Aim: This class implements the class Ehrhart Polynomial which is related to lattice point enumeration in bounded lattice polytopes. More...

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

Public Types
typedef EhrhartPolynomial< TSpace, TInteger >	Self

typedef TSpace	Space

typedef TInteger	Integer

typedef BoundedLatticePolytope< Space >	LatticePolytope

typedef LagrangeInterpolation< Integer >	Lagrange

typedef Lagrange::Polynomial	Polynomial

typedef std::size_t	Size

Public Member Functions
	~EhrhartPolynomial ()=default

	EhrhartPolynomial ()

	EhrhartPolynomial (const LatticePolytope &polytope)

	EhrhartPolynomial (const EhrhartPolynomial &other)=default

	EhrhartPolynomial (EhrhartPolynomial &&other)=default

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

EhrhartPolynomial &	operator= (EhrhartPolynomial &&other)=default

Polynomial	numerator () const

Integer	denominator () const
	The (integral) denominator of the Ehrhart polynomial. More...

void	init (const LatticePolytope &polytope)

Integer	count (Integer t) const

Integer	remainder (Integer t) const

Integer	countInterior (Integer t) const

Integer	remainderInterior (Integer t) const

void	selfDisplay (std::ostream &that_stream) const

bool	OK () const

Protected Attributes
Polynomial	myE
	The Ehrhart polynomial (integral numerator part) More...

Integer	myD
	The (integral) denominator of the Ehrhart polynomial. More...

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

	BOOST_CONCEPT_ASSERT ((concepts::CInteger< TInteger >))

Detailed Description

template<typename TSpace, typename TInteger>
class DGtal::EhrhartPolynomial< TSpace, TInteger >

Aim: This class implements the class Ehrhart Polynomial which is related to lattice point enumeration in bounded lattice polytopes.

Description of class 'EhrhartPolynomial'

See also: testEhrhartPolynomial.cpp

Template Parameters

TSpace	an arbitrary model of CSpace, which tells in which space live bounded lattice polytopes.
TInteger	an arbitrary model of CInteger, which must be precise enough to compute the Ehrhart polynomial (either int64_t or BigInteger).

Definition at line 66 of file EhrhartPolynomial.h.

Member Typedef Documentation

◆ Integer

template<typename TSpace , typename TInteger >

typedef TInteger DGtal::EhrhartPolynomial< TSpace, TInteger >::Integer

Definition at line 75 of file EhrhartPolynomial.h.

◆ Lagrange

template<typename TSpace , typename TInteger >

typedef LagrangeInterpolation< Integer > DGtal::EhrhartPolynomial< TSpace, TInteger >::Lagrange

Definition at line 77 of file EhrhartPolynomial.h.

◆ LatticePolytope

template<typename TSpace , typename TInteger >

typedef BoundedLatticePolytope<Space> DGtal::EhrhartPolynomial< TSpace, TInteger >::LatticePolytope

Definition at line 76 of file EhrhartPolynomial.h.

◆ Polynomial

template<typename TSpace , typename TInteger >

typedef Lagrange::Polynomial DGtal::EhrhartPolynomial< TSpace, TInteger >::Polynomial

Definition at line 78 of file EhrhartPolynomial.h.

◆ Self

template<typename TSpace , typename TInteger >

typedef EhrhartPolynomial< TSpace, TInteger > DGtal::EhrhartPolynomial< TSpace, TInteger >::Self

Definition at line 73 of file EhrhartPolynomial.h.

◆ Size

template<typename TSpace , typename TInteger >

typedef std::size_t DGtal::EhrhartPolynomial< TSpace, TInteger >::Size

Definition at line 79 of file EhrhartPolynomial.h.

◆ Space

template<typename TSpace , typename TInteger >

typedef TSpace DGtal::EhrhartPolynomial< TSpace, TInteger >::Space

Definition at line 74 of file EhrhartPolynomial.h.

Constructor & Destructor Documentation

◆ ~EhrhartPolynomial()

template<typename TSpace , typename TInteger >

DGtal::EhrhartPolynomial< TSpace, TInteger >::~EhrhartPolynomial ( )

default

Destructor.

◆ EhrhartPolynomial() [1/4]

template<typename TSpace , typename TInteger >

DGtal::EhrhartPolynomial< TSpace, TInteger >::EhrhartPolynomial ( )

inline

Constructor. The object is invalid.

Definition at line 92 of file EhrhartPolynomial.h.

93 : myE(), myD( NumberTraits< Integer >::ZERO )

94 {}

DGtal::EhrhartPolynomial::myD

Integer myD

The (integral) denominator of the Ehrhart polynomial.

Definition: EhrhartPolynomial.h:223

DGtal::EhrhartPolynomial::myE

Polynomial myE

The Ehrhart polynomial (integral numerator part)

Definition: EhrhartPolynomial.h:221

◆ EhrhartPolynomial() [2/4]

template<typename TSpace , typename TInteger >

DGtal::EhrhartPolynomial< TSpace, TInteger >::EhrhartPolynomial ( const LatticePolytope & polytope )

inline

Constructs the Ehrhart polynomial from a lattice polytope

Parameters

polytope the lattice polytope

Definition at line 99 of file EhrhartPolynomial.h.

    {
      init( polytope );
    }

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

◆ EhrhartPolynomial() [3/4]

template<typename TSpace , typename TInteger >

DGtal::EhrhartPolynomial< TSpace, TInteger >::EhrhartPolynomial ( const EhrhartPolynomial< TSpace, TInteger > & other )

default

Copy constructor.

Parameters

other the object to clone.

◆ EhrhartPolynomial() [4/4]

template<typename TSpace , typename TInteger >

DGtal::EhrhartPolynomial< TSpace, TInteger >::EhrhartPolynomial ( EhrhartPolynomial< TSpace, TInteger > && other )

default

Move constructor.

Parameters

other the object to clone.

Member Function Documentation

◆ BOOST_CONCEPT_ASSERT() [1/2]

template<typename TSpace , typename TInteger >

DGtal::EhrhartPolynomial< TSpace, TInteger >::BOOST_CONCEPT_ASSERT ( (concepts::CInteger< TInteger >) )

private

◆ BOOST_CONCEPT_ASSERT() [2/2]

template<typename TSpace , typename TInteger >

DGtal::EhrhartPolynomial< TSpace, TInteger >::BOOST_CONCEPT_ASSERT ( (concepts::CSpace< TSpace >) )

private

◆ count()

template<typename TSpace , typename TInteger >

Integer DGtal::EhrhartPolynomial< TSpace, TInteger >::count ( Integer t ) const

inline

Definition at line 166 of file EhrhartPolynomial.h.

    {
      return myE( t ) / myD;
    }

References DGtal::EhrhartPolynomial< TSpace, TInteger >::myD, and DGtal::EhrhartPolynomial< TSpace, TInteger >::myE.

◆ countInterior()

template<typename TSpace , typename TInteger >

Integer DGtal::EhrhartPolynomial< TSpace, TInteger >::countInterior ( Integer t ) const

inline

Definition at line 181 of file EhrhartPolynomial.h.

                                             { 
      return myE.degree() % 2 == 0 
        ?  ( myE( -t ) / myD ) 
        :  - ( myE( -t ) / myD ); 
    }

References DGtal::MPolynomial< n, TRing, TAlloc >::degree(), DGtal::EhrhartPolynomial< TSpace, TInteger >::myD, and DGtal::EhrhartPolynomial< TSpace, TInteger >::myE.

◆ denominator()

template<typename TSpace , typename TInteger >

Integer DGtal::EhrhartPolynomial< TSpace, TInteger >::denominator ( ) const

inline

The (integral) denominator of the Ehrhart polynomial.

Definition at line 137 of file EhrhartPolynomial.h.

    {
      return myD;
    }

References DGtal::EhrhartPolynomial< TSpace, TInteger >::myD.

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

◆ init()

template<typename TSpace , typename TInteger >

void DGtal::EhrhartPolynomial< TSpace, TInteger >::init ( const LatticePolytope & polytope )

inline

Initializes the Ehrhart polynomial with a lattice polytope.

Parameters

polytope the lattice polytope

Definition at line 145 of file EhrhartPolynomial.h.

    {
      std::vector< Integer > X( Space::dimension + 1 );
      std::vector< Integer > Y( Space::dimension + 1 );
      X[ 0 ] = NumberTraits< Integer >::ZERO;
      Y[ 0 ] = NumberTraits< Integer >::ONE;
      for ( Integer i = 1; i <= Space::dimension; ++i )
        {
          LatticePolytope Q = i * polytope;
          Integer nb = Q.count();
          X[ i ] = i;
          Y[ i ] = nb;
        }
      // Compute polynomial (degree d)
      Lagrange L( X );
      myE = L.polynomial( Y );
      myD = L.denominator();
    }

References DGtal::BoundedLatticePolytope< TSpace >::count(), DGtal::LagrangeInterpolation< TEuclideanRing >::denominator(), DGtal::SpaceND< dim, TInteger >::dimension, DGtal::EhrhartPolynomial< TSpace, TInteger >::myD, DGtal::EhrhartPolynomial< TSpace, TInteger >::myE, and DGtal::LagrangeInterpolation< TEuclideanRing >::polynomial().

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

◆ numerator()

template<typename TSpace , typename TInteger >

Polynomial DGtal::EhrhartPolynomial< TSpace, TInteger >::numerator ( ) const

inline

Returns: the Ehrhart polynomial (integral numerator part)

Definition at line 131 of file EhrhartPolynomial.h.

    {
      return myE;
    }

References DGtal::EhrhartPolynomial< TSpace, TInteger >::myE.

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

◆ OK()

template<typename TSpace , typename TInteger >

bool DGtal::EhrhartPolynomial< TSpace, TInteger >::OK ( ) const

inline

Checks the validity/consistency of the object.

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

Definition at line 212 of file EhrhartPolynomial.h.

    {
      return myD != NumberTraits< Integer >::ZERO;
    }

References DGtal::EhrhartPolynomial< TSpace, TInteger >::myD.

◆ operator=() [1/2]

template<typename TSpace , typename TInteger >

EhrhartPolynomial & DGtal::EhrhartPolynomial< TSpace, TInteger >::operator= ( const EhrhartPolynomial< TSpace, TInteger > & other )

default

Assignment.

Parameters

other the object to copy.

Returns: a reference on 'this'.

◆ operator=() [2/2]

template<typename TSpace , typename TInteger >

EhrhartPolynomial & DGtal::EhrhartPolynomial< TSpace, TInteger >::operator= ( EhrhartPolynomial< TSpace, TInteger > && other )

default

Move assignment.

Parameters

other the object to move.

Returns: a reference on 'this'.

◆ remainder()

template<typename TSpace , typename TInteger >

Integer DGtal::EhrhartPolynomial< TSpace, TInteger >::remainder ( Integer t ) const

inline

Definition at line 173 of file EhrhartPolynomial.h.

    {
      return myE( t ) % myD;
    }

References DGtal::EhrhartPolynomial< TSpace, TInteger >::myD, and DGtal::EhrhartPolynomial< TSpace, TInteger >::myE.

◆ remainderInterior()

template<typename TSpace , typename TInteger >

Integer DGtal::EhrhartPolynomial< TSpace, TInteger >::remainderInterior ( Integer t ) const

inline

Definition at line 189 of file EhrhartPolynomial.h.

                                                 { 
      return myE.degree() % 2 == 0 
        ?  ( myE( -t ) % myD ) 
        :  - ( myE( -t ) % myD ); 
    }

References DGtal::MPolynomial< n, TRing, TAlloc >::degree(), DGtal::EhrhartPolynomial< TSpace, TInteger >::myD, and DGtal::EhrhartPolynomial< TSpace, TInteger >::myE.

◆ selfDisplay()

template<typename TSpace , typename TInteger >

void DGtal::EhrhartPolynomial< TSpace, TInteger >::selfDisplay ( std::ostream & that_stream ) const

inline

Writes/Displays the object on an output stream.

Parameters

that_stream the output stream where the object is written.

Definition at line 202 of file EhrhartPolynomial.h.

    {
      that_stream << "[EhrhartPolynomial num=" << numerator()
                  << " den=" << denominator() << " ]" << std::endl;
    }

References DGtal::EhrhartPolynomial< TSpace, TInteger >::denominator(), and DGtal::EhrhartPolynomial< TSpace, TInteger >::numerator().

Referenced by DGtal::operator<<().

Field Documentation

◆ myD

template<typename TSpace , typename TInteger >

Integer DGtal::EhrhartPolynomial< TSpace, TInteger >::myD

protected

The (integral) denominator of the Ehrhart polynomial.

Definition at line 223 of file EhrhartPolynomial.h.

Referenced by DGtal::EhrhartPolynomial< TSpace, TInteger >::count(), DGtal::EhrhartPolynomial< TSpace, TInteger >::countInterior(), DGtal::EhrhartPolynomial< TSpace, TInteger >::denominator(), DGtal::EhrhartPolynomial< TSpace, TInteger >::init(), DGtal::EhrhartPolynomial< TSpace, TInteger >::OK(), DGtal::EhrhartPolynomial< TSpace, TInteger >::remainder(), and DGtal::EhrhartPolynomial< TSpace, TInteger >::remainderInterior().

◆ myE

template<typename TSpace , typename TInteger >

Polynomial DGtal::EhrhartPolynomial< TSpace, TInteger >::myE

protected

The Ehrhart polynomial (integral numerator part)

Definition at line 221 of file EhrhartPolynomial.h.

Referenced by DGtal::EhrhartPolynomial< TSpace, TInteger >::count(), DGtal::EhrhartPolynomial< TSpace, TInteger >::countInterior(), DGtal::EhrhartPolynomial< TSpace, TInteger >::init(), DGtal::EhrhartPolynomial< TSpace, TInteger >::numerator(), DGtal::EhrhartPolynomial< TSpace, TInteger >::remainder(), and DGtal::EhrhartPolynomial< TSpace, TInteger >::remainderInterior().

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

EhrhartPolynomial.h

Public Types

Public Member Functions

Protected Attributes

Private Member Functions

Detailed Description

Member Typedef Documentation

◆ Integer

◆ Lagrange

◆ LatticePolytope

◆ Polynomial

◆ Self

◆ Size

◆ Space

Constructor & Destructor Documentation

◆ ~EhrhartPolynomial()

◆ EhrhartPolynomial() [1/4]

◆ EhrhartPolynomial() [2/4]

◆ EhrhartPolynomial() [3/4]

◆ EhrhartPolynomial() [4/4]

Member Function Documentation

◆ BOOST_CONCEPT_ASSERT() [1/2]

◆ BOOST_CONCEPT_ASSERT() [2/2]

◆ count()

◆ countInterior()

◆ denominator()

◆ init()

◆ numerator()

◆ OK()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ remainder()

◆ remainderInterior()

◆ selfDisplay()

Field Documentation

◆ myD

◆ myE