Aim: Represents an nD lattice polytope, i.e. a convex polyhedron bounded with vertices with integer coordinates, as a set of inequalities. Otherwise said, it is a H-representation of a polytope (as an intersection of half-spaces). A limitation is that we model only bounded polytopes, i.e. polytopes that can be included in a finite bounding box. More...

#include <DGtal/geometry/volumes/BoundedLatticePolytope.h>

Data Structures
struct	LeftStrictUnitCell

struct	LeftStrictUnitSegment

struct	RightStrictUnitCell

struct	RightStrictUnitSegment

struct	UnitCell

struct	UnitSegment

Public Types
typedef BoundedLatticePolytope< TSpace >	Self

typedef TSpace	Space

typedef Space::Integer	Integer

typedef Space::Point	Point

typedef Space::Vector	Vector

typedef std::vector< Vector >	InequalityMatrix

typedef std::vector< Integer >	InequalityVector

typedef HyperRectDomain< Space >	Domain

typedef ClosedIntegerHalfPlane< Space >	HalfSpace

typedef DGtal::BigInteger	BigInteger

Public Member Functions
Standard services (construction, initialization, assignment, interior, closure)
	~BoundedLatticePolytope ()=default

	BoundedLatticePolytope ()=default

	BoundedLatticePolytope (const Self &other)=default

	BoundedLatticePolytope (std::initializer_list< Point > l)

template<typename PointIterator >
	BoundedLatticePolytope (PointIterator itB, PointIterator itE)

template<typename HalfSpaceIterator >
	BoundedLatticePolytope (const Domain &domain, HalfSpaceIterator itB, HalfSpaceIterator itE, bool valid_edge_constraints=false, bool check_duplicate_constraints=false)

template<typename HalfSpaceIterator >
void	init (const Domain &domain, HalfSpaceIterator itB, HalfSpaceIterator itE, bool valid_edge_constraints=false, bool check_duplicate_constraints=false)

template<typename PointIterator >
bool	init (PointIterator itB, PointIterator itE)

Self &	operator= (const Self &other)=default

void	clear ()
	Clears the polytope.

BoundedLatticePolytope	interiorPolytope () const

BoundedLatticePolytope	closurePolytope () const

Accessor services
const Domain &	getDomain () const

unsigned int	nbHalfSpaces () const

const Vector &	getA (unsigned int i) const

Integer	getB (unsigned int i) const

bool	isLarge (unsigned int i) const

const InequalityMatrix &	getA () const

const InequalityVector &	getB () const

const std::vector< bool > &	getI () const

bool	canBeSummed () const

Check point services (is inside test)
bool	isInside (const Point &p) const

bool	isDomainPointInside (const Point &p) const

bool	isInterior (const Point &p) const

bool	isBoundary (const Point &p) const

Global modification services (cut, swap, Minkowski sum)
unsigned int	cut (Dimension k, bool pos, Integer b, bool large=true)

unsigned int	cut (const Vector &a, Integer b, bool large=true, bool valid_edge_constraint=false)

unsigned int	cut (const HalfSpace &hs, bool large=true, bool valid_edge_constraint=false)

void	swap (BoundedLatticePolytope &other)

Self &	operator*= (Integer t)

Self &	operator+= (UnitSegment s)

Self &	operator+= (UnitCell c)

Self &	operator+= (RightStrictUnitSegment s)

Self &	operator+= (RightStrictUnitCell c)

Self &	operator+= (LeftStrictUnitSegment s)

Self &	operator+= (LeftStrictUnitCell c)

Enumeration services (counting, get points in polytope)
Integer	count () const

Integer	countInterior () const

Integer	countBoundary () const

Integer	countWithin (Point low, Point hi) const

Integer	countUpTo (Integer max) const

void	getPoints (std::vector< Point > &pts) const

void	getKPoints (std::vector< Point > &pts, const Point &alpha_shift) const

void	getInteriorPoints (std::vector< Point > &pts) const

void	getBoundaryPoints (std::vector< Point > &pts) const

template<typename PointSet >
void	insertPoints (PointSet &pts_set) const

template<typename PointSet >
void	insertKPoints (PointSet &pts_set, const Point &alpha_shift) const

Enumeration services by scanning (counting, get points in polytope)
Integer	countByScanning () const

Integer	countInteriorByScanning () const

Integer	countBoundaryByScanning () const

Integer	countWithinByScanning (Point low, Point hi) const

Integer	countUpToByScanning (Integer max) const

void	getPointsByScanning (std::vector< Point > &pts) const

void	getInteriorPointsByScanning (std::vector< Point > &pts) const

void	getBoundaryPointsByScanning (std::vector< Point > &pts) const

template<typename PointSet >
void	insertPointsByScanning (PointSet &pts_set) const

Interface services
void	selfDisplay (std::ostream &out) const

bool	isValid () const

std::string	className () const

Static Public Attributes
static const Dimension	dimension = Space::dimension

Protected Attributes
InequalityMatrix	A
	The matrix A in the polytope representation \( Ax \le B \).

InequalityVector	B
	The vector B in the polytope representation \( Ax \le B \).

Domain	D
	Tight bounded box.

std::vector< bool >	I
	Are inequalities large ?

bool	myValidEdgeConstraints
	Indicates if Minkowski sums with segments will be valid.

Private Member Functions
	BOOST_CONCEPT_ASSERT ((concepts::CSpace< TSpace >))

bool	internalInitFromTriangle3D (Point a, Point b, Point c)

bool	internalInitFromSegment3D (Point a, Point b)

bool	internalInitFromSegment2D (Point a, Point b)

Detailed Description

template<typename TSpace>
class DGtal::BoundedLatticePolytope< TSpace >

Aim: Represents an nD lattice polytope, i.e. a convex polyhedron bounded with vertices with integer coordinates, as a set of inequalities. Otherwise said, it is a H-representation of a polytope (as an intersection of half-spaces). A limitation is that we model only bounded polytopes, i.e. polytopes that can be included in a finite bounding box.

Description of template class 'BoundedLatticePolytope'

It is a model of boost::CopyConstructible, boost::DefaultConstructible, boost::Assignable.

Template Parameters

TSpace an arbitrary model of CSpace.

Definition at line 73 of file BoundedLatticePolytope.h.

Member Typedef Documentation

◆ BigInteger

template<typename TSpace >

typedef DGtal::BigInteger DGtal::BoundedLatticePolytope< TSpace >::BigInteger

Definition at line 88 of file BoundedLatticePolytope.h.

◆ Domain

template<typename TSpace >

typedef HyperRectDomain< Space > DGtal::BoundedLatticePolytope< TSpace >::Domain

Definition at line 85 of file BoundedLatticePolytope.h.

◆ HalfSpace

template<typename TSpace >

typedef ClosedIntegerHalfPlane< Space > DGtal::BoundedLatticePolytope< TSpace >::HalfSpace

Definition at line 86 of file BoundedLatticePolytope.h.

◆ InequalityMatrix

template<typename TSpace >

typedef std::vector<Vector> DGtal::BoundedLatticePolytope< TSpace >::InequalityMatrix

Definition at line 83 of file BoundedLatticePolytope.h.

◆ InequalityVector

template<typename TSpace >

typedef std::vector<Integer> DGtal::BoundedLatticePolytope< TSpace >::InequalityVector

Definition at line 84 of file BoundedLatticePolytope.h.

◆ Integer

template<typename TSpace >

typedef Space::Integer DGtal::BoundedLatticePolytope< TSpace >::Integer

Definition at line 80 of file BoundedLatticePolytope.h.

◆ Point

template<typename TSpace >

typedef Space::Point DGtal::BoundedLatticePolytope< TSpace >::Point

Definition at line 81 of file BoundedLatticePolytope.h.

◆ Self

template<typename TSpace >

typedef BoundedLatticePolytope<TSpace> DGtal::BoundedLatticePolytope< TSpace >::Self

Definition at line 78 of file BoundedLatticePolytope.h.

◆ Space

template<typename TSpace >

typedef TSpace DGtal::BoundedLatticePolytope< TSpace >::Space

Definition at line 79 of file BoundedLatticePolytope.h.

◆ Vector

template<typename TSpace >

typedef Space::Vector DGtal::BoundedLatticePolytope< TSpace >::Vector

Definition at line 82 of file BoundedLatticePolytope.h.

Constructor & Destructor Documentation

◆ ~BoundedLatticePolytope()

template<typename TSpace >

DGtal::BoundedLatticePolytope< TSpace >::~BoundedLatticePolytope ( )

default

Destructor.

◆ BoundedLatticePolytope() [1/5]

template<typename TSpace >

DGtal::BoundedLatticePolytope< TSpace >::BoundedLatticePolytope ( )

default

Constructor.

◆ BoundedLatticePolytope() [2/5]

template<typename TSpace >

DGtal::BoundedLatticePolytope< TSpace >::BoundedLatticePolytope ( const Self & other )

default

Copy constructor.

Parameters

other the object to clone.

◆ BoundedLatticePolytope() [3/5]

template<typename TSpace >

DGtal::BoundedLatticePolytope< TSpace >::BoundedLatticePolytope ( std::initializer_list< Point > l )

Constructs the polytope from a simplex given as an initializer_list.

Parameters

l	any list of no more than d+1 points in general positions.

◆ BoundedLatticePolytope() [4/5]

template<typename TSpace >

template<typename PointIterator >

DGtal::BoundedLatticePolytope< TSpace >::BoundedLatticePolytope	(	PointIterator	itB,
		PointIterator	itE )

Constructs the polytope from a simplex given as a range [itB,itE) of lattice points.

Template Parameters

PointIterator any model of forward iterator on Point.

Parameters

itB	the start of the range of no more than n+1 points defining the simplex.
itE	past the end the range of no more than n+1 points defining the simplex.

◆ BoundedLatticePolytope() [5/5]

template<typename TSpace >

template<typename HalfSpaceIterator >

DGtal::BoundedLatticePolytope< TSpace >::BoundedLatticePolytope	(	const Domain &	domain,
		HalfSpaceIterator	itB,
		HalfSpaceIterator	itE,
		bool	valid_edge_constraints = false,
		bool	check_duplicate_constraints = false )

Constructs a polytope from a domain and a collection of half-spaces.

Template Parameters

HalfSpaceIterator any model of forward iterator on HalfSpace.

Parameters

domain	a bounded lattice domain.
itB	the start of the range of half-spaces.
itE	past the end of the range of half-spaces.
valid_edge_constraints	when 'true', tells that there are half-spaces that represents th constraints on edges (n-2 cells) lying between two faces (n-1 cells) pointing to different orthants.
check_duplicate_constraints	when 'true', the initialization checks if the given range of half-spaces contains axis-aligned half-space constraints already defined by the domain and if so it merges the duplicated constraints, otherwise it accepts and stores the constraints as is.

Member Function Documentation

◆ BOOST_CONCEPT_ASSERT()

template<typename TSpace >

DGtal::BoundedLatticePolytope< TSpace >::BOOST_CONCEPT_ASSERT ( (concepts::CSpace< TSpace >) )

private

◆ canBeSummed()

template<typename TSpace >

bool DGtal::BoundedLatticePolytope< TSpace >::canBeSummed ( ) const

Returns: 'true' if we can perform exact Minkowksi sums on this polytope. This is related to the presence of valid edge constraints (n-k cells for k >= 2) in-between face constraints (n-1 cells) that change orthants.

◆ className()

template<typename TSpace >

std::string DGtal::BoundedLatticePolytope< TSpace >::className ( ) const

Returns: the class name. It is notably used for drawing this object.

◆ clear()

template<typename TSpace >

void DGtal::BoundedLatticePolytope< TSpace >::clear ( )

Clears the polytope.

◆ closurePolytope()

template<typename TSpace >

BoundedLatticePolytope DGtal::BoundedLatticePolytope< TSpace >::closurePolytope ( ) const

Returns: the closure (in the topological sense) of this polytope, by making all constraints large.

◆ count()

template<typename TSpace >

Integer DGtal::BoundedLatticePolytope< TSpace >::count ( ) const

Computes the number of integer points lying within the polytope.

Returns: the number of integer points lying within the polytope.

Note: Quite fast: obtained by line intersection, see BoundedLatticePolytopeCounter

Referenced by DGtal::EhrhartPolynomial< TSpace, TInteger >::init().

◆ countBoundary()

template<typename TSpace >

Integer DGtal::BoundedLatticePolytope< TSpace >::countBoundary ( ) const

Computes the number of integer points lying on the boundary of the polytope.

Returns: the number of integer points lying on the boundary of the polytope.

Note: Quite fast: obtained by line intersection, see BoundedLatticePolytopeCounter; count() <= countInterior() + countBoundary() with equality when the polytope is closed.

◆ countBoundaryByScanning()

template<typename TSpace >

Integer DGtal::BoundedLatticePolytope< TSpace >::countBoundaryByScanning ( ) const

Computes the number of integer points lying on the boundary of the polytope.

Returns: the number of integer points lying on the boundary of the polytope.

Note: Quite slow: obtained by checking every point of the polytope domain.; count() <= countInterior() + countBoundary() with equality when the polytope is closed.

◆ countByScanning()

template<typename TSpace >

Integer DGtal::BoundedLatticePolytope< TSpace >::countByScanning ( ) const

Computes the number of integer points lying within the polytope.

Returns: the number of integer points lying within the polytope.

Note: Quite slow: obtained by checking every point of the polytope domain.

◆ countInterior()

template<typename TSpace >

Integer DGtal::BoundedLatticePolytope< TSpace >::countInterior ( ) const

Computes the number of integer points lying within the interior of the polytope.

Returns: the number of integer points lying within the interior of the polytope.

Note: Quite fast: obtained by line intersection, see BoundedLatticePolytopeCounter; count() <= countInterior() + countBoundary() with equality when the polytope is closed.

◆ countInteriorByScanning()

template<typename TSpace >

Integer DGtal::BoundedLatticePolytope< TSpace >::countInteriorByScanning ( ) const

Computes the number of integer points lying within the interior of the polytope.

Returns: the number of integer points lying within the interior of the polytope.

Note: Quite slow: obtained by checking every point of the polytope domain.; count() <= countInterior() + countBoundary() with equality when the polytope is closed.

◆ countUpTo()

template<typename TSpace >

Integer DGtal::BoundedLatticePolytope< TSpace >::countUpTo ( Integer max ) const

Computes the number of integer points within the polytope up to some maximum number max.

Note: For instance, a d-dimensional simplex that contains no integer points in its interior contains only d+1 points. If there is more, you know that the simplex has a non empty interior.

Parameters

[in] max the maximum number of points that are counted, the method exists when this number of reached.

Returns: the number of integer points within the polytope up to .

Note: Quite fast: obtained by line intersection, see BoundedLatticePolytopeCounter

◆ countUpToByScanning()

template<typename TSpace >

Integer DGtal::BoundedLatticePolytope< TSpace >::countUpToByScanning ( Integer max ) const

Computes the number of integer points within the polytope up to some maximum number max.

Note: For instance, a d-dimensional simplex that contains no integer points in its interior contains only d+1 points. If there is more, you know that the simplex has a non empty interior.

Parameters

[in] max the maximum number of points that are counted, the method exists when this number of reached.

Returns: the number of integer points within the polytope up to .

Note: Quite slow: obtained by checking every point of the polytope domain.

◆ countWithin()

template<typename TSpace >

Integer DGtal::BoundedLatticePolytope< TSpace >::countWithin	(	Point	low,
		Point	hi ) const

Computes the number of integer points within the polytope and the domain bounded by low and hi.

Parameters

[in]	low	the lowest point of the domain.
[in]	hi	the highest point of the domain.

Returns: the number of integer points within the polytope.

Note: Quite fast: obtained by line intersection, see BoundedLatticePolytopeCounter

◆ countWithinByScanning()

template<typename TSpace >

Integer DGtal::BoundedLatticePolytope< TSpace >::countWithinByScanning	(	Point	low,
		Point	hi ) const

Computes the number of integer points within the polytope and the domain bounded by low and hi.

Parameters

[in]	low	the lowest point of the domain.
[in]	hi	the highest point of the domain.

Returns: the number of integer points within the polytope.

Note: Quite slow: obtained by checking every point of the polytope domain.

◆ cut() [1/3]

template<typename TSpace >

unsigned int DGtal::BoundedLatticePolytope< TSpace >::cut	(	const HalfSpace &	hs,
		bool	large = true,
		bool	valid_edge_constraint = false )

Cuts the lattice polytope with the given half-space constraint.

Parameters

hs	any half-space constraint.
large	tells if the inequality is large (true) or strict (false).
valid_edge_constraint	when 'true', tells that the half-spaces that represents constraints on edges (n-2 cells) lying between two faces (n-1 cells) pointing to different orthants are still valid after this operation.

Returns: the index of the constraint in the polytope.

Note: For now complexity is O(n) where n=A.rows() because it checks if a parallel closed half-space defines already a face of the polytope.

◆ cut() [2/3]

template<typename TSpace >

unsigned int DGtal::BoundedLatticePolytope< TSpace >::cut	(	const Vector &	a,
		Integer	b,
		bool	large = true,
		bool	valid_edge_constraint = false )

Cut the polytope by the given half space a.x <= b or a.x < b.

Parameters

a	any integer vector
b	any integer number
large	tells if the inequality is large (true) or strict (false).
valid_edge_constraint	when 'true', tells that the half-spaces that represents constraints on edges (n-2 cells) lying between two faces (n-1 cells) pointing to different orthants are still valid after this operation.

Returns: the index of the constraint in the polytope.

Note: For now complexity is O(n) where n=A.rows() because it checks if a parallel closed half-space defines already a face of the polytope.

◆ cut() [3/3]

template<typename TSpace >

unsigned int DGtal::BoundedLatticePolytope< TSpace >::cut	(	Dimension	k,
		bool	pos,
		Integer	b,
		bool	large = true )

Cut the polytope by the given half space a.x <= b or a.x < b where a is some axis vector.

Parameters

k	the dimension of the axis vector \( +/- e_k \)
pos	'true' is positive, 'false' is negative for the axis vector \( +/- e_k \)
b	any integer number
large	tells if the inequality is large (true) or strict (false).

Returns: the index of the constraint in the polytope.

Referenced by DGtal::detail::BoundedLatticePolytopeSpecializer< 3, TInteger >::addEdgeConstraint().

◆ getA() [1/2]

template<typename TSpace >

const InequalityMatrix & DGtal::BoundedLatticePolytope< TSpace >::getA ( ) const

Returns: the matrix A in the polytope representation \( Ax \le B \).

◆ getA() [2/2]

template<typename TSpace >

const Vector & DGtal::BoundedLatticePolytope< TSpace >::getA ( unsigned int i ) const

Parameters

i	the index of the half-space constraint between 0 and `nbHalfSpaces()` (excluded).

Returns: the normal vector of the i-th half space constraint (i.e. A in constraint Ax <= b).

◆ getB() [1/2]

template<typename TSpace >

const InequalityVector & DGtal::BoundedLatticePolytope< TSpace >::getB ( ) const

Returns: the vector B in the polytope representation \( Ax \le B \).

◆ getB() [2/2]

template<typename TSpace >

Integer DGtal::BoundedLatticePolytope< TSpace >::getB ( unsigned int i ) const

Parameters

i	the index of the half-space constraint between 0 and `nbHalfSpaces()` (excluded).

Returns: the offset of the i-th half space constraint (i.e. b in constraint Ax <= b).

◆ getBoundaryPoints()

template<typename TSpace >

void DGtal::BoundedLatticePolytope< TSpace >::getBoundaryPoints ( std::vector< Point > & pts ) const

Computes the integer points boundary to the polytope.

Parameters

[out] pts the integer points boundary to the polytope.

Note: Quite fast: obtained by line intersection, see BoundedLatticePolytopeCounter; At output, pts.size() == this->countBoundary()

◆ getBoundaryPointsByScanning()

template<typename TSpace >

void DGtal::BoundedLatticePolytope< TSpace >::getBoundaryPointsByScanning ( std::vector< Point > & pts ) const

Computes the integer points boundary to the polytope.

Parameters

[out] pts the integer points boundary to the polytope.

Note: Quite slow: obtained by checking every point of the polytope domain.; At output, pts.size() == this->countBoundary()

◆ getDomain()

template<typename TSpace >

const Domain & DGtal::BoundedLatticePolytope< TSpace >::getDomain ( ) const

Returns: the domain of the current polytope.

◆ getI()

template<typename TSpace >

const std::vector< bool > & DGtal::BoundedLatticePolytope< TSpace >::getI ( ) const

Returns: the vector I telling if inequalities are large in the polytope representation \( Ax \prec B \), with \( \prec \in \{ <, \le \} \).

◆ getInteriorPoints()

template<typename TSpace >

void DGtal::BoundedLatticePolytope< TSpace >::getInteriorPoints ( std::vector< Point > & pts ) const

Computes the integer points interior to the polytope.

Parameters

[out] pts the integer points interior to the polytope.

Note: Quite fast: obtained by line intersection, see BoundedLatticePolytopeCounter; At output, pts.size() == this->countInterior()

◆ getInteriorPointsByScanning()

template<typename TSpace >

void DGtal::BoundedLatticePolytope< TSpace >::getInteriorPointsByScanning ( std::vector< Point > & pts ) const

Computes the integer points interior to the polytope.

Parameters

[out] pts the integer points interior to the polytope.

Note: Quite slow: obtained by checking every point of the polytope domain.; At output, pts.size() == this->countInterior()

◆ getKPoints()

template<typename TSpace >

void DGtal::BoundedLatticePolytope< TSpace >::getKPoints	(	std::vector< Point > &	pts,
		const Point &	alpha_shift ) const

Computes the integer points within the polytope and converts them to cells represented with their Khalimsky coordinates.

Parameters

[out]	pts	the integer points within the polytope, with coordinates equal to `2*p - alpha_shift`, for `p` in the polytope.
[in]	alpha_shift	the translation applied to each point.

Note: Quite fast: obtained by line intersection, see BoundedLatticePolytopeCounter; At output, pts.size() == this->count(); Usefull for full convexity checks.

Referenced by SCENARIO(), and SCENARIO().

◆ getPoints()

template<typename TSpace >

void DGtal::BoundedLatticePolytope< TSpace >::getPoints ( std::vector< Point > & pts ) const

Computes the integer points within the polytope.

Parameters

[out] pts the integer points within the polytope.

Note: Quite fast: obtained by line intersection, see BoundedLatticePolytopeCounter; At output, pts.size() == this->count()

Examples: geometry/volumes/checkFullConvexityTheorems.cpp.

Referenced by checkCvxHPlusHypercubeFullConvexity(), checkFullConvexityCharacterization(), SCENARIO(), SCENARIO(), SCENARIO(), and SCENARIO().

◆ getPointsByScanning()

template<typename TSpace >

void DGtal::BoundedLatticePolytope< TSpace >::getPointsByScanning ( std::vector< Point > & pts ) const

Computes the integer points within the polytope.

Parameters

[out] pts the integer points within the polytope.

Note: Quite slow: obtained by checking every point of the polytope domain.; At output, pts.size() == this->count()

◆ init() [1/2]

template<typename TSpace >

template<typename HalfSpaceIterator >

void DGtal::BoundedLatticePolytope< TSpace >::init	(	const Domain &	domain,
		HalfSpaceIterator	itB,
		HalfSpaceIterator	itE,
		bool	valid_edge_constraints = false,
		bool	check_duplicate_constraints = false )

Initializes a polytope from a domain and a vector of half-spaces.

Template Parameters

HalfSpaceIterator any model of forward iterator on HalfSpace.

Parameters

domain	a bounded lattice domain.
itB	the start of the range of half-spaces.
itE	past the end of the range of half-spaces.
valid_edge_constraints	when 'true', tells that there are half-spaces that represents the constraints on edges (n-2 cells) lying between two faces (n-1 cells) pointing to different orthants.
check_duplicate_constraints	when 'true', the initialization checks if the given range of half-spaces contains axis-aligned half-space constraints already defined by the domain and if so it merges the duplicated constraints, otherwise it accepts and stores the constraints as is.

◆ init() [2/2]

template<typename TSpace >

template<typename PointIterator >

bool DGtal::BoundedLatticePolytope< TSpace >::init	(	PointIterator	itB,
		PointIterator	itE )

Initializes the polytope from a simplex given as a range [itB,itE) of points.

Parameters

itB	the start of the range of no more than n+1 points defining the simplex.
itE	past the end the range of no more than n+1 points defining the simplex.

Returns: 'true' if [itB,itE) was a valid simplex, otherwise return 'false' and the polytope is empty.

Note: Use DGtal SimpleMatrix::determinant which is not efficient when the dimension is high. Does not use Eigen to avoid dependency.

◆ insertKPoints()

template<typename TSpace >

template<typename PointSet >

void DGtal::BoundedLatticePolytope< TSpace >::insertKPoints	(	PointSet &	pts_set,
		const Point &	alpha_shift ) const

Computes the integer points within the polytope and converts them to cells represented with their Khalimsky coordinates.

Template Parameters

PointSet any model of set with a method insert( Point ).

Parameters

[in,out]	pts_set	the set of points where points within this polytope are inserted. Each point is inserted with coordinates equal to `2*p - alpha_shift`, for `p` in the polytope.
[in]	alpha_shift	the translation applied to each point.

Note: Quite fast: obtained by line intersection, see BoundedLatticePolytopeCounter

◆ insertPoints()

template<typename TSpace >

template<typename PointSet >

void DGtal::BoundedLatticePolytope< TSpace >::insertPoints ( PointSet & pts_set ) const

Computes the integer points within the polytope.

Template Parameters

PointSet any model of set with a method insert( Point ).

Parameters

[in,out] pts_set the set of points where points within this polytope are inserted.

Note: Quite fast: obtained by line intersection, see BoundedLatticePolytopeCounter

◆ insertPointsByScanning()

template<typename TSpace >

template<typename PointSet >

void DGtal::BoundedLatticePolytope< TSpace >::insertPointsByScanning ( PointSet & pts_set ) const

Computes the integer points within the polytope.

Template Parameters

PointSet any model of set with a method insert( Point ).

Parameters

[in,out] pts_set the set of points where points within this polytope are inserted.

Note: Quite slow: obtained by checking every point of the polytope domain.

◆ interiorPolytope()

template<typename TSpace >

BoundedLatticePolytope DGtal::BoundedLatticePolytope< TSpace >::interiorPolytope ( ) const

Returns: the interior (in the topological sense) of this polytope, by making all constraints strict.

◆ internalInitFromSegment2D()

template<typename TSpace >

bool DGtal::BoundedLatticePolytope< TSpace >::internalInitFromSegment2D	(	Point	a,
		Point	b )

private

In 2D, builds a valid lattice polytope with empty interior from 2 points.

Parameters

a	any point
b	any point

Returns: 'true'

◆ internalInitFromSegment3D()

template<typename TSpace >

bool DGtal::BoundedLatticePolytope< TSpace >::internalInitFromSegment3D	(	Point	a,
		Point	b )

private

In 3D, builds a valid lattice polytope with empty interior from 2 points.

Parameters

a	any point
b	any point

Returns: 'true'

◆ internalInitFromTriangle3D()

template<typename TSpace >

bool DGtal::BoundedLatticePolytope< TSpace >::internalInitFromTriangle3D	(	Point	a,
		Point	b,
		Point	c )

private

In 3D, builds a valid lattice polytope with empty interior from 3 non-colinear points.

Parameters

a	any point such that a, b, and c are not colinear.
b	any point such that a, b, and c are not colinear.
c	any point such that a, b, and c are not colinear.

Returns: 'true' if they were not colinear, false otherwise.

◆ isBoundary()

template<typename TSpace >

bool DGtal::BoundedLatticePolytope< TSpace >::isBoundary ( const Point & p ) const

Parameters

p	any point of the space.

Returns: 'true' if and only if p lies on the boundary of this polytope.

◆ isDomainPointInside()

template<typename TSpace >

bool DGtal::BoundedLatticePolytope< TSpace >::isDomainPointInside ( const Point & p ) const

Parameters

p	any point inside the domain of this polytope.

Returns: 'true' if and only if p is inside this polytope.

Note: Slightly faster than isInside.

◆ isInside()

template<typename TSpace >

bool DGtal::BoundedLatticePolytope< TSpace >::isInside ( const Point & p ) const

Parameters

p	any point of the space.

Returns: 'true' if and only if p is inside this polytope.

◆ isInterior()

template<typename TSpace >

bool DGtal::BoundedLatticePolytope< TSpace >::isInterior ( const Point & p ) const

Parameters

p	any point of the space.

Returns: 'true' if and only if p is strictly inside this polytope.

◆ isLarge()

template<typename TSpace >

bool DGtal::BoundedLatticePolytope< TSpace >::isLarge ( unsigned int i ) const

Parameters

i	the index of the half-space constraint between 0 and `nbHalfSpaces()` (excluded).

Returns: 'true' if the i-th half space constraint is of the form Ax <= b, 'false' if it is of the form Ax < b.

◆ isValid()

template<typename TSpace >

bool DGtal::BoundedLatticePolytope< TSpace >::isValid ( ) const

Checks the validity/consistency of the object. If the polytope has been default constructed, it is invalid.

Returns: 'true' if the object is valid, 'false' otherwise.

◆ nbHalfSpaces()

template<typename TSpace >

unsigned int DGtal::BoundedLatticePolytope< TSpace >::nbHalfSpaces ( ) const

Returns: the number of half-space constraints.

◆ operator*=()

template<typename TSpace >

Self & DGtal::BoundedLatticePolytope< TSpace >::operator*= ( Integer t )

Dilates this polytope P by t.

Parameters

t	any integer.

Returns: a reference to 'this', which is now the polytope tP.

◆ operator+=() [1/6]

template<typename TSpace >

Self & DGtal::BoundedLatticePolytope< TSpace >::operator+= ( LeftStrictUnitCell c )

Minkowski sum of this polytope with an axis-aligned strict unit cell.

Parameters

c	any strict unit cell.

Returns: a reference to 'this'.

◆ operator+=() [2/6]

template<typename TSpace >

Self & DGtal::BoundedLatticePolytope< TSpace >::operator+= ( LeftStrictUnitSegment s )

Minkowski sum of this polytope with a strict unit segment aligned with some axis.

Parameters

s	any strict unit segment.

Returns: a reference to 'this'.

◆ operator+=() [3/6]

template<typename TSpace >

Self & DGtal::BoundedLatticePolytope< TSpace >::operator+= ( RightStrictUnitCell c )

Minkowski sum of this polytope with an axis-aligned strict unit cell.

Parameters

c	any strict unit cell.

Returns: a reference to 'this'.

◆ operator+=() [4/6]

template<typename TSpace >

Self & DGtal::BoundedLatticePolytope< TSpace >::operator+= ( RightStrictUnitSegment s )

Minkowski sum of this polytope with a strict unit segment aligned with some axis.

Parameters

s	any strict unit segment.

Returns: a reference to 'this'.

◆ operator+=() [5/6]

template<typename TSpace >

Self & DGtal::BoundedLatticePolytope< TSpace >::operator+= ( UnitCell c )

Minkowski sum of this polytope with an axis-aligned unit cell.

Parameters

c	any unit cell.

Returns: a reference to 'this'.

◆ operator+=() [6/6]

template<typename TSpace >

Self & DGtal::BoundedLatticePolytope< TSpace >::operator+= ( UnitSegment s )

Minkowski sum of this polytope with a unit segment aligned with some axis.

Parameters

s	any unit segment.

Returns: a reference to 'this'.

◆ operator=()

template<typename TSpace >

Self & DGtal::BoundedLatticePolytope< TSpace >::operator= ( const Self & other )

default

Assignment.

Parameters

other the object to copy.

Returns: a reference on 'this'.

◆ selfDisplay()

template<typename TSpace >

void DGtal::BoundedLatticePolytope< TSpace >::selfDisplay ( std::ostream & out ) const

Writes/Displays the object on an output stream.

Parameters

out	the output stream where the object is written.

◆ swap()

template<typename TSpace >

void DGtal::BoundedLatticePolytope< TSpace >::swap ( BoundedLatticePolytope< TSpace > & other )

Swaps the content of this object with other. O(1) complexity.

Parameters

other any other BoundedLatticePolytope.

Field Documentation

◆ A

template<typename TSpace >

InequalityMatrix DGtal::BoundedLatticePolytope< TSpace >::A

protected

The matrix A in the polytope representation \( Ax \le B \).

Definition at line 836 of file BoundedLatticePolytope.h.

◆ B

template<typename TSpace >

InequalityVector DGtal::BoundedLatticePolytope< TSpace >::B

protected

The vector B in the polytope representation \( Ax \le B \).

Definition at line 838 of file BoundedLatticePolytope.h.

◆ D

template<typename TSpace >

Domain DGtal::BoundedLatticePolytope< TSpace >::D

protected

Tight bounded box.

Definition at line 840 of file BoundedLatticePolytope.h.

◆ dimension

template<typename TSpace >

const Dimension DGtal::BoundedLatticePolytope< TSpace >::dimension = Space::dimension

static

Definition at line 92 of file BoundedLatticePolytope.h.

◆ I

template<typename TSpace >

std::vector<bool> DGtal::BoundedLatticePolytope< TSpace >::I

protected

Are inequalities large ?

Definition at line 842 of file BoundedLatticePolytope.h.

◆ myValidEdgeConstraints

template<typename TSpace >

bool DGtal::BoundedLatticePolytope< TSpace >::myValidEdgeConstraints

protected

Indicates if Minkowski sums with segments will be valid.

Definition at line 844 of file BoundedLatticePolytope.h.

The documentation for this class was generated from the following file:

BoundedLatticePolytope.h

Data Structures

Public Types

Public Member Functions

Static Public Attributes

Protected Attributes

Private Member Functions

Detailed Description

Member Typedef Documentation

◆ BigInteger

◆ Domain

◆ HalfSpace

◆ InequalityMatrix

◆ InequalityVector

◆ Integer

◆ Point

◆ Self

◆ Space

◆ Vector

Constructor & Destructor Documentation

◆ ~BoundedLatticePolytope()

◆ BoundedLatticePolytope() [1/5]

◆ BoundedLatticePolytope() [2/5]

◆ BoundedLatticePolytope() [3/5]

◆ BoundedLatticePolytope() [4/5]

◆ BoundedLatticePolytope() [5/5]

Member Function Documentation

◆ BOOST_CONCEPT_ASSERT()

◆ canBeSummed()

◆ className()

◆ clear()

◆ closurePolytope()

◆ count()

◆ countBoundary()

◆ countBoundaryByScanning()

◆ countByScanning()

◆ countInterior()

◆ countInteriorByScanning()

◆ countUpTo()

◆ countUpToByScanning()

◆ countWithin()

◆ countWithinByScanning()

◆ cut() [1/3]

◆ cut() [2/3]

◆ cut() [3/3]

◆ getA() [1/2]

◆ getA() [2/2]

◆ getB() [1/2]

◆ getB() [2/2]

◆ getBoundaryPoints()

◆ getBoundaryPointsByScanning()

◆ getDomain()

◆ getI()

◆ getInteriorPoints()

◆ getInteriorPointsByScanning()

◆ getKPoints()

◆ getPoints()

◆ getPointsByScanning()

◆ init() [1/2]

◆ init() [2/2]

◆ insertKPoints()

◆ insertPoints()

◆ insertPointsByScanning()

◆ interiorPolytope()

◆ internalInitFromSegment2D()

◆ internalInitFromSegment3D()

◆ internalInitFromTriangle3D()

◆ isBoundary()

◆ isDomainPointInside()

◆ isInside()

◆ isInterior()

◆ isLarge()

◆ isValid()

◆ nbHalfSpaces()

◆ operator*=()

◆ operator+=() [1/6]

◆ operator+=() [2/6]

◆ operator+=() [3/6]

◆ operator+=() [4/6]

◆ operator+=() [5/6]

◆ operator+=() [6/6]