CGAL 6.1.1 - Linear Cell Complex

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User Manual

Author
  Guillaume Damiand

# Introduction

A dD linear cell complex allows to represent an orientable subdivided dD object having linear geometry: each vertex of the subdivision is associated with a point. The geometry of each edge is a segment whose end points are associated with the two vertices of the edge, the geometry of each 2-cell is obtained from all the segments associated to the edges describing the boundary of the 2-cell and so on.

The combinatorial part of a linear cell complex is described either by a dD combinatorial map or by a dD generalized map (it is strongly recommended to first read the Combinatorial maps user manual or Generalized maps user manual for definitions). To add the linear geometrical embedding, a point (a model of `Point_2` or `Point_3` or `Point_d`) is associated to each vertex of the combinatorial data-structure.

Figure 32.1 Examples of objects with linear geometry. Left: A 2D object composed of three 2-cells, nine 1-cells and seven points associated to the seven 0-cells . Right: A 3D object composed of three 3-cells, twelve 2-cells, sixteen 1-cells and eight points associated to the eight 0-cells.

If we reconsider the example introduced in the combinatorial map package, recalled in Figure 32.1 (Right), the combinatorial part of the 3D object is described by a 3D combinatorial map. As illustrated in Figure 32.2, the geometrical part of the object is described by associating a point to each vertex of the map.

Figure 32.2 Example of 3D linear cell complex describing the object given in Figure 32.1 (Right). Left: The 3D linear cell complex which contains 54 darts (18 for each 3-cell) where each vertex is associated with a point, here a `Point_3`. Blue segments represent \( \beta_3\) relations. Middle: Zoom around the central edge which details the six darts belonging to the edge and the associations between darts and points. Right: Zoom around the facet between light gray and white 3-cells, which details the eight darts belonging to the facet and the associations between darts and points (given by red segments).

Things are similar for generalized map, as illustrated in Figure 32.3. In this example, a 2D generalized map is used as underlying data-structure to describe the object given in Figure 32.1 (Left). The 2D linear cell complex shown in Figure 32.3 (Left) is obtained from this 2D generalized map by associating a point to each vertex of the map. We can compare in Figure 32.1 (Right) this 2D linear cell complex with the 2D linear cell complex describing the same 2D object but using 2D combinatorial maps (instead of generalized map). The only difference comes from the original definitions of combinatorial and generalized maps. Combinatorial maps have twice less darts than generalized maps, and thus the corresponding 2D linear cell complex has twice less association between darts and points.

Figure 32.3 Example of 2D linear cell complexes describing the object given in Figure 32.1 (Left). (Left) Based on a 2D generalized map. (Right) Based on a 2D combinatorial map. In both cases, each vertex of the combinatorial data-structure is associated with a point, here a `Point_2`. Associations between darts and points is drawn by red segments.

Note that the dimension of the combinatorial or the generalized map d is not necessarily equal to the dimension of the ambient space d2. Indeed, we can use for example a 2D combinatorial map in a 2D ambient space to describe a planar graph (d=d2=2), or a 2D combinatorial map in a 3D ambient space to describe a surface in 3D space (d=2, d2=3) (case of the `Polyhedron_3` package), or a 3D generalized map in a 3D ambient space (d=d2=3) and so on.

# Software Design

The diagram in Figure 32.4 shows the main classes of the package. `Linear_cell_complex_for_combinatorial_map` is the main class if you use combinatorial maps as combinatorial data-structure, and `Linear_cell_complex_for_generalized_map` is the main class if you use generalized maps as combinatorial data-structure (see Section Linear Cell Complex). `Linear_cell_complex_for_combinatorial_map` inherits from the `Combinatorial_map` class and `Linear_cell_complex_for_generalized_map` inherits from the `Generalized_map` class. Attributes can be associated to some cells of the linear cell complex thanks to an items class (see Section Linear Cell Complex Items), which defines the information associated to darts, and the attributes types. These types may be different for different dimensions of cells, and they may also be void. In the class `Linear_cell_complex_for_combinatorial_map`, it is required that specific attributes are associated to all vertices of the combinatorial or generalized map. These attributes must contain a point (a model of `Point_2` or `Point_3` or `Point_d`), and can be represented by instances of class `Cell_attribute_with_point` (see Section Cell Attributes).

Figure 32.4 UML diagram of the main classes of the package. Gray elements come from the Combinatorial maps and Generalized maps packages.

# Linear Cell Complex

The `Linear_cell_complex_for_combinatorial_map<d,d2,LCCTraits,Items,Alloc>` class is a model of the `LinearCellComplex` concept that uses a combinatorial map as underlying combinatorial data-structure. Similarly, the `Linear_cell_complex_for_generalized_map<d,d2,LCCTraits,Items,Alloc>` class is a model of the `LinearCellComplex` concept that uses a generalized map as underlying combinatorial data-structure. These two classes guarantee that each vertex is associated with an attribute containing a point. These classes can be used in geometric algorithms (they play the same role as `Polyhedron_3` for Halfedge Data Structures).

These classes has five template parameters standing for the dimension of the map, the dimension of the ambient space, a traits class (a model of the `LinearCellComplexTraits` concept, see Section Linear Cell Complex Traits), an items class (a model of the `LinearCellComplexItems` concept, see Section Linear Cell Complex Items), and an allocator which must be a model of the allocator concept of STL. Default classes are provided for the traits, items, and for the allocator classes, and by default `d2=d`.

A linear cell complex is valid, if it is a valid combinatorial or generalized map where each dart is associated with an attribute containing a point (i.e. an instance of a model of the `CellAttributeWithPoint` concept). Note that there are no validity constraints on the geometry (test on self intersection, planarity of 2-cells...). We can see two examples of `Linear_cell_complex_for_combinatorial_map` in Figure 32.5.

Figure 32.5 Examples of `Linear_cell_complex_for_combinatorial_map`. Gray disks show the attributes associated to vertices. Associations between darts and attributes are drawn by small lines between darts and disks. Left: Example of `Linear_cell_complex_for_combinatorial_map<2,2>`. Right: Example of `Linear_cell_complex_for_combinatorial_map<3,3>`.

## Cell Attributes

The `Cell_attribute_with_point<LCC,Info_,Tag,OnMerge,OnSplit>` class is a model of the `CellAttributeWithPoint` concept, which is a refinement of the `CellAttribute` concept. It represents an attribute associated with a cell, which can contain an information (depending on whether `Info_==void` or not), but which always contains a point, an instance of `LCC::Point`.

## Linear Cell Complex Traits

The `LinearCellComplexTraits` geometric traits concept defines the required types and functors used in the `Linear_cell_complex_for_combinatorial_map` class. For example it defines `Point`, the type of points used, and `Vector`, the corresponding vector type. It also defines all the required functors used for constructions and operations, as for example `Construct_translated_point` or `Construct_sum_of_vectors`.

The class `Linear_cell_complex_traits<d,K>` is a model of `LinearCellComplexTraits`. It defines the different types which are obtained from `K` that, depending on `d`, is a model of the concept `Kernel` if `d==2` or `d==3`, and a model of the concept `Kernel_d` otherwise.

## Linear Cell Complex Items

The `LinearCellComplexItems` concept refines the `GenericMapItems` concept by adding the requirement that 0-attributes are enabled, and associated with a type of attribute being a model of the `CellAttributeWithPoint` concept. Like for combinatorial and generalized maps, it is possible to use indices instead of handles as descriptors, by defining CGAL::Tag_true for Use_index type in the item class.

The class `Linear_cell_complex_min_items<d>` is a model of `LinearCellComplexItems`. It defines `void` as information associated to darts, and instances of `Cell_attribute_with_point` (which contain no information) associated to each vertex. All other attributes are `void`.

# Operations

Several operations defined in the combinatorial maps or generalized maps package can be used on a linear cell complex. This is the case for all the iteration operations that do not modify the model (see example in Section A 3D Linear Cell Complex). This is also the case for all the operations that do not create new 0-cells: `sew`, `unsew`, `remove_cell`, `insert_cell_1_in_cell_2`, `insert_cell_1_between_two_cells_2` or `insert_cell_2_in_cell_3`. Indeed, all these operations update non `void` attributes, and thus update vertex attributes of a linear cell complex. Note that some existing 0-attributes can be duplicated by the `unsew` method, but these 0-attributes are not new but copies of existing old 0-attributes.

However, operations that create a new 0-cell can not be directly used since the new 0-cell would not be associated with a vertex attribute. Indeed, it is not possible for these operations to automatically decide which point to create. These operations are: `insert_cell_0_in_cell_1`, `insert_cell_0_in_cell_2`, `insert_dangling_cell_1_in_cell_2`, plus all the creation operations. For these operations, new versions are proposed taking some points as additional parameters. Lastly, some new operations are defined, which use the geometry (see sections Construction Operations and Modification Operations).

All the operations given in this section guarantee that given a valid linear cell complex and a possible operation, the result is a valid linear cell complex. As for a combinatorial or generalized map, it is also possible to use low level operations but additional operations may be needed to restore the validity conditions.

## Sewing and Unsewing

As explained in the combinatorial map and generalized map user manuals, it is possible to glue two i-cells along an (i-1)-cell by using the `sew<i>` method. Since this method updates non void attributes, and since points are specific attributes, they are automatically updated during the `sew<i>` method. Thus the sewing of two i-cells could deform the geometry of the concerned objects.

For example, in Figure 32.6, we want to 3-sew the two initial 3-cells. `sew<3>(1,5)` links by \( \beta_3\) the pairs of darts (1,5), (2,8), (3,7) and (4,6). The eight vertex attributes around the facet between the two 3-cells before the sew are merged by pair during the sew operation (and the `On_merge` functor is called four times). Thus, after the sew, there are only four 0-attributes around the facet. By default, the attributes associated with the first dart of the sew operation are kept (but this can be modified by defining your own functor in the attribute class as explained in the packages combinatorial map and generalized map. Intuitively, the geometry of the second 2-cell is deformed to fit to the first 2-cell.

Figure 32.6 Example of 3-sew operation for linear cell complex. Left: A 3D linear cell complex containing two 3-cells that are not connected. Vertex attributes are drawn with circles containing point coordinates. Associations between darts and attributes are drawn with small lines between darts and disks. Right: The 3D linear cell complex obtained as result of `sew<3>(1,5)` (or `sew<3>(2,8)`, or `sew<3>(3,7)`, or `sew<3>(4,6)`). The eight 0-attributes around the facet between the two 3-cells before the sew operation, are merged into four 0-attributes after. The geometry of the pyramid is deformed since its base is fitted on the 2-cell of the cube.

This is similar for `unsew<i>` operation, which removes i-links of all the darts in a given (i-1)-cell, and updates non void attributes which are no more associated to a same cell due to the unlinks. If we take the linear cell complex given in Figure 32.6 (Right), and we call `unsew<3>(2)`, we obtain the linear cell complex in Figure 32.6 (Left) except for the coordinates of the new four vertices, which by default are copies of original vertices (this behavior can be modified thanks to the functor `On_split` in the attribute class). The `unsew<3>` operation has removed the four \( \beta_3\) links, and has duplicated the 0-attributes since vertices are split in two after the unsew operation.

Advanced

If `set_automatic_attributes_management(false)` is called, all the future sew and unsew operations will not update non void attributes. These attributes will be updated latter by the call to `set_automatic_attributes_management(true)`.

## Construction Operations

There are several member functions allowing to insert specific configurations of darts into a linear cell complex. These functions return a `Dart_descriptor` to the new object. Note that the dimension of the linear cell complex must be large enough: darts must contain all the applications ( \( \alpha\) or \( \beta\)) used by the operation. All these methods add new darts in the current linear cell complex, existing darts are not modified. These functions are `make_segment`, `make_triangle`, `make_tetrahedron` and `make_hexahedron`.

There are two functions allowing to build a linear cell complex from two other CGAL data types:

- `import_from_triangulation_3(lcc,atr)`: adds in `lcc` all the tetrahedra present in `atr`, a `Triangulation_3`;
- `import_from_polyhedron_3(lcc,ap)`: adds in `lcc` all the cells present in `ap`, a `Polyhedron_3`.

Lastly, the function `import_from_plane_graph(lcc,ais)` adds in `lcc` all the cells reconstructed from the planar graph read in `ais`, a `std::istream` (see the `reference manual` for the file format).

## Modification Operations

Some methods are defined in `LinearCellComplex` to modify a linear cell complex and update the vertex attributes. In the following, we denote by `d0`, `d1`, `d2` for three dart descriptors.

Figure 32.7 Example of `insert_barycenter_in_cell<1>` and `remove_cell<0>` operations. Left: Initial linear cell complex. Right: After the insertion of a point in the barycenter of the 1-cell containing dart d1. Now if we remove the 0-cell containing dart d2, we obtain a linear cell complex isomorphic to the initial one.

`lcc.insert_barycenter_in_cell<unsigned int i>(d0)` adds the barycenter of the i-cell containing dart `d0`. This operation is possible if `d0` \( \in\)`lcc.darts()` (see examples on Figure 32.7 and Figure 32.8).

`lcc.insert_point_in_cell<unsigned int i>(d0,p)` is an operation similar to the previous operation, the only difference being that the coordinates of the new point are here given by `p` instead of being computed as the barycenter of the i-cell. Currently, these two operations are only defined for `i=1` to insert a point in an edge, or `i=2` to insert a point in a facet.

Figure 32.8 Examples of `insert_barycenter_in_cell<2>` operation.

`lcc.insert_dangling_cell_1_in_cell_2(d0,p)` adds a 1-cell in the 2-cell containing dart `d0`, the 1-cell being attached by only one of its vertex to the 0-cell containing dart `d0`. The second vertex of the new edge is associated with a new 0-attribute containing a copy of `p` as point. This operation is possible if `d0` \( \in\)`lcc.darts()` (see example on Figure 32.9).

Figure 32.9 Example of `insert_dangling_cell_1_in_cell_2`, `insert_cell_1_in_cell_2` and `remove_cell<1>` operations. Left: Initial linear cell complex. Right: After the insertion of a dangling 1-cell in the 2-cell containing dart d1, and of a 1-cell in the 2-cell containing dart d2. Now if we remove the 1-cells containing dart d4 and d5, we obtain a linear cell complex isomorphic to the initial one.

Some examples of use of these operations are given in Section A 4D Linear Cell Complex.

Advanced

If `set_automatic_attributes_management(false)` is called, all the future insertion or removal operations will not update non void attributes. These attributes will be updated latter by the call to `set_automatic_attributes_management(true)`. This can be useful to speed up an algorithm which uses several successive insertion and removal operations. See example Automatic attributes management.

## Incremental Builder

A utility class `Linear_cell_complex_incremental_builder_3` helps in creating 2D and 3D linear cell complexes from a list of points followed by a list of facets that are represented as indices into the point list. Note that, compared to `Polyhedron_incremental_builder_3` it has only absolute indexing and no rollback mechanism.

# Examples

## A 3D Linear Cell Complex

This example uses a 3-dimensional linear cell complex based on combinatorial maps. It creates two tetrahedra and displays all the points of the linear cell complex thanks to a `Vertex_attribute_const_range`. Then, the two tetrahedra are `3-sewn` and we translate all the points of the second tetrahedron along vector `v(3,1,1)`. Since the two tetrahedra are 3-sewn, this translation moves also the 2-cell of the first tetrahedron shared with the second one. This is illustrated by displaying all the points of each 3-cell. For that we use a `std::for_each` and the `Display_vol_vertices` functor.

File Linear_cell_complex/linear_cell_complex_3.cpp

#include <CGAL/Linear_cell_complex_for_combinatorial_map.h>
#include <iostream>
#include <algorithm>
typedef CGAL::Linear_cell_complex_for_combinatorial_map<3> LCC_3;
typedef LCC_3::Dart_descriptor Dart_descriptor;
typedef LCC_3::Point Point;
typedef LCC_3::FT FT;
// Functor used to display all the vertices of a given volume.
template<class LCC>
struct Display_vol_vertices : public CGAL::cpp98::unary_function<LCC, void>
{
Display_vol_vertices(const LCC& alcc) :
lcc(alcc),
nb_volume(0)
{}
void operator() (typename LCC::Dart& d)
{
std::cout<<"Volume "<<++nb_volume<<" : ";
for (typename LCC::template One_dart_per_incident_cell_range<0,3>::
const_iterator it=lcc.template one_dart_per_incident_cell<0,3>
(lcc.dart_descriptor(d)).begin(),
itend=lcc.template one_dart_per_incident_cell<0,3>
(lcc.dart_descriptor(d)).end();
it!=itend; ++it)
{
std::cout << lcc.point(it) << "; ";
}
std::cout<<std::endl;
}
private:
const LCC& lcc;
unsigned int nb_volume;
};
int main()
{
LCC_3 lcc;
// Create two tetrahedra.
Dart_descriptor d1 = lcc.make_tetrahedron(Point(-1, 0, 0), Point(0, 2, 0),
Point(1, 0, 0), Point(1, 1, 2));
Dart_descriptor d2 = lcc.make_tetrahedron(Point(0, 2, -1),
Point(-1, 0, -1),
Point(1, 0, -1),
Point(1, 1, -3));
// Display all the vertices of the lcc by iterating on the
// Vertex_attribute container.
CGAL::IO::set_ascii_mode(std::cout);
std::cout<<"Vertices: ";
for (LCC_3::Vertex_attribute_const_range::iterator
v=lcc.vertex_attributes().begin(),
vend=lcc.vertex_attributes().end();
v!=vend; ++v)
std::cout << lcc.point_of_vertex_attribute(v) << "; ";
std::cout<<std::endl;
// Display the vertices of each volume by iterating on darts.
std::for_each(lcc.one_dart_per_cell<3>().begin(),
lcc.one_dart_per_cell<3>().end(),
Display_vol_vertices<LCC_3>(lcc));
// 3-Sew the 2 tetrahedra along one facet
lcc.sew<3>(d1, d2);
// Display the vertices of each volume by iterating on darts.
std::for_each(lcc.one_dart_per_cell<3>().begin(),
lcc.one_dart_per_cell<3>().end(),
Display_vol_vertices<LCC_3>(lcc));
// Translate the second tetrahedra by a given vector
LCC_3::Vector v(3,1,1);
for (LCC_3::One_dart_per_incident_cell_range<0,3>::iterator
it=lcc.one_dart_per_incident_cell<0,3>(d2).begin(),
itend=lcc.one_dart_per_incident_cell<0,3>(d2).end();
it!=itend; ++it)
{
lcc.point(it)=LCC_3::Traits::Construct_translated_point_3()
(lcc.point(it),v);
}
// Display the vertices of each volume by iterating on darts.
std::for_each(lcc.one_dart_per_cell<3>().begin(),
lcc.one_dart_per_cell<3>().end(),
Display_vol_vertices<LCC_3>(lcc));
// We display the lcc characteristics.
std::cout<<"LCC characteristics: ";
lcc.display_characteristics(std::cout) << ", valid=" << lcc.is_valid()
<< std::endl;
return EXIT_SUCCESS;
}
CGAL::Linear_cell_complex_for_combinatorial_map
The class Linear_cell_complex_for_combinatorial_map represents a linear cell complex in dimension d,...
Definition: Linear_cell_complex_for_combinatorial_map.h:29
CGAL::IO::set_ascii_mode
Mode set_ascii_mode(std::ios &s)
CGAL::cpp98::unary_function

The output is:

    Vertices: 1 1 2; 1 0 0; 0 2 0; -1 0 0; 1 1 -3; 1 0 -1; -1 0 -1; 0 2 -1;
    Volume 1 : -1 0 0; 0 2 0; 1 0 0; 1 1 2;
    Volume 2 : 0 2 -1; -1 0 -1; 1 0 -1; 1 1 -3;
    Volume 1 : -1 0 0; 0 2 0; 1 0 0; 1 1 2;
    Volume 2 : 0 2 0; -1 0 0; 1 0 0; 1 1 -3;
    Volume 1 : 2 1 1; 3 3 1; 4 1 1; 1 1 2;
    Volume 2 : 3 3 1; 2 1 1; 4 1 1; 4 2 -2;
    LCC characteristics: #Darts=24, #0-cells=5, #1-cells=9, #2-cells=7, #3-cells=2, #ccs=1, valid=1

The first line gives the points of the linear cell complex before the `sew<3>`. There are eight points, four for each tetrahedron. After the sew, six vertices are merged two by two, thus there are five vertices. We can see the points of each 3-cell (lines Volume 1 and Volume 2) before the sew, after the sew and after the translation of the second volume. We can see that this translation has also modified the three common points between the two 3-cells. The last line shows the number of cells of the linear cell complex, the number of connected components, and finally a Boolean to show the validity of the linear cell complex.

## A 4D Linear Cell Complex

This example uses a 4-dimensional linear cell complex embedded in a 5-dimensional ambient space and based on generalized maps. It creates two tetrahedra having 5D points and sews the two tetrahedra by \( \beta_4\). Then we use some high level operations, display the number of cells of the linear cell complex, and check its validity. Last we use the reverse operations to get back to the initial configuration.

File Linear_cell_complex/linear_cell_complex_4.cpp

#include <CGAL/Linear_cell_complex_for_generalized_map.h>
#include <iostream>
#include <vector>
#include <cassert>
typedef CGAL::Linear_cell_complex_for_generalized_map<4,5> LCC_4;
typedef LCC_4::Dart_descriptor Dart_descriptor;
typedef LCC_4::Point Point;
typedef LCC_4::Vector Vector;
typedef LCC_4::FT FT;
int main()
{
LCC_4 lcc;
// Create two tetrahedra.
FT p1[5]={ 0, 0, 0, 0, 0}; std::vector<FT> v1(p1,p1+5);
FT p2[5]={ 0, 2, 0, 0, 0}; std::vector<FT> v2(p2,p2+5);
FT p3[5]={ 0, 1, 2, 0, 0}; std::vector<FT> v3(p3,p3+5);
FT p4[5]={ 2, 1, 0, 0, 0}; std::vector<FT> v4(p4,p4+5);
FT p5[5]={-1, 0, 0, 0, 0}; std::vector<FT> v5(p5,p5+5);
FT p6[5]={-1, 2, 0, 0, 0}; std::vector<FT> v6(p6,p6+5);
FT p7[5]={-1, 1, 2, 0, 0}; std::vector<FT> v7(p7,p7+5);
FT p8[5]={-3, 1, 2, 0, 0}; std::vector<FT> v8(p8,p8+5);
Dart_descriptor d1 = lcc.make_tetrahedron(Point(5, v1.begin(), v1.end()),
Point(5, v2.begin(), v2.end()),
Point(5, v3.begin(), v3.end()),
Point(5, v4.begin(), v4.end()));
Dart_descriptor d2 = lcc.make_tetrahedron(Point(5, v5.begin(), v5.end()),
Point(5, v6.begin(), v6.end()),
Point(5, v7.begin(), v7.end()),
Point(5, v8.begin(), v8.end()));
lcc.display_characteristics(std::cout);
std::cout<<", valid="<<lcc.is_valid()<<std::endl;
lcc.sew<4>(d1,d2);
lcc.display_characteristics(std::cout);
std::cout<<", valid="<<lcc.is_valid()<<std::endl;
// Add one vertex on the middle of the edge containing dart d1.
Dart_descriptor d3 = lcc.insert_barycenter_in_cell<1>(d1);
assert( lcc.is_valid() );
lcc.display_characteristics(std::cout);
std::cout<<", valid="<<lcc.is_valid()<<std::endl;
// Add one edge to cut the face containing dart d3 in two.
Dart_descriptor d4 = lcc.insert_cell_1_in_cell_2(d3, lcc.alpha(d1, 1, 0, 1));
assert( lcc.is_valid() );
lcc.display_characteristics(std::cout);
std::cout<<", valid="<<lcc.is_valid()<<std::endl;
// We use removal operations to get back to the initial configuration.
lcc.remove_cell<1>(d4);
assert( lcc.is_valid() );
lcc.remove_cell<0>(d3);
assert( lcc.is_valid() );
lcc.unsew<4>(d1);
lcc.display_characteristics(std::cout);
std::cout<<", valid="<<lcc.is_valid()<<std::endl;
return EXIT_SUCCESS;
}
CGAL::Linear_cell_complex_for_generalized_map
The class Linear_cell_complex_for_generalized_map represents a linear cell complex in dimension d,...
Definition: Linear_cell_complex_for_generalized_map.h:29
Vector

The output is:

    #Darts=48, #0-cells=8, #1-cells=12, #2-cells=8, #3-cells=2, #4-cells=2, #ccs=2, orientable=true, valid=1
    #Darts=48, #0-cells=4, #1-cells=6, #2-cells=4, #3-cells=1, #4-cells=2, #ccs=1, orientable=true, valid=1
    #Darts=56, #0-cells=5, #1-cells=7, #2-cells=4, #3-cells=1, #4-cells=2, #ccs=1, orientable=true, valid=1
    #Darts=64, #0-cells=5, #1-cells=8, #2-cells=5, #3-cells=1, #4-cells=2, #ccs=1, orientable=true, valid=1
    #Darts=48, #0-cells=8, #1-cells=12, #2-cells=8, #3-cells=2, #4-cells=2, #ccs=2, orientable=true, valid=1

## A 3D Linear Cell Complex with Colored Vertices

This example illustrates the way to use a 3D linear cell complex by adding another information to vertices. For that, we need to define our own items class. The difference with the `Linear_cell_complex_min_items` class is about the definition of the vertex attribute where we use a `Cell_attribute_with_point` with a non `void` info. In this example, the vertex color is just given by an `int` (the second template parameter of the `Cell_attribute_with_point`). Lastly, we define the `Average_functor` class in order to set the color of a vertex resulting of the merging of two vertices to the average of the two initial values. This functor is associated with the vertex attribute by passing it as template parameter. Using this items class instead of the default one is done during the instantiation of template parameters of the `Linear_cell_complex_for_combinatorial_map` class.

Now we can use `LCC_3` in which each vertex is associated with an attribute containing both a point and an information. In the following example, we create two cubes, and set the color of the vertices of the first cube to 1 and of the second cube to 19 (by iterating through two `One_dart_per_incident_cell_range<0, 3>` ranges). Then we `3-sew` the two cubes along one facet. This operation merges some vertices (as in the example of Figure 32.6). We insert a vertex in the common 2-cell between the two cubes, and set the information of the new 0-attribute to 5. In the last loop, we display the point and the information of each vertex of the linear cell complex.

File Linear_cell_complex/linear_cell_complex_3_with_colored_vertices.cpp

#include <CGAL/Linear_cell_complex_for_combinatorial_map.h>
#include <iostream>
#include <algorithm>
struct Average_functor
{
template<class CellAttribute>
void operator()(CellAttribute& ca1, CellAttribute& ca2)
{ ca1.info()=(ca1.info()+ ca2.info())/2; }
};
struct Myitem
{
template<class Refs>
struct Dart_wrapper
{
typedef CGAL::Cell_attribute_with_point< Refs, int, CGAL::Tag_true,
Average_functor >
Vertex_attribute;
typedef std::tuple<Vertex_attribute> Attributes;
};
};
typedef CGAL::Linear_cell_complex_traits
<3, CGAL::Exact_predicates_inexact_constructions_kernel> Traits;
typedef CGAL::Linear_cell_complex_for_combinatorial_map<3,3,Traits,Myitem> LCC_3;
typedef LCC_3::Dart_descriptor Dart_descriptor;
typedef LCC_3::Point Point;
typedef LCC_3::FT FT;
Dart_descriptor make_iso_cuboid(LCC_3& lcc, const Point& basepoint, FT lg)
{
return lcc.make_hexahedron(basepoint,
Traits::Construct_translated_point()
(basepoint,Traits::Vector(lg,0,0)),
Traits::Construct_translated_point()
(basepoint,Traits::Vector(lg,lg,0)),
Traits::Construct_translated_point()
(basepoint,Traits::Vector(0,lg,0)),
Traits::Construct_translated_point()
(basepoint,Traits::Vector(0,lg,lg)),
Traits::Construct_translated_point()
(basepoint,Traits::Vector(0,0,lg)),
Traits::Construct_translated_point()
(basepoint,Traits::Vector(lg,0,lg)),
Traits::Construct_translated_point()
(basepoint,Traits::Vector(lg,lg,lg)));
}
int main()
{
LCC_3 lcc;
// Create two iso_cuboids.
Dart_descriptor d1 = make_iso_cuboid(lcc, Point(-2, 0, 0), 1);
Dart_descriptor d2 = make_iso_cuboid(lcc, Point(0, 0, 0), 1);
// Set the "color" of all vertices of the first cube to 1.
for (LCC_3::One_dart_per_incident_cell_range<0, 3>::iterator
it=lcc.one_dart_per_incident_cell<0,3>(d1).begin(),
itend=lcc.one_dart_per_incident_cell<0,3>(d1).end(); it!=itend; ++it)
{ lcc.info<0>(it)=1; }
// Set the "color" of all vertices of the second cube to 19.
for (LCC_3::One_dart_per_incident_cell_range<0, 3>::iterator it=
lcc.one_dart_per_incident_cell<0,3>(d2).begin(),
itend=lcc.one_dart_per_incident_cell<0,3>(d2).end(); it!=itend; ++it)
{ lcc.info<0>(it)=19; }
// 3-Sew the two cubes along one facet.
lcc.sew<3>(lcc.beta(d1, 1, 1, 2), lcc.beta(d2, 2));
// Barycentric triangulation of the facet between the two cubes.
Dart_descriptor d3=lcc.insert_barycenter_in_cell<2>(lcc.beta(d2, 2));
// Set the color of the new vertex to 5.
lcc.info<0>(d3)=5;
// Display all the vertices of the map.
for (LCC_3::Vertex_attribute_range::iterator
it=lcc.vertex_attributes().begin(),
itend=lcc.vertex_attributes().end();
it!=itend; ++it)
{
std::cout<<"point: "<<lcc.point_of_vertex_attribute(it)<<", "<<"color: "
<<lcc.info_of_attribute<0>(it)<<std::endl;
}
return EXIT_SUCCESS;
}
CGAL::Cell_attribute_with_point
The class Cell_attribute_with_point represents an attribute containing a point and containing an info...
Definition: Cell_attribute_with_point.h:23
CGAL::Exact_predicates_inexact_constructions_kernel
CellAttribute
CellAttribute::info
Info & info()
CGAL::Tag_true
CGAL::Boolean_tag< true > Tag_true
CGAL::Linear_cell_complex_traits
This geometric traits concept is used in the Linear_cell_complex_for_combinatorial_map and Linear_cel...
Definition: Linear_cell_complex_traits.h:21

The output is:

    point: -1 1 1, color: 10
    point: -1 0 1, color: 10
    point: -2 0 1, color: 1
    point: -2 1 1, color: 1
    point: -2 1 0, color: 1
    point: -1 1 0, color: 10
    point: -1 0 0, color: 10
    point: -2 0 0, color: 1
    point: 1 1 1, color: 19
    point: 1 0 1, color: 19
    point: -1 0.5 0.5, color: 5
    point: 1 1 0, color: 19
    point: 1 0 0, color: 19

Before applying the sew operation, the eight vertices of the first cube are colored by `1`, and the eight vertices of the second cube by `19`. After the sew operation, there are eight vertices which are merged two by two, and due to the average functor, the color of the four resulting vertices is now 10. Then we insert a vertex in the center of the common 2-cell between the two cubes. The coordinates of this vertex are initialized with the barycenter of the 2-cell (-1,0.5,0.5), and its color is not initialized by the method, thus we set its color manually by using the result of `insert_barycenter_in_cell<2>` which is a dart incident to the new vertex.

## Automatic Attribute Management

The following example illustrates the use of the automatic attributes management for a linear cell complex. An off file is loaded into a 2D linear cell complex embedded in 3D. Then, a certain percentage of edges is removed from the linear cell complex. The same method is applied twice: the first time by using the automatic attributes management (which is the default behavior) and the second time by calling first `set_automatic_attributes_management(false)` to disable the automatic updating of attributes.

We can observe that the second run is faster than the first one. Indeed, updating attribute for each edge removal give a bigger complexity. Moreover, the gain increases when the percentage of removed edges increases.

File Linear_cell_complex/linear_cell_complex_3_attributes_management.cpp

#include <CGAL/Linear_cell_complex_for_combinatorial_map.h>
#include <CGAL/Linear_cell_complex_constructors.h>
#include <CGAL/Timer.h>
#include <iostream>
#include <fstream>
typedef CGAL::Linear_cell_complex_for_combinatorial_map<2,3> LCC_3;
typedef LCC_3::Dart_descriptor Dart_descriptor;
typedef LCC_3::Point Point;
typedef LCC_3::FT FT;
void load_and_simplify_off(LCC_3& lcc, const std::string& filename,
bool updateattribs, int percent)
{
std::ifstream ifile(filename.c_str());
if (ifile)
{
CGAL::load_off(lcc, ifile);
CGAL::Timer timer;
Dart_descriptor d;
std::size_t nb=(lcc.number_of_darts()*percent)/200;
timer.start();
if (!updateattribs) lcc.set_automatic_attributes_management(false);
for (LCC_3::Dart_range::iterator it=lcc.darts().begin(),
itend=lcc.darts().end(); it!=itend && nb>0; )
{
d=it++;
if ( it!=itend && it==lcc.beta<2>(d) ) ++it;
lcc.remove_cell<1>(d);
--nb;
}
if ( !updateattribs ) lcc.set_automatic_attributes_management(true);
timer.stop();
lcc.display_characteristics(std::cout);
std::cout<<", valid="<< lcc.is_valid()
<<" time: "<<timer.time()<<" seconds." << std::endl;
}
}
int main(int narg, char** argv)
{
if (narg>1 && (!strcmp(argv[1],"-h") || !strcmp(argv[1],"-?")) )
{
std::cout<<"Usage: a.out file.off [percentage]"<<std::endl;
return EXIT_FAILURE;
}
std::string filename;
if ( narg==1 )
{
filename=std::string(CGAL::data_file_path("meshes/armadillo.off"));
std::cout<<"No filename given: use data/armadillo.off by default."<<std::endl;
}
else filename=std::string(argv[1]);
int percent = 30; // remove 30 percent of edges
if ( narg>2 ) { percent = atoi(argv[2]); }
std::cout<<percent<<"% edges to remove."<<std::endl;
LCC_3 lcc;
std::cout<<"Update attribute DURING operations: ";
load_and_simplify_off(lcc, filename, true, percent);
LCC_3 lcc2;
std::cout<<"Update attribute AFTER operations: ";
load_and_simplify_off(lcc2, filename, false, percent);
return EXIT_SUCCESS;
}

## Draw a Linear Cell Complex

A linear cell complex can be visualized by calling the CGAL::draw<LCC>() function as shown in the following example. This function opens a new window showing the given linear cell complex. A call to this function is blocking, that is the program continues as soon as the user closes the window.

File Linear_cell_complex/draw_linear_cell_complex.cpp

#include <CGAL/Linear_cell_complex_for_combinatorial_map.h>
#include <CGAL/draw_linear_cell_complex.h>
typedef CGAL::Linear_cell_complex_for_combinatorial_map<3> LCC;
typedef LCC::Dart_descriptor Dart_descriptor;
typedef LCC::Point Point;
int main()
{
LCC lcc;
Dart_descriptor d1=
lcc.make_hexahedron(Point(0,0,0), Point(5,0,0),
Point(5,5,0), Point(0,5,0),
Point(0,5,4), Point(0,0,4),
Point(5,0,4), Point(5,5,4));
Dart_descriptor d2=
lcc.make_hexahedron(Point(5,0,0), Point(10,0,0),
Point(10,5,0), Point(5,5,0),
Point(5,5,4), Point(5,0,4),
Point(10,0,4), Point(10,5,4));
lcc.sew<3>(lcc.beta(d1, 1, 1, 2), lcc.beta(d2, 2));
lcc.display_characteristics(std::cout)<<", valid="
<<lcc.is_valid()<<std::endl;
CGAL::draw(lcc);
return EXIT_SUCCESS;
}
CGAL::draw
void draw(const P &p, const GSOptions &gso)

This function requires `CGAL_Qt6`, and is only available if the macro `CGAL_USE_BASIC_VIEWER` is defined. Linking with the cmake target `CGAL::CGAL_Basic_viewer` will link with `CGAL_Qt6` and add the definition `CGAL_USE_BASIC_VIEWER`.

Figure 32.10 Result of the run of the draw_linear_cell_complex program. A window shows two 3D cubes and allows to navigate through the 3D scene.

## Incremental Builder

The following example shows the incremental builder.

File Linear_cell_complex/linear_cell_complex_3_incremental_builder.cpp

#include <CGAL/Linear_cell_complex_for_combinatorial_map.h>
#include <CGAL/Linear_cell_complex_incremental_builder_3.h>
#include <CGAL/draw_linear_cell_complex.h>
typedef CGAL::Linear_cell_complex_for_combinatorial_map<3, 3> LCC_3;
using Point=LCC_3::Point;
int main()
{
LCC_3 lcc;
CGAL::Linear_cell_complex_incremental_builder_3<LCC_3> ib(lcc);
ib.add_vertex(Point(0,0,0)); // vertex 0
ib.add_vertex(Point(1,0,0)); // vertex 1
ib.add_vertex(Point(1,1,0)); // vertex 2
ib.add_vertex(Point(0,1,0)); // vertex 3
ib.add_vertex(Point(0,1,1)); // vertex 4
ib.add_vertex(Point(0,0,1)); // vertex 5
ib.add_vertex(Point(1,0,1)); // vertex 6
ib.add_vertex(Point(1,1,1)); // vertex 7
// Create a cube
ib.begin_surface();
ib.add_facet({0,1,2,3}); // Create a new facet version 1: given all of its indices
ib.add_facet({1,0,5,6});
ib.add_facet({2,1,6,7});
ib.add_facet({3,2,7,4});
ib.add_facet({5,4,7,6});
ib.begin_facet(); // Create a new facet version 2: begin facet
ib.add_vertex_to_facet(0); // add successively its indices
ib.add_vertex_to_facet(3);
ib.add_vertex_to_facet(4);
ib.add_vertex_to_facet(5);
ib.end_facet();
ib.end_surface();
ib.add_vertex(Point(-1, 0.5, 0.5)); // vertex 8
// Create a pyramid, sharing one of its facets with a facet of the cube
ib.begin_surface();
ib.add_facet({3,0,5,4});
ib.add_facet({0,3,8});
ib.add_facet({3,4,8});
ib.add_facet({4,5,8});
ib.add_facet({5,0,8});
ib.end_surface();
LCC_3::One_dart_per_cell_range<3,3> cells = lcc.one_dart_per_cell<3>();
for (auto c = cells.begin(); c != cells.end(); ++c) {
std::cout << "a cell"<< std::endl;
LCC_3::One_dart_per_incident_cell_range<2,3> faces = lcc.one_dart_per_incident_cell<2,3>(c);
for (auto f = faces.begin(); f != faces.end(); ++f) {
std::cout << " a face"<< std::endl;
LCC_3::One_dart_per_incident_cell_range<0,2> vertices = lcc.one_dart_per_incident_cell<0,2>(f);
for(auto v = vertices.begin(); v!= vertices.end(); ++v){
std::cout << " " << lcc.point(v) << std::endl;
}
}
}
// Draw the lcc and display its characteristics
lcc.display_characteristics(std::cout)<<std::endl;
CGAL::draw(lcc);
return EXIT_SUCCESS;
}
CGAL::Linear_cell_complex_incremental_builder_3
The auxiliary class Linear_cell_complex_incremental_builder_3 supports the incremental construction o...
Definition: Linear_cell_complex_incremental_builder_3.h:15

## Insert an Edge Between Two Different Faces

The following example shows the use of `insert_cell_1_between_two_cells_2` operation that inserts an edge between two different faces, thus creating an hole in the first face.

File Linear_cell_complex/linear_cell_complex_3_insert.cpp

#include <CGAL/Linear_cell_complex_for_combinatorial_map.h>
#include <CGAL/Linear_cell_complex_for_generalized_map.h>
#include <CGAL/draw_linear_cell_complex.h>
typedef CGAL::Linear_cell_complex_for_combinatorial_map<3> LCC1;
typedef CGAL::Linear_cell_complex_for_generalized_map<3> LCC2;
template<typename LCC>
void test()
{
LCC lcc;
using Point=typename LCC::Point;
typename LCC::Dart_descriptor d1=
lcc.make_hexahedron(Point(0,0,0), Point(5,0,0),
Point(5,5,0), Point(0,5,0),
Point(0,5,4), Point(0,0,4),
Point(5,0,4), Point(5,5,4));
typename LCC::Dart_descriptor d2=
lcc.make_quadrangle(Point(5,2,2), Point(5,1,2),
Point(5,1,1), Point(5,2,1));
lcc.insert_cell_1_between_two_cells_2
(lcc.template opposite<2>(lcc.next(lcc.next(d1))),
lcc.next(lcc.next(d2)));
CGAL::draw(lcc);
}
int main()
{
test<LCC1>();
test<LCC2>();
return EXIT_SUCCESS;
}

Figure 32.11 Result of the run of the linear_cell_complex_3_insert program. A window shows the 3D cube where one face has a hole.

# Design and Implementation History

This package was developed by Guillaume Damiand, with the help of Andreas Fabri, Sébastien Loriot and Laurent Rineau. Monique Teillaud and Bernd Gärtner contributed to the manual.