh<K_exact::Point_3>

Definition at line 72 of file intersections.cc.

CGALSegment3

using CGALWrappers::CGALSegment3 = typedef K::Segment_3

Definition at line 73 of file intersections.cc.

CGALSegment3_exact

using CGALWrappers::CGALSegment3_exact = typedef K_exact::Segment_3

Definition at line 74 of file intersections.cc.

CGALTetra

using CGALWrappers::CGALTetra = typedef K::Tetrahedron_3

Definition at line 75 of file intersections.cc.

CGALTetra_exact

using CGALWrappers::CGALTetra_exact = typedef K_exact::Tetrahedron_3

Definition at line 76 of file intersections.cc.

Triangulation2

using CGALWrappers::Triangulation2 = typedef CGAL::Triangulation_2<K>

Definition at line 77 of file intersections.cc.

Triangulation3

using CGALWrappers::Triangulation3 = typedef CGAL::Triangulation_3<K>

Definition at line 78 of file intersections.cc.

Triangulation3_exact

using CGALWrappers::Triangulation3_exact = typedef CGAL::Triangulation_3<K_exact>

Definition at line 79 of file intersections.cc.

Vb

using CGALWrappers::Vb = typedef CGAL::Triangulation_vertex_base_2<K>

Definition at line 92 of file intersections.cc.

Fbb

using CGALWrappers::Fbb = typedef CGAL::Triangulation_face_base_with_info_2<FaceInfo2, K>

Definition at line 93 of file intersections.cc.

CFb

using CGALWrappers::CFb = typedef CGAL::Constrained_triangulation_face_base_2<K, Fbb>

Definition at line 94 of file intersections.cc.

Fb

using CGALWrappers::Fb = typedef CGAL::Delaunay_mesh_face_base_2<K, CFb>

Definition at line 95 of file intersections.cc.

Tds

using CGALWrappers::Tds = typedef CGAL::Triangulation_data_structure_2<Vb, Fb>

Definition at line 96 of file intersections.cc.

Itag

using CGALWrappers::Itag = typedef CGAL::Exact_predicates_tag

Definition at line 97 of file intersections.cc.

CDT

using CGALWrappers::CDT = typedef CGAL::Constrained_Delaunay_triangulation_2<K, Tds, Itag>

Definition at line 98 of file intersections.cc.

Criteria

using CGALWrappers::Criteria = typedef CGAL::Delaunay_mesh_size_criteria_2<CDT>

Definition at line 99 of file intersections.cc.

Vertex_handle

using CGALWrappers::Vertex_handle = typedef CDT::Vertex_handle

Definition at line 100 of file intersections.cc.

Face_handle

using CGALWrappers::Face_handle = typedef CDT::Face_handle

Definition at line 101 of file intersections.cc.

Enumeration Type Documentation

FacetTopology

enum class CGALWrappers::FacetTopology

strong

Enumerator
facet_vertices_on_surface	Each vertex of the facet have to be on the surface, on a curve, or on a corner.
facet_vertices_on_same_surface_patch	The three vertices of a facet belonging to a surface patch s have to be on the same surface patch s, on a curve or on a corner.
facet_vertices_on_same_surface_patch_with_adjacency_check	The three vertices of a facet belonging to a surface patch s have to be on the same surface patch s, or on a curve incident to the surface patch s or on a corner incident to the surface patch s.

Definition at line 31 of file additional_data.h.

BooleanOperation

enum class CGALWrappers::BooleanOperation

strong

An enum type given to functions that compute boolean operations between geometrical objects, defined by triangulated surface grids.

As an example, we show all supported boolean operations applied to a hedra (the union of two pyramids) and a cube.

Enumerator
compute_corefinement	Given two triangulated surfaces, refine the first surface until its intersection with the second surface is captured exactly by the refinement
compute_difference	Given two triangulated surfaces, compute the boolean difference of the first surface minus the second surface
compute_intersection	Given two triangulated surfaces, compute their intersection
compute_union	Given two triangulated surfaces, compute their union

Definition at line 94 of file utilities.h.

Function Documentation

compute_intersection_of_cells() [1/2]

template<int structdim0, int structdim1, int spacedim>

std::vector< std::array< Point< spacedim >, structdim1+1 > > CGALWrappers::compute_intersection_of_cells	(	const typename Triangulation< structdim0, spacedim >::cell_iterator &	cell0,
		const typename Triangulation< structdim1, spacedim >::cell_iterator &	cell1,
		const Mapping< structdim0, spacedim > &	mapping0,
		const Mapping< structdim1, spacedim > &	mapping1,
		const double	tol = 1e-9
	)

Given two deal.II affine cells, compute their intersection and return a vector of simplices, each one identified by an array of deal.II Points. All the simplices together are a subdivision of the intersection. If cells are non-affine, a geometrical error is introduced. If the measure of one of the simplices is below a certain threshold which defaults to 1e-9, then it is discarded. In case the two cells are disjoint, an empty array is returned.

Note

If the two cells have different intrinsic dimensions, than the first iterator is assumed to have with higher intrinsic dimension. The same holds for the two Mapping objects.

Parameters

cell0	Iterator to the first cell.
cell1	Iterator to the second cell.
mapping0	Mapping for the first cell.
mapping1	Mapping for the second cell.
tol	Threshold to decide whether or not a simplex is included.

Returns

Vector of arrays, where each array identify a simplex by its vertices.

Definition at line 842 of file intersections.cc.

compute_intersection_of_cells() [2/2]

template<int structdim0, int structdim1, int spacedim>

std::vector< std::array< Point< spacedim >, structdim1+1 > > CGALWrappers::compute_intersection_of_cells	(	const ArrayView< const Point< spacedim > > &	vertices0,
		const ArrayView< const Point< spacedim > > &	vertices1,
		const double	tol = 1e-9
	)

Same function as above, but working directly with vertices. It's assumed that the first vertices are coming from a Triangulation<dim0,spacedim>, while the other vertices are from a Triangulation<dim1,spacedim>, with dim0 > dim1.

Note

The vertices have to be given in CGAL order.

Definition at line 772 of file intersections.cc.

dealii_cell_to_cgal_surface_mesh()

template<typename CGALPointType , int dim, int spacedim>

void CGALWrappers::dealii_cell_to_cgal_surface_mesh	(	const typename ::Triangulation< dim, spacedim >::cell_iterator &	cell,
		const ::Mapping< dim, spacedim > &	mapping,
		CGAL::Surface_mesh< CGALPointType > &	mesh
	)

Build a CGAL::Surface_mesh from a deal.II cell.

The class Surface_mesh implements a halfedge data structure and can be used to represent polyhedral surfaces. It is an edge-centered data structure capable of maintaining incidence information of vertices, edges, and faces. Each edge is represented by two halfedges with opposite orientation. The orientation of a face is chosen so that the halfedges around a face are oriented counterclockwise.

More information on this class is available at https://doc.cgal.org/latest/Surface_mesh/index.html

The function will throw an exception in dimension one. In dimension two, it generates a surface mesh of the quadrilateral cell or of the triangle cell, while in dimension three it will generate the surface mesh of the cell, i.e., a polyhedral mesh containing the faces of the input cell.

The generated mesh is useful when performing geometric operations using CGAL::Polygon_mesh_processing, i.e., to compute boolean operations on cells, splitting, cutting, slicing, etc.

For examples on how to use the resulting CGAL::Surface_mesh see https://doc.cgal.org/latest/Polygon_mesh_processing/

Parameters

[in]	cell	The input deal.II cell iterator
[in]	mapping	The mapping used to map the vertices of the cell
[out]	mesh	The output CGAL::Surface_mesh

dealii_tria_to_cgal_surface_mesh()

template<typename CGALPointType , int dim, int spacedim>

void CGALWrappers::dealii_tria_to_cgal_surface_mesh	(	const ::Triangulation< dim, spacedim > &	triangulation,
		CGAL::Surface_mesh< CGALPointType > &	mesh
	)

Convert a deal.II triangulation to a CGAL::Surface_mesh. The output depends on the intrinsic dimension of the input deal.II triangulation.

In 2d, i.e. with a Triangulation<2> or a Triangulation<2,3>, the output is the CGAL::Surface_mesh describing the whole triangulation.

In 3d, the boundary the of the deal.II Triangulation is converted to a CGAL::Surface_mesh by looping over all the boundary faces.

Parameters

[in]	triangulation	The input deal.II triangulation.
[out]	mesh	The output CGAL::Surface_mesh.

cgal_surface_mesh_to_cgal_triangulation()

template<typename C3t3 >

void CGALWrappers::cgal_surface_mesh_to_cgal_triangulation	(	CGAL::Surface_mesh< typename C3t3::Point::Point > &	surface_mesh,
		C3t3 &	triangulation,
		const AdditionalData< 3 > &	data = AdditionalData< 3 >{}
	)

Given a closed CGAL::Surface_mesh, this function fills the region bounded by the surface with tets.

Parameters

[in]	surface_mesh	The (closed) surface mesh bounding the volume that has to be filled.
[out]	triangulation	The output triangulation filled with tetrahedra.
[in]	data	AdditionalData object to pass to the CGAL::make_mesh_3 function. See the documentation of that struct for a description of those parameters.

compute_boolean_operation()

template<typename CGALPointType >

void CGALWrappers::compute_boolean_operation	(	const CGAL::Surface_mesh< CGALPointType > &	surface_mesh_1,
		const CGAL::Surface_mesh< CGALPointType > &	surface_mesh_2,
		const BooleanOperation &	boolean_operation,
		CGAL::Surface_mesh< CGALPointType > &	output_surface_mesh
	)

Given two triangulated surface meshes that bound two volumes, execute a boolean operation on them, and store the result in a third surface mesh.

Quoting from CGAL documentation (https://doc.cgal.org/latest/Polygon_mesh_processing/index.html#title14):

‍Given a closed triangulated surface mesh, each connected component splits the 3d space into two subspaces. The vertex sequence of each face of a component is seen either clockwise or counterclockwise from these two subspaces. The subspace that sees the sequence clockwise (resp. counterclockwise) is on the negative (resp. positive) side of the component.

‍Given a closed triangulated surface mesh surface with no self-intersections, we say that surface bounds a volume if each subspace lies exclusively on the positive (or negative) side of all the incident connected components of surface. The volume bounded by surface is the union of all subspaces that are on negative sides of their incident connected components of surface.

‍There is no restriction on the topology of the input volumes. However, there are some requirements on the input to guarantee that the operation is possible. First, the input meshes must not self-intersect. Second, the operation is possible only if the output can be bounded by a manifold triangulated surface mesh. In particular this means that the output volume has no part with zero thickness. Mathematically speaking, the intersection with an infinitesimally small ball centered in the output volume is a topological ball. At the surface level this means that no non-manifold vertex or edge is allowed in the output. For example, it is not possible to compute the union of two cubes that are disjoint but sharing an edge.

See BooleanOperation for a list of available operations, with the corresponding examples.

Parameters

[in]	surface_mesh_1	The first surface mesh.
[in]	surface_mesh_2	The second surface mesh.
[in]	boolean_operation	See BooleanOperation for the list of the allowed operations.
[out]	output_surface_mesh	The surface mesh with the result of the boolean operation.

compute_quadrature()

template<typename CGALTriangulationType >

::Quadrature< CGALTriangulationType::Point::Ambient_dimension::value > CGALWrappers::compute_quadrature	(	const CGALTriangulationType &	tria,
		const unsigned int	degree
	)

Given a CGAL Triangulation describing a polyhedral region, create a Quadrature rule to integrate over the polygon by looping through all the vertices and exploiting QGaussSimplex.

Parameters

[in]	tria	The CGAL triangulation object describing the polyhedral region.
[in]	degree	Desired degree of the Quadrature rule on each element of the polyhedral.

Returns

A global Quadrature rule that allows to integrate over the polyhedron.

compute_quadrature_on_boolean_operation()

template<int dim0, int dim1, int spacedim>

::Quadrature< spacedim > CGALWrappers::compute_quadrature_on_boolean_operation	(	const typename ::Triangulation< dim0, spacedim >::cell_iterator &	cell0,
		const typename ::Triangulation< dim1, spacedim >::cell_iterator &	cell1,
		const unsigned int	degree,
		const BooleanOperation &	bool_op,
		const Mapping< dim0, spacedim > &	mapping0 = (ReferenceCells::get_hypercube< dim0 >() .template get_default_linear_mapping< dim0, spacedim >()),
		const Mapping< dim1, spacedim > &	mapping1 = (ReferenceCells::get_hypercube< dim1 >() .template get_default_linear_mapping< dim1, spacedim >())
	)

Compute a Quadrature formula over the polygonal/polyhedral region described by a BooleanOperation between two deal.II cell_iterator objects. The degree of the Quadrature rule is specified by the argument degree. This function splits the polygon/polyhedron into simplices and collects QGaussSimplex quadrature rules.

Parameters

[in]	cell0	A cell_iterator to the first deal.II cell.
[in]	cell1	A cell_iterator to the second deal.II cell.
[in]	degree	The degree of accuracy you wish to get for the global quadrature formula.
[in]	bool_op	The BooleanOperation to be performed.
[in]	mapping0	Mapping object for the first cell.
[in]	mapping1	Mapping object for the first cell.

Returns

The global quadrature rule on the polygon/polyhedron.

compute_quadrature_on_intersection()

template<int dim0, int dim1, int spacedim>

::Quadrature< spacedim > CGALWrappers::compute_quadrature_on_intersection	(	const typename ::Triangulation< dim0, spacedim >::cell_iterator &	cell0,
		const typename ::Triangulation< dim1, spacedim >::cell_iterator &	cell1,
		const unsigned int	degree,
		const Mapping< dim0, spacedim > &	mapping0,
		const Mapping< dim1, spacedim > &	mapping1
	)

A specialization of the function above when the BooleanOperation is an intersection. The rationale behind this specialization is that deal.II affine cells are convex sets, and as the intersection of convex sets is itself convex, this function internally exploits this to use a cheaper way to mesh the inside.

Parameters

[in]	cell0	A cell_iterator to the first deal.II cell.
[in]	cell1	A cell_iterator to the second deal.II cell.
[in]	mapping0	Mapping object for the first cell.
[in]	mapping1	Mapping object for the first cell.
[in]	degree	The degree of accuracy you wish to get for the global quadrature formula.

Returns

The global quadrature rule on the polygon/polyhedron.

remesh_surface()

template<typename CGALPointType >

void CGALWrappers::remesh_surface	(	CGAL::Surface_mesh< CGALPointType > &	surface_mesh,
		const AdditionalData< 3 > &	data = AdditionalData< 3 >{}
	)

Remesh a CGAL::Surface_mesh.

If the domain has 1-dimensional exposed features, the criteria includes a sizing field to guide the sampling of 1-dimensional features with protecting balls centers.

• CGAL::parameters::edge_size.

This utility should be exploited to improve the quality of a mesh coming from boolean operations. As an example, consider two CGAL::Surface_mesh objects describing two hyper_spheres that intersect non-trivially. After computing their corefinement and union with compute_boolean_operation(), the resulting mesh is the following:

Clearly, elements (triangles) like the ones arising along the intersection of the two spheres should be avoided as they're quite degenerate and would decrease the accuracy of the results. A call to the present function will try to remesh the surface, according to the given named parameters, to improve its quality. In the present case, the result is the following:

Parameters

	surface_mesh	The input CGAL::Surface_mesh.
[in]	data	AdditionalData object to pass to the CGAL::make_mesh_3 function. See the documentation of that struct for a description of those parameters.

get_if_() [1/2]

template<class T , class... Types>

const T * CGALWrappers::get_if_

(

const std::variant< Types... > *

v

)

Definition at line 105 of file intersections.cc.

get_if_() [2/2]

template<class T , class... Types>

const T * CGALWrappers::get_if_

(

const boost::variant< Types... > *

v

)

Definition at line 112 of file intersections.cc.