Definition

The concept AbstractChainComplex describes the requirements for (topological) chain complexes associated to abstract complexes used in the concept CGAL::HDVF.

It provides methods to:

get the dimension of the complex, the number of cells in each dimension
get the boundary and co-boundary of cell(s)
get the vertices of a given cell
output the complex in text format

Cells are indexed along each dimension and thus identified by their index together with their dimension.

Has models
: CGAL::HDVF::Abstract_simplicial_chain_complex<Ring>
: CGAL::HDVF::Sub_chain_complex_mask<Ring, AbstractChainComplex>

Types
typedef OSM::Sparse_chain< CoefficientType, OSM::COLUMN >	CChain
	Type of column-major chains (returned by the boundary operator)

typedef OSM::Sparse_chain< CoefficientType, OSM::ROW >	RChain
	Type of row-major chains (returned by the co-boundary operator)

typedef OSM::Sparse_matrix< CoefficientType, OSM::COLUMN >	CMatrix
	Type of column-major sparse matrices (used to store the boundary operator)

Operators
AbstractChainComplex &	operator= (const AbstractChainComplex &complex)
	Affectation operator.

Access functions
int	dim ()
	Returns the dimension of the complex, that is, the largest dimension of cells.

size_t	nb_cells (int q)
	Returns the number of cells of dimension `q`.

const vector< CMatrix > &	get_bnd_matrices () const
	Returns all boundary matrices.

const CMatrix &	get_bnd_matrix (int q) const
	Returns the boundary matrix of dimension `q` (ie.

CChain	d (size_t id_cell, int q)
	Returns the boundary of the cell of index `id_cell` in dimension `q`.

RChain	cod (size_t id_cell, int q)
	Returns the co-boundary of the cell of index `id_cell` in dimension `q`.

std::vector< size_t >	bottom_faces (size_t id_cell, int q) const
	Returns the vertices of a given cell (that is, the indices of its faces of dimension 0).

template<typename CoefficientT , int ChainTypeF>
CChain	cofaces_chain (SparseChain< CoefficientT, ChainTypeF > chain, int q) const
	Returns the cofaces of a given chain in dimension `q`.

Output functions
std::ostream &	print_complex (std::ostream &out=std::cout) const
	Outputs the chain complex in text mode.

Member Function Documentation

◆ cod()

RChain AbstractChainComplex::cod	(	size_t	id_cell,
		int	q
	)

Returns the co-boundary of the cell of index id_cell in dimension q.

This boundary is a finite linear combination of cells of dimension q+1. It is encoded as a row-major chain (which maps each cell with a non-zero coefficient to this coefficient). This co-boundary is thus the id_cell-th row of the boundary matrix in dimension q+1.

◆ cofaces_chain()

template<typename CoefficientT , int ChainTypeF>

CChain AbstractChainComplex::cofaces_chain	(	SparseChain< CoefficientT, ChainTypeF >	chain,
		int	q
	)		const

Returns the cofaces of a given chain in dimension q.

The resulting chain lies in dimension q+1 and is null if this dimension exceeds the dimension of the complex.

◆ d()

CChain AbstractChainComplex::d	(	size_t	id_cell,
		int	q
	)

Returns the boundary of the cell of index id_cell in dimension q.

This boundary is a finite linear combination of cells of dimension q-1. It is encoded as a column-major chain (which maps each cell with a non-zero coefficient to this coefficient). This boundary is thus the id_cell-th column of the boundary matrix in dimension q.

◆ get_bnd_matrices()

const vector< CMatrix > & AbstractChainComplex::get_bnd_matrices ( ) const

Returns all boundary matrices.

The function returns constant reference to a vector of column-major sparse matrices. The q-th element of this vector is the matrix of \(\partial_q\), which gives the boundary of cells of dimension q(as a linear combination of q-1 cells).

◆ get_bnd_matrix()

const CMatrix & AbstractChainComplex::get_bnd_matrix ( int q ) const

Returns the boundary matrix of dimension q (ie.

the matrix of \(\partial_q}\)).

The function returns a column-major sparse matrices.

◆ nb_cells()

size_t AbstractChainComplex::nb_cells ( int q )

Returns the number of cells of dimension q.

If q is negative of larger than the dimension of the complex, returns 0.

◆ operator=()

AbstractChainComplex & AbstractChainComplex::operator= ( const AbstractChainComplex & complex )

Affectation operator.

The operator creates a "fresh" copy of complex.

◆ print_complex()

std::ostream & AbstractChainComplex::print_complex ( std::ostream & out = std::cout ) const

Outputs the chain complex in text mode.

By default, outputs the complex to std::cout.