Aim: Represents an nD rational polytope, i.e. a convex polyhedron bounded by vertices with rational 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/BoundedRationalPolytope.h>

Data Structures
struct	Rational

struct	UnitCell

struct	UnitSegment

Public Types
typedef BoundedRationalPolytope< 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)
	~BoundedRationalPolytope ()=default

	BoundedRationalPolytope ()

	BoundedRationalPolytope (const Self &other)=default

	BoundedRationalPolytope (std::initializer_list< Point > l)

template<typename PointIterator >
	BoundedRationalPolytope (Integer d, PointIterator itB, PointIterator itE)

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

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

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

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

void	clear ()
	Clears the polytope.

Accessor services
const Domain &	getDomain () const

const Domain &	getLatticeDomain () const

const Domain &	getRationalDomain () const

unsigned int	nbHalfSpaces () const

Integer	denominator () 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)
Note These services consider the rational polytope embedded as continuous polytope. Therefore even vertices of the rational polytope may not be lattice points if their coordinates do not reduce to an integer when divided by the denominator of the rational polytope.
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)
BoundedRationalPolytope	interiorPolytope () const

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 (BoundedRationalPolytope &other)

Self &	operator*= (Integer t)

Self &	operator*= (Rational r)

Self &	operator+= (UnitSegment s)

Self &	operator+= (UnitCell 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	getInteriorPoints (std::vector< Point > &pts) const

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

template<typename PointSet >
void	insertPoints (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
Integer	q

InequalityMatrix	A

InequalityVector	B

Domain	rationalD

Domain	latticeD

std::vector< bool >	I

bool	myValidEdgeConstraints

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)

Domain	computeLatticeDomain (const Domain &d)

Domain	computeRationalDomain (const Domain &d)

Detailed Description

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

Aim: Represents an nD rational polytope, i.e. a convex polyhedron bounded by vertices with rational 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 'BoundedRationalPolytope'

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

Template Parameters

TSpace an arbitrary model of CSpace.

Definition at line 74 of file BoundedRationalPolytope.h.

Member Typedef Documentation

◆ BigInteger

template<typename TSpace >

typedef DGtal::BigInteger DGtal::BoundedRationalPolytope< TSpace >::BigInteger

Definition at line 88 of file BoundedRationalPolytope.h.

◆ Domain

template<typename TSpace >

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

Definition at line 86 of file BoundedRationalPolytope.h.

◆ HalfSpace

template<typename TSpace >

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

Definition at line 87 of file BoundedRationalPolytope.h.

◆ InequalityMatrix

template<typename TSpace >

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

Definition at line 84 of file BoundedRationalPolytope.h.

◆ InequalityVector

template<typename TSpace >

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

Definition at line 85 of file BoundedRationalPolytope.h.

◆ Integer

template<typename TSpace >

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

Definition at line 81 of file BoundedRationalPolytope.h.

◆ Point

template<typename TSpace >

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

Definition at line 82 of file BoundedRationalPolytope.h.

◆ Self

template<typename TSpace >

typedef BoundedRationalPolytope<TSpace> DGtal::BoundedRationalPolytope< TSpace >::Self

Definition at line 79 of file BoundedRationalPolytope.h.

◆ Space

template<typename TSpace >

typedef TSpace DGtal::BoundedRationalPolytope< TSpace >::Space

Definition at line 80 of file BoundedRationalPolytope.h.

◆ Vector

template<typename TSpace >

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

Definition at line 83 of file BoundedRationalPolytope.h.

Constructor & Destructor Documentation

◆ ~BoundedRationalPolytope()

template<typename TSpace >

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

default

Destructor.

◆ BoundedRationalPolytope() [1/5]

template<typename TSpace >

DGtal::BoundedRationalPolytope< TSpace >::BoundedRationalPolytope ( )

Constructor.

◆ BoundedRationalPolytope() [2/5]

template<typename TSpace >

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

default

Copy constructor.

Parameters

other the object to clone.

◆ BoundedRationalPolytope() [3/5]

template<typename TSpace >

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

Constructs the polytope from a simplex given as an initializer_list.

Parameters

l	any list where the first point give the denominator and then no more than d+1 points in general positions.

Note: If your list is l = { (4,x), (3,2), (1,7), (6,6) }, then the denominator is d = 4 and your polytope has vertices { (3/4,2/4), (1/4,7/4), (6/4,6/4) }.

◆ BoundedRationalPolytope() [4/5]

template<typename TSpace >

template<typename PointIterator >

DGtal::BoundedRationalPolytope< TSpace >::BoundedRationalPolytope	(	Integer	d,
		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

d	the common denominator of all given lattice point coordinates.
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.

Note: If your range is [itB,itE) = { (3,2), (1,7), (6,6) } and the denominator d = 4, then your polytope has vertices { (3/4,2/4), (1/4,7/4), (6/4,6/4) }.

◆ BoundedRationalPolytope() [5/5]

template<typename TSpace >

template<typename HalfSpaceIterator >

DGtal::BoundedRationalPolytope< TSpace >::BoundedRationalPolytope	(	Integer	d,
		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

d	the common denominator of all given lattice point coordinates.
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.

Note: Asumme your real domain is { (-2.25,-1), (3.75, 4.25) } and real halfspaces are ‘{ (1.5*x+2.5*y <= 3), (0.5*x-1.25*y <= 2.5) }. Then for denominator 'd = 4’, you must pass domain={ (-9,-4), (15,17) }, and [itB,itE) = { (6*x+10*y <= 12), (2*x-5*y <= 10) }.

Member Function Documentation

◆ BOOST_CONCEPT_ASSERT()

template<typename TSpace >

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

private

◆ canBeSummed()

template<typename TSpace >

bool DGtal::BoundedRationalPolytope< 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::BoundedRationalPolytope< TSpace >::className ( ) const

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

◆ clear()

template<typename TSpace >

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

Clears the polytope.

◆ computeLatticeDomain()

template<typename TSpace >

Domain DGtal::BoundedRationalPolytope< TSpace >::computeLatticeDomain ( const Domain & d )

private

Computes the lattice domain from the given rational domain, i.e. d/q

Parameters

d	a domain where integer coordinates (x,y,z) means rational coordinates (x/q,y/q,z/q) where q is the denominator of the rational polytope.

◆ computeRationalDomain()

template<typename TSpace >

Domain DGtal::BoundedRationalPolytope< TSpace >::computeRationalDomain ( const Domain & d )

private

Computes the rational domain from the given lattice domain, i.e. d*q

Parameters

d	a domain where integer coordinates (x,y,z) means lattice coordinates.

◆ count()

template<typename TSpace >

Integer DGtal::BoundedRationalPolytope< 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 slow: obtained by checking every point of the polytope domain.

◆ countBoundary()

template<typename TSpace >

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

◆ countInterior()

template<typename TSpace >

Integer DGtal::BoundedRationalPolytope< 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 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::BoundedRationalPolytope< 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 slow: obtained by checking every point of the polytope domain.

◆ countWithin()

template<typename TSpace >

Integer DGtal::BoundedRationalPolytope< 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 lattice point of the domain.
[in]	hi	the highest lattice 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::BoundedRationalPolytope< 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::BoundedRationalPolytope< 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::BoundedRationalPolytope< 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::BoundedRationalPolytopeSpecializer< 3, TInteger >::addEdgeConstraint().

◆ denominator()

template<typename TSpace >

Integer DGtal::BoundedRationalPolytope< TSpace >::denominator ( ) const

Returns: the denominator of the polytope, i.e. vertices lattice coordinates must be divided by this value, or otherwise said, constraints are multiplied by this factor.

◆ getA() [1/2]

template<typename TSpace >

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

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

◆ getA() [2/2]

template<typename TSpace >

const Vector & DGtal::BoundedRationalPolytope< 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::BoundedRationalPolytope< TSpace >::getB ( ) const

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

◆ getB() [2/2]

template<typename TSpace >

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

◆ getDomain()

template<typename TSpace >

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

Returns: the lattice domain of the current polytope.

◆ getI()

template<typename TSpace >

const std::vector< bool > & DGtal::BoundedRationalPolytope< 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::BoundedRationalPolytope< 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 slow: obtained by checking every point of the polytope domain.; At output, pts.size() == this->countInterior()

◆ getLatticeDomain()

template<typename TSpace >

const Domain & DGtal::BoundedRationalPolytope< TSpace >::getLatticeDomain ( ) const

Returns: the lattice domain of the current polytope.

Note: same as getDomain

◆ getPoints()

template<typename TSpace >

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

◆ getRationalDomain()

template<typename TSpace >

const Domain & DGtal::BoundedRationalPolytope< TSpace >::getRationalDomain ( ) const

Returns: the rational domain of the current polytope.

Note: when divided by the polytope denominator, it is the lattice domain.

◆ init() [1/2]

template<typename TSpace >

template<typename HalfSpaceIterator >

void DGtal::BoundedRationalPolytope< TSpace >::init	(	Integer	d,
		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

d	the common denominator of all given lattice point coordinates.
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.

Note: Asumme your real domain is { (-2.25,-1), (3.75, 4.25) } and real halfspaces are ‘{ (1.5*x+2.5*y <= 3), (0.5*x-1.25*y <= 2.5) }. Then for denominator 'd = 4’, you must pass domain={ (-9,-4), (15,17) }, and [itB,itE) = { (6*x+10*y <= 12), (2*x-5*y <= 10) }.

◆ init() [2/2]

template<typename TSpace >

template<typename PointIterator >

bool DGtal::BoundedRationalPolytope< TSpace >::init	(	Integer	d,
		PointIterator	itB,
		PointIterator	itE
	)

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

Parameters

d	the common denominator of all given lattice point coordinates.
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: If your range is [itB,itE) = { (3,2), (1,7), (6,6) } and the denominator d = 4, then your polytope is bounded by { (3/4,2/4), (1/4,7/4), (6/4,6/4) }.; Use DGtal SimpleMatrix::determinant which is not efficient when the dimension is high. Does not use Eigen to avoid dependency.

◆ insertPoints()

template<typename TSpace >

template<typename PointSet >

void DGtal::BoundedRationalPolytope< 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 slow: obtained by checking every point of the polytope domain.

◆ interiorPolytope()

template<typename TSpace >

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

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

◆ internalInitFromSegment2D()

template<typename TSpace >

bool DGtal::BoundedRationalPolytope< 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::BoundedRationalPolytope< 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::BoundedRationalPolytope< 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::BoundedRationalPolytope< 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::BoundedRationalPolytope< 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::BoundedRationalPolytope< 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::BoundedRationalPolytope< 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::BoundedRationalPolytope< 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::BoundedRationalPolytope< 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::BoundedRationalPolytope< TSpace >::nbHalfSpaces ( ) const

Returns: the number of half-space constraints.

◆ operator*=() [1/2]

template<typename TSpace >

Self & DGtal::BoundedRationalPolytope< 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*=() [2/2]

template<typename TSpace >

Self & DGtal::BoundedRationalPolytope< TSpace >::operator*= ( Rational r )

Dilates this polytope P by rational r.p/r.q.

Parameters

r	any rational.

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

◆ operator+=() [1/2]

template<typename TSpace >

Self & DGtal::BoundedRationalPolytope< 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+=() [2/2]

template<typename TSpace >

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

default

Assignment.

Parameters

other the object to copy.

Returns: a reference on 'this'.

◆ selfDisplay()

template<typename TSpace >

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

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

Parameters

other any other BoundedRationalPolytope.

Field Documentation

◆ A

template<typename TSpace >

InequalityMatrix DGtal::BoundedRationalPolytope< TSpace >::A

protected

Definition at line 637 of file BoundedRationalPolytope.h.

◆ B

template<typename TSpace >

InequalityVector DGtal::BoundedRationalPolytope< TSpace >::B

protected

Definition at line 639 of file BoundedRationalPolytope.h.

◆ dimension

template<typename TSpace >

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

static

Definition at line 90 of file BoundedRationalPolytope.h.

◆ I

template<typename TSpace >

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

protected

Definition at line 645 of file BoundedRationalPolytope.h.

◆ latticeD

template<typename TSpace >

Domain DGtal::BoundedRationalPolytope< TSpace >::latticeD

protected

Definition at line 643 of file BoundedRationalPolytope.h.

◆ myValidEdgeConstraints

template<typename TSpace >

bool DGtal::BoundedRationalPolytope< TSpace >::myValidEdgeConstraints

protected

Definition at line 647 of file BoundedRationalPolytope.h.

◆ q

template<typename TSpace >

Integer DGtal::BoundedRationalPolytope< TSpace >::q

protected

Definition at line 635 of file BoundedRationalPolytope.h.

◆ rationalD

template<typename TSpace >

Domain DGtal::BoundedRationalPolytope< TSpace >::rationalD

protected

Definition at line 641 of file BoundedRationalPolytope.h.

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

BoundedRationalPolytope.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

◆ ~BoundedRationalPolytope()

◆ BoundedRationalPolytope() [1/5]

◆ BoundedRationalPolytope() [2/5]

◆ BoundedRationalPolytope() [3/5]

◆ BoundedRationalPolytope() [4/5]

◆ BoundedRationalPolytope() [5/5]

Member Function Documentation

◆ BOOST_CONCEPT_ASSERT()

◆ canBeSummed()

◆ className()

◆ clear()

◆ computeLatticeDomain()

◆ computeRationalDomain()

◆ count()

◆ countBoundary()

◆ countInterior()

◆ countUpTo()

◆ countWithin()

◆ cut() [1/3]

◆ cut() [2/3]

◆ cut() [3/3]

◆ denominator()

◆ getA() [1/2]

◆ getA() [2/2]

◆ getB() [1/2]

◆ getB() [2/2]

◆ getBoundaryPoints()

◆ getDomain()

◆ getI()

◆ getInteriorPoints()

◆ getLatticeDomain()

◆ getPoints()

◆ getRationalDomain()

◆ init() [1/2]

◆ init() [2/2]

◆ insertPoints()

◆ interiorPolytope()

◆ internalInitFromSegment2D()

◆ internalInitFromSegment3D()

◆ internalInitFromTriangle3D()

◆ isBoundary()

◆ isDomainPointInside()

◆ isInside()

◆ isInterior()

◆ isLarge()

◆ isValid()

◆ nbHalfSpaces()

◆ operator*=() [1/2]

◆ operator*=() [2/2]

◆ operator+=() [1/2]

◆ operator+=() [2/2]

◆ operator=()

◆ selfDisplay()

◆ swap()

Field Documentation

◆ A

◆ B

◆ dimension

◆ I

◆ latticeD

◆ myValidEdgeConstraints