#include <CGAL/HDVF/Hdvf.h>

Inherits from

CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex, OSM::Sparse_chain, OSM::Sparse_matrix >.

Definition

template<typename ChainComplex>
class CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >

The class Hdvf implements homology and cohomology computation via homological discrete vector fields (HDVF for short).

It derives from Hdvf_core and shares all its data and methods.

But besides construction operations and methods (using the A() operation), the Hdvf class implements four other HDVF operations: R, M, W and MW together with appropriate "find_pair()" functions. These operations change the HDVF (that is change homology / cohomology generators) and thus provide a convenient tool to move inside the "space of homology/cohomology computations".

R() operation is the "dual" of the A pairing operation (it cancels the pairing and turns back a PRIMARY/SECONDARY pair into a pair of CRITICAL cells)
M() operation exchanges a PRIMARY \(\pi\) and a CRITICAL cell \(\gamma\) (under conditions) and modifies the homology generator associated to \(\gamma\) (while preserving is associated cohomology generator)
W() operation exchanges a SECONDARY \(\sigma\) and a CRITICAL cell \(\gamma\) (under conditions) and modifies the cohomology generator associated to \(\gamma\) (while preserving is associated homology generator)
MW() operation exchanges a PRIMARY \(\pi\) and a SECONDARY cell \(\sigma\) (under conditions). See the introduction to HDVF for more details on this operation.

Using appropriate combinations of such operations, one can change a HDVF until corresponding homology or cohomology generators meet a given basis or delineate a hole.

Let us consider the following simple cubical complex and a perfect HDVF (top) together with the three corresponding homology generators (bottom, highlighted in pink):

A W operation between cells of Khalimsky coordinates \(\sigma = (3,2)\) (CRITICAL) and \(\tau=(5,2)\) (SECONDARY) produces the following HDVF (homology generators did not change; note that cohomology generators are modified):

Then, a MW operation between cells of Khalimsky coordinates \(\sigma' = (4,1)\) (PRIMARY) and \(\tau=(3,2)\) (SECONDARY) produces the following HDVF where homology generators become "minimal":

Perfect HDVFs provide various topological results:

Betti numbers: are well known topological descriptors (providing the number of holes in each dimension). For a perfect HDVF, Betti numbers equal the number of critical cells (that can be obtained through psc_flags() with the CRITICALPSC_flag as argument).
Homology/cohomology generators are actually algebraic objects, namely chains. Methods homology_chain() and cohomology_chain() return the homology and cohomology generator chain associated to a given critical cell. VTK export functions output all the cells of such chains with non zero coefficients. Figures here above illustrate such homology and co-homology generators.
Homology/cohomology annotation: use get_annotation() or get_coannotation() to get the annotation/co-annotation of a cycle/co-cycle in the homology/cohomology basis (as a chain of critical cells).
Homology/cohomology comparison: use are_same_cycles() or are_same_cocycles() to check if two cycles (or co-cycles) belong to the same homology / cohomology class.

Figure 94.1 Illustration of cycles annotation and comparison. Left: Generator \(g(\sigma)\) associated to the critical cell \(\sigma\); Middle: A cycle \(\alpha\) ; Right: A second cycle \(\beta\).

For \(\mathbb Z_2\) homology, the annotation of \(\alpha\) is \(\sigma\) ( \(\alpha\) equals \(g(\sigma)\) up to a boundary), while the annotation of \(\beta\) is \(\sigma+\tau+\gamma\) ( \(\beta\) equals \(g(\sigma)+g(\tau)+g(\gamma)\) up to a boundary).

Therefore, are_same_cycles will return true for \(\alpha\) and \(g(\sigma)\), but false for \(\beta\) and \(g(\sigma)\).

Is model of: HDVF

Template Parameters

ChainComplex a model of the AbstractChainComplex concept, providing the type of abstract chain complex used.

Public Types
typedef ChainComplex	Chain_complex
	Chain complex type.

typedef Chain_complex::Coefficient_ring	Coefficient_ring
	Type of coefficients used to compute homology.

typedef Hdvf_core< Chain_complex, OSM::Sparse_chain, OSM::Sparse_matrix >	Base
	Type of parent Hdvf_core class.

using	Column_chain = Base::Column_chain

using	Row_chain = Base::Row_chain

using	Column_matrix = Base::Column_matrix

using	Row_matrix = Base::Row_matrix

Public Types inherited from CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex, OSM::Sparse_chain, OSM::Sparse_matrix >
typedef ChainComplex::Coefficient_ring	Coefficient_ring
	Type of coefficients used to compute homology.

typedef OSM::Sparse_chain< Coefficient_ring, CGAL::OSM::COLUMN >	Column_chain
	Type of column-major chains.

typedef OSM::Sparse_chain< Coefficient_ring, CGAL::OSM::ROW >	Row_chain
	Type of row-major chains.

typedef OSM::Sparse_matrix< Coefficient_ring, CGAL::OSM::COLUMN >	Column_matrix
	Type of column-major sparse matrices.

typedef OSM::Sparse_matrix< Coefficient_ring, CGAL::OSM::ROW >	Row_matrix
	Type of row-major sparse matrices.

Public Member Functions
	Hdvf (const Chain_complex &K, int hdvf_opt=OPT_FULL)
	Default constructor.

	Hdvf (const Hdvf &hdvf)

	~Hdvf ()

Cell_pair	find_pair_M (int q, bool &found) const
	Finds a valid Cell_pair of dimension q for M.

Cell_pair	find_pair_M (int q, bool &found, size_t tau) const
	Finds a valid Cell_pair of dimension q for M cointaining `tau`.

std::vector< Cell_pair >	find_pairs_M (int q, bool &found) const
	Finds all valid Cell_pair of dimension q for M.

std::vector< Cell_pair >	find_pairs_M (int q, bool &found, size_t tau) const
	Finds all valid Cell_pair of dimension q for M cointaining `tau`.

Cell_pair	find_pair_W (int q, bool &found) const
	Finds a valid Cell_pair of dimension q for W.

Cell_pair	find_pair_W (int q, bool &found, size_t tau) const
	Finds a valid Cell_pair of dimension q for W cointaining `tau`.

std::vector< Cell_pair >	find_pairs_W (int q, bool &found) const
	Finds all valid Cell_pair of dimension q for W.

std::vector< Cell_pair >	find_pairs_W (int q, bool &found, size_t tau) const
	Finds all valid Cell_pair of dimension q for W cointaining `tau`.

Cell_pair	find_pair_MW (int q, bool &found) const
	Finds a valid Cell_pair of dimension q for MW.

Cell_pair	find_pair_MW (int q, bool &found, size_t tau) const
	Finds a valid Cell_pair of dimension q for MW cointaining `tau`.

std::vector< Cell_pair >	find_pairs_MW (int q, bool &found) const
	Finds all valid Cell_pair of dimension q for MW.

std::vector< Cell_pair >	find_pairs_MW (int q, bool &found, size_t tau) const
	Finds all valid Cell_pair of dimension q for W cointaining `tau`.

void	R (size_t pi, size_t sigma, int q)
	R operation (cancels a A operation).

void	M (size_t pi, size_t gamma, int q)
	M operation.

void	W (size_t sigma, size_t gamma, int q)
	W operation.

void	MW (size_t pi, size_t sigma, int q)
	MW operation.

Column_chain	get_annotation (Column_chain chain, int dim) const
	Gets the annotation of a cycle in the homology basis.

Row_chain	get_coannotation (Row_chain chain, int dim) const
	Gets the co-annotation of a co-cycle in the cohomology basis.

bool	are_same_cycles (Column_chain chain1, Column_chain chain2, int dim)
	Checks if two cycles belong to the same homology class.

bool	are_same_cocycles (Row_chain chain1, Row_chain chain2, int dim)
	Checks if two co-cycles belong to the same cohomology class.

Public Member Functions inherited from CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex, OSM::Sparse_chain, OSM::Sparse_matrix >
	Hdvf_core (const ChainComplex &K, int hdvf_opt=OPT_FULL)
	Default constructor.

	Hdvf_core (const Hdvf_core &hdvf)

	~Hdvf_core ()

virtual Cell_pair	find_pair_A (int q, bool &found) const
	Finds a valid Cell_pair of dimension q / q+1 for A.

virtual Cell_pair	find_pair_A (int q, bool &found, size_t gamma) const
	Finds a valid Cell_pair for A containing `gamma` (a cell of dimension `q`)

virtual std::vector< Cell_pair >	find_pairs_A (int q, bool &found) const
	Finds all valid Cell_pair of dimension q / q+1 for A.

virtual std::vector< Cell_pair >	find_pairs_A (int q, bool &found, size_t gamma) const
	Finds all valid Cell_pair for A containing `gamma` (a cell of dimension `q`)

void	A (size_t gamma1, size_t gamma2, int q)
	A operation: pairs critical cells.

std::vector< Cell_pair >	compute_perfect_hdvf (bool verbose=false)
	Computes a perfect HDVF.

std::vector< Cell_pair >	compute_rand_perfect_hdvf (bool verbose=false)
	Computes a random perfect HDVF.

bool	is_perfect_hdvf ()
	Tests if a HDVF is perfect.

virtual std::vector< std::vector< size_t > >	psc_flags (PSC_flag flag) const
	Gets cells with a given `PSC_flag` in any dimension.

virtual std::vector< size_t >	psc_flags (PSC_flag flag, int q) const
	Gets cells with a given `PSC_flag` in dimension `q`.

PSC_flag	psc_flag (int q, size_t tau) const
	Gets the PSC_flag of the cell `tau` in dimension `q`.

int	hdvf_opts () const
	Gets HDVF computation option.

const Row_matrix &	matrix_f (int q) const
	Gets the row-major matrix of \(f\) (from the reduction associated to the HDVF).

const Column_matrix &	matrix_g (int q) const
	Gets the column-major matrix of \(g\) (from the reduction associated to the HDVF).

const Column_matrix &	matrix_h (int q) const
	Gets the column-major matrix of \(h\) (from the reduction associated to the HDVF).

const Column_matrix &	matrix_dd (int q) const
	Gets the column-major matrix of \(\partial'\), reduced boundary operator (from the reduction associated to the HDVF).

std::ostream &	write_matrices (std::ostream &out=std::cout) const
	Prints the matrices of the reduction.

std::ostream &	write_reduction (std::ostream &out=std::cout) const
	Writes the homology and cohomology reduction information.

virtual std::vector< std::vector< int > >	psc_labels () const
	Exports primary/secondary/critical labels (in particular for vtk export)

virtual Column_chain	homology_chain (size_t cell_index, int q) const
	Gets homology generators associated to `cell` (critical cell) of dimension `q` (used by vtk export).

virtual Column_chain	cohomology_chain (size_t cell_index, int dim) const
	Gets cohomology generators associated to `cell_index` (critical cell) of dimension `q` (used by vtk export).

std::ostream &	write_hdvf_reduction (std::ostream &out)
	Writes a HDVF together with the associated reduction (f, g, h, d matrices)

void	write_hdvf_reduction (std::string filename)
	Writes a HDVF together with the associated reduction to a file (f, g, h, d matrices).

std::istream &	read_hdvf_reduction (std::istream &in_stream)
	Loads a HDVF together with the associated reduction (f, g, h, d matrices)

void	read_hdvf_reduction (std::string filename)
	Loads a HDVF together with the associated reduction from a file (f, g, h, d matrices)

bool	compare (const Hdvf_core &other, bool full_compare=false)
	Compares the HDVF with another HDVF over the same underlying complex.

bool	operator== (const Hdvf_core &other)
	Comparison operator.

Additional Inherited Members
Protected Member Functions inherited from CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex, OSM::Sparse_chain, OSM::Sparse_matrix >
OSM::Sparse_chain< Coefficient_ring, StorageFormat >	projection (const OSM::Sparse_chain< Coefficient_ring, StorageFormat > &chain, PSC_flag flag, int q) const

void	progress_bar (size_t i, size_t n)

Protected Attributes inherited from CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex, OSM::Sparse_chain, OSM::Sparse_matrix >
std::vector< std::vector< PSC_flag > >	_flag

std::vector< size_t >	_nb_P

std::vector< size_t >	_nb_S

std::vector< size_t >	_nb_C

std::vector< Row_matrix >	_F_row

std::vector< Column_matrix >	_G_col

std::vector< Column_matrix >	_H_col

std::vector< Column_matrix >	_DD_col

const ChainComplex &	_K

int	_hdvf_opt

Constructor & Destructor Documentation

◆ Hdvf()

template<typename ChainComplex >

CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::Hdvf	(	const Chain_complex &	K,
		int	hdvf_opt = `OPT_FULL`
	)

Default constructor.

Builds an" empty" HDVF associated to K (with all cells critical). By default, the HDVF option is set to OPT_FULL (full reduction computed).

Parameters

K	A chain complex (a model of `AbstractChainComplex`)
hdvf_opt	Option for HDVF computation (`OPT_BND`, `OPT_F`, `OPT_G` or `OPT_FULL`)

Member Function Documentation

◆ are_same_cocycles()

template<typename ChainComplex >

bool CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::are_same_cocycles	(	Row_chain	chain1,
		Row_chain	chain2,
		int	dim
	)

Checks if two co-cycles belong to the same cohomology class.

Warning: Will raise an error is chains provided are not co-cycles.; The HDVF must be perfect.

Parameters

chain1	First co-cycle.
chain2	Second co-cycle.
dim	Dimension of both co-cycles.

◆ are_same_cycles()

template<typename ChainComplex >

bool CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::are_same_cycles	(	Column_chain	chain1,
		Column_chain	chain2,
		int	dim
	)

Checks if two cycles belong to the same homology class.

Warning: Will raise an error is chains provided are not cycles.; The HDVF must be perfect.

Parameters

chain1	First cycle.
chain2	Second cycle.
dim	Dimension of both cycles.

◆ find_pair_M() [1/2]

template<typename ChainComplex >

Cell_pair CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::find_pair_M	(	int	q,
		bool &	found
	)		const

Finds a valid Cell_pair of dimension q for M.

The function searches a pair of cells \((\pi, \gamma)\) with \(\pi\) PRIMARY and \(\gamma\) CRITICAL, valid for M (ie. such that \(\langle f(\pi), \gamma \rangle\) invertible). It returns the first valid pair found by iterators.

Parameters

q	Dimension of the pair searched.
found	Reference to a Boolean variable. The method sets `found` to `true` if a valid pair is found, `false` otherwise.

◆ find_pair_M() [2/2]

template<typename ChainComplex >

Cell_pair CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::find_pair_M	(	int	q,
		bool &	found,
		size_t	tau
	)		const

Finds a valid Cell_pair of dimension q for M cointaining tau.

The function searches a pair of cells \((\pi, \gamma)\) with \(\pi\) PRIMARY and \(\gamma\) CRITICAL (one of them is tau), valid for M (ie. such that \(\langle f(\pi), \gamma \rangle\) invertible). It returns the first valid pair found by iterators.

Parameters

q	Dimension of the pair searched.
found	Reference to a Boolean variable. The method sets `found` to `true` if a valid pair is found, `false` otherwise.
tau	Cell of dimension `q` to pair.

◆ find_pair_MW() [1/2]

template<typename ChainComplex >

Cell_pair CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::find_pair_MW	(	int	q,
		bool &	found
	)		const

Finds a valid Cell_pair of dimension q for MW.

The function searches a pair of cells \((\pi, \sigma)\) with \(\pi\) PRIMARY and \(\sigma\) SECONDARY, valid for MW (ie. such that \(\langle h_{q-1}\partial_q(\pi), \sigma \rangle\) invertible and \(\langle \partial_{q+1} h_q(\sigma), \pi \rangle\) invertible). It returns the first valid pair found by iterators.

Parameters

q	Dimension of the pair searched.
found	Reference to a Boolean variable. The method sets `found` to `true` if a valid pair is found, `false` otherwise.

◆ find_pair_MW() [2/2]

template<typename ChainComplex >

Cell_pair CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::find_pair_MW	(	int	q,
		bool &	found,
		size_t	tau
	)		const

Finds a valid Cell_pair of dimension q for MW cointaining tau.

The function searches a pair of cells \((\pi, \sigma)\) with \(\pi\) PRIMARY and \(\sigma\) SECONDARY (one of them is tau), valid for MW (ie. such that \(\langle h_{q-1}\partial_q(\pi), \sigma \rangle\) invertible and \(\langle \partial_{q+1} h_q(\sigma), \pi \rangle\) invertible). It returns the first valid pair found by iterators.

Parameters

q	Dimension of the pair searched.
found	Reference to a Boolean variable. The method sets `found` to `true` if a valid pair is found, `false` otherwise.
tau	Cell of dimension `q` to pair.

◆ find_pair_W() [1/2]

template<typename ChainComplex >

Cell_pair CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::find_pair_W	(	int	q,
		bool &	found
	)		const

Finds a valid Cell_pair of dimension q for W.

The function searches a pair of cells \((\sigma, \gamma)\) with \(\sigma\) SECONDARY and \(\gamma\) CRITICAL, valid for W (ie. such that \(\langle g(\gamma), \sigma \rangle\) invertible). It returns the first valid pair found by iterators.

Parameters

q	Dimension of the pair searched.
found	Reference to a Boolean variable. The method sets `found` to `true` if a valid pair is found, `false` otherwise.

◆ find_pair_W() [2/2]

template<typename ChainComplex >

Cell_pair CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::find_pair_W	(	int	q,
		bool &	found,
		size_t	tau
	)		const

Finds a valid Cell_pair of dimension q for W cointaining tau.

The function searches a pair of cells \((\sigma, \gamma)\) with \(\sigma\) SECONDARY and \(\gamma\) CRITICAL (one of them is tau), valid for W (ie. such that \(\langle g(\gamma), \sigma \rangle\) invertible). It returns the first valid pair found by iterators.

Parameters

q	Dimension of the pair searched.
found	Reference to a Boolean variable. The method sets `found` to `true` if a valid pair is found, `false` otherwise.
tau	Cell of dimension `q` to pair.

◆ find_pairs_M() [1/2]

template<typename ChainComplex >

std::vector< Cell_pair > CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::find_pairs_M	(	int	q,
		bool &	found
	)		const

Finds all valid Cell_pair of dimension q for M.

The function searches all pairs of cells \((\pi, \gamma)\) with \(\pi\) PRIMARY and \(\gamma\) CRITICAL, valid for M (ie. such that \(\langle f(\pi), \gamma \rangle\) invertible). It returns a vector of such pairs.

Parameters

q	Dimension of the pair searched.
found	Reference to a Boolean variable. The method sets `found` to `true` if a valid pair is found, `false` otherwise.

◆ find_pairs_M() [2/2]

template<typename ChainComplex >

std::vector< Cell_pair > CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::find_pairs_M	(	int	q,
		bool &	found,
		size_t	tau
	)		const

Finds all valid Cell_pair of dimension q for M cointaining tau.

The function searches all pairs of cells \((\pi, \gamma)\) with \(\pi\) PRIMARY and \(\gamma\) CRITICAL (one of them is tau), valid for M (ie. such that \(\langle f(\pi), \gamma \rangle\) invertible). It returns a vector of such pairs.

Parameters

q	Dimension of the pair searched.
found	Reference to a Boolean variable. The method sets `found` to `true` if a valid pair is found, `false` otherwise.
tau	Cell of dimension `q` to pair.

◆ find_pairs_MW() [1/2]

template<typename ChainComplex >

std::vector< Cell_pair > CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::find_pairs_MW	(	int	q,
		bool &	found
	)		const

Finds all valid Cell_pair of dimension q for MW.

The function searches all pairs of cells \((\pi, \sigma)\) with \(\pi\) PRIMARY and \(\sigma\) SECONDARY, valid for MW (ie. such that \(\langle h_{q-1}\partial_q(\pi), \sigma \rangle\) invertible and \(\langle \partial_{q+1} h_q(\sigma), \pi \rangle\) invertible). It returns a vector of such pairs.

Parameters

q	Dimension of the pair searched.
found	Reference to a Boolean variable. The method sets `found` to `true` if a valid pair is found, `false` otherwise.

◆ find_pairs_MW() [2/2]

template<typename ChainComplex >

std::vector< Cell_pair > CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::find_pairs_MW	(	int	q,
		bool &	found,
		size_t	tau
	)		const

Finds all valid Cell_pair of dimension q for W cointaining tau.

The function searches all pairs of cells \((\pi, \sigma)\) with \(\pi\) PRIMARY and \(\sigma\) SECONDARY (one of them is tau), valid for MW (ie. such that \(\langle h_{q-1}\partial_q(\pi), \sigma \rangle\) invertible and \(\langle \partial_{q+1} h_q(\sigma), \pi \rangle\) invertible). It returns a vector of such pairs.

Parameters

q	Dimension of the pair searched.
found	Reference to a Boolean variable. The method sets `found` to `true` if a valid pair is found, `false` otherwise.
tau	Cell of dimension `q` to pair.

◆ find_pairs_W() [1/2]

template<typename ChainComplex >

std::vector< Cell_pair > CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::find_pairs_W	(	int	q,
		bool &	found
	)		const

Finds all valid Cell_pair of dimension q for W.

The function searches all pairs of cells \((\sigma, \gamma)\) with \(\sigma\) SECONDARY and \(\gamma\) CRITICAL, valid for W (ie. such that \(\langle g(\gamma), \sigma \rangle\) invertible). It returns a vector of such pairs.

Parameters

q	Dimension of the pair searched.
found	Reference to a Boolean variable. The method sets `found` to `true` if a valid pair is found, `false` otherwise.

◆ find_pairs_W() [2/2]

template<typename ChainComplex >

std::vector< Cell_pair > CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::find_pairs_W	(	int	q,
		bool &	found,
		size_t	tau
	)		const

Finds all valid Cell_pair of dimension q for W cointaining tau.

The function searches all pairs of cells \((\sigma, \gamma)\) with \(\sigma\) SECONDARY and \(\gamma\) CRITICAL (one of them is tau), valid for W (ie. such that \(\langle g(\gamma), \sigma \rangle\) invertible). It returns the first valid pair found by iterators. It returns a vector of such pairs.

Parameters

q	Dimension of the pair searched.
found	Reference to a Boolean variable. The method sets `found` to `true` if a valid pair is found, `false` otherwise.
tau	Cell of dimension `q` to pair.

◆ get_annotation()

template<typename ChainComplex >

Column_chain CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::get_annotation	(	Column_chain	chain,
		int	dim
	)		const

Gets the annotation of a cycle in the homology basis.

The method returns the image of a cycle by the morphism \(f\), that is, a linear combination of critical cells (corresponding to the decomposition of the cycle in the homology basis). If the annotation has a single non zero coefficient for a given critical cell \(\sigma\), then the cycle belongs to the class of the homology generator \(g(\sigma)\).

Warning: Will raise an error is the chain provided is not a cycle.; The HDVF must be perfect.

Parameters

chain	The cycle to annotate in the homology basis.
dim	Dimension of the cycle.

◆ get_coannotation()

template<typename ChainComplex >

Row_chain CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::get_coannotation	(	Row_chain	chain,
		int	dim
	)		const

Gets the co-annotation of a co-cycle in the cohomology basis.

The method returns the image of a co-cycle by the morphism \(g^*\), that is, a linear combination of critical cells (corresponding to the decomposition of the co-cycle in the cohomology basis). If the annotation has a single non zero coefficient for a given critical cell \(\sigma\), then the co-cycle belongs to the class of the cohomology generator \(f^*(\sigma)\).

Warning: Will raise an error is the chain provided is not a co-cycle.; The HDVF must be perfect.

Parameters

chain	The co-cycle to annotate in the homology basis.
dim	Dimension of the co-cycle.

◆ M()

template<typename ChainComplex >

void CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::M	(	size_t	pi,
		size_t	gamma,
		int	q
	)

M operation.

A pair of cells \((\pi, \gamma)\) of dimension q, with \(\pi\) PRIMARY and \(\gamma\) CRITICAL, is valid for M if \(\langle f(\pi), \gamma \rangle\) is invertible. After the M operation, \(\pi\) becomes CRITICAL and \(\gamma\) become PRIMARY. The M method updates the reduction accordingly (in time \(\mathcal O(n^2)\)).

Parameters

pi	First cell of the pair (dimension `q`)
gamma	Second cell of the pair (dimension `q`)
q	Dimension of the pair

◆ MW()

template<typename ChainComplex >

void CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::MW	(	size_t	pi,
		size_t	sigma,
		int	q
	)

MW operation.

A pair of cells \((\pi, \sigma)\) of dimension q, with \(\pi\) PRIMARY and \(\sigma\) SECONDARY, is valid for MW if \(\langle h_{q-1}\partial_q(\pi), \sigma \rangle\) is invertible and \(\langle \partial_{q+1} h_q(\sigma), \pi \rangle\) is invertible. After the MW operation, \(\pi\) becomes SECONDARY and \(\sigma\) become PRIMARY. The MW method updates the reduction accordingly (in time \(\mathcal O(n^2)\)).

Parameters

pi	First cell of the pair (dimension `q`)
sigma	Second cell of the pair (dimension `q`)
q	Dimension of the pair

◆ R()

template<typename ChainComplex >

void CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::R	(	size_t	pi,
		size_t	sigma,
		int	q
	)

R operation (cancels a A operation).

A pair of cells \((\pi, \sigma)\) of respective dimension q and q+1, with \(\pi\) PRIMARY and \(\sigma\) SECONDARY, is valid for R if \(\langle h(\pi), \sigma \rangle\) is invertible. After the R operation, \(\pi\) and \(\sigma\) become CRITICAL. The R method updates the reduction accordingly (in time \(\mathcal O(n^2)\)).

Parameters

pi	First cell of the pair (dimension `q`)
sigma	Second cell of the pair (dimension `q+1`)
q	Dimension of the pair

◆ W()

template<typename ChainComplex >

void CGAL::Homological_discrete_vector_field::Hdvf< ChainComplex >::W	(	size_t	sigma,
		size_t	gamma,
		int	q
	)

W operation.

A pair of cells \((\sigma, \gamma)\) of dimension q, with \(\sigma\) SECONDARY and \(\gamma\) CRITICAL, is valid for W if \(\langle g(\gamma), \sigma \rangle\) is invertible. After the W operation, \(\sigma\) becomes CRITICAL and \(\gamma\) become SECONDARY. The W method updates the reduction accordingly (in time \(\mathcal O(n^2)\)).

Parameters

sigma	First cell of the pair (dimension `q`)
gamma	Second cell of the pair (dimension `q`)
q	Dimension of the pair

Inherits from

Definition

Public Types

Public Member Functions

Additional Inherited Members

Constructor & Destructor Documentation

◆ Hdvf()

Member Function Documentation

◆ are_same_cocycles()

◆ are_same_cycles()

◆ find_pair_M() [1/2]

◆ find_pair_M() [2/2]

◆ find_pair_MW() [1/2]

◆ find_pair_MW() [2/2]

◆ find_pair_W() [1/2]

◆ find_pair_W() [2/2]

◆ find_pairs_M() [1/2]

◆ find_pairs_M() [2/2]

◆ find_pairs_MW() [1/2]

◆ find_pairs_MW() [2/2]

◆ find_pairs_W() [1/2]

◆ find_pairs_W() [2/2]

◆ get_annotation()

◆ get_coannotation()

◆ M()

◆ MW()

◆ R()

◆ W()