Aim: A helper class to build polytopes from digital sets and to check digital k-convexity and full convexity. More...

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

Public Types
typedef DigitalConvexity< TKSpace >	Self

typedef TKSpace	KSpace

typedef KSpace::Integer	Integer

typedef KSpace::Point	Point

typedef KSpace::Vector	Vector

typedef KSpace::Cell	Cell

typedef KSpace::Space	Space

typedef std::size_t	Size

typedef DGtal::BoundedLatticePolytope< Space >	Polytope

typedef DGtal::BoundedLatticePolytope< Space >	LatticePolytope

typedef DGtal::BoundedRationalPolytope< Space >	RationalPolytope

typedef DGtal::CellGeometry< KSpace >	CellGeometry

typedef std::vector< Point >	PointRange

typedef std::unordered_set< Point >	PointSet

typedef DGtal::BoundedLatticePolytopeCounter< Space >	Counter

typedef Counter::Interval	Interval

typedef DGtal::LatticeSetByIntervals< Space >	LatticeSet

Public Member Functions
Standard services (construction, initialization, assignment)
	~DigitalConvexity ()=default

	DigitalConvexity ()=default

	DigitalConvexity (const Self &other)=default

	DigitalConvexity (Clone< KSpace > K, bool safe=false)

	DigitalConvexity (Point lo, Point hi, bool safe=false)

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

const KSpace &	space () const

Cell geometry services
template<typename PointIterator >
CellGeometry	makeCellCover (PointIterator itB, PointIterator itE, Dimension i=0, Dimension k=KSpace::dimension) const

CellGeometry	makeCellCover (const LatticePolytope &P, Dimension i=0, Dimension k=KSpace::dimension) const

CellGeometry	makeCellCover (const RationalPolytope &P, Dimension i=0, Dimension k=KSpace::dimension) const

Morphological services
LatticePolytope	makePolytope (const PointRange &X, bool make_minkowski_summable=false) const

PointRange	U (Dimension i, const PointRange &X) const

bool	is0Convex (const PointRange &X) const

bool	isFullyConvex (const PointRange &X, bool convex0=false) const

bool	isFullyConvexFast (const PointRange &X) const

bool	isFullySubconvex (const PointRange &Y, const LatticeSet &StarX) const

bool	isKSubconvex (const Point &a, const Point &b, const CellGeometry &C, const Dimension k) const

bool	isFullySubconvex (const Point &a, const Point &b, const CellGeometry &C) const

bool	isFullySubconvex (const Point &a, const Point &b, const LatticeSet &StarX) const

LatticePolytope	CvxH (const PointRange &X) const

PointRange	ExtrCvxH (const PointRange &X) const

LatticeSet	StarCvxH (const PointRange &X, Dimension axis=dimension) const

Integer	sizeStarCvxH (const PointRange &X) const

LatticeSet	Star (const PointRange &X, Dimension axis=dimension) const

LatticeSet	StarCells (const PointRange &C, Dimension axis=dimension) const

PointRange	Extr (const PointRange &C) const

PointRange	Extr (const LatticeSet &C) const

LatticeSet	Skel (const LatticeSet &C) const

PointRange	ExtrSkel (const LatticeSet &C) const

LatticeSet	toLatticeSet (const PointRange &X, Dimension axis=dimension) const

PointRange	toPointRange (const LatticeSet &L) const

Convexity services for lattice polytopes
bool	isKConvex (const LatticePolytope &P, const Dimension k) const

bool	isFullyConvex (const LatticePolytope &P) const

bool	isKSubconvex (const LatticePolytope &P, const CellGeometry &C, const Dimension k) const

bool	isFullySubconvex (const LatticePolytope &P, const CellGeometry &C) const

bool	isFullySubconvex (const LatticePolytope &P, const LatticeSet &StarX) const

Convexity services for rational polytopes
bool	isKConvex (const RationalPolytope &P, const Dimension k) const

bool	isFullyConvex (const RationalPolytope &P) const

bool	isKSubconvex (const RationalPolytope &P, const CellGeometry &C, const Dimension k) const

bool	isFullySubconvex (const RationalPolytope &P, const CellGeometry &C) const

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

bool	isValid () const

Static Public Member Functions
Lattice and rational polytope services
static PointRange	insidePoints (const LatticePolytope &polytope)

static PointRange	interiorPoints (const LatticePolytope &polytope)

static PointRange	insidePoints (const RationalPolytope &polytope)

static PointRange	interiorPoints (const RationalPolytope &polytope)

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

Protected Attributes
KSpace	myK
	The cellular grid space where computations are done. More...

bool	mySafe

Size	myDepthLastFCE
	The number of iterations of the last FullyConvexEnvelope operation. More...

Private Member Functions
	BOOST_CONCEPT_ASSERT ((concepts::CCellularGridSpaceND< TKSpace >))

PointRange	FC_direct (const PointRange &Z) const

PointRange	FC_LatticeSet (const PointRange &Z) const

Static Private Member Functions
static void	eraseInterval (Interval I, std::vector< Interval > &V)

template<typename Predicate >
static PointRange	filter (const PointRange &E, const Predicate &Pred)

Simplex services
enum class	SimplexType { INVALID , DEGENERATED , UNITARY , COMMON }
	The possible types for simplices. More...

template<typename PointIterator >
static LatticePolytope	makeSimplex (PointIterator itB, PointIterator itE)

static LatticePolytope	makeSimplex (std::initializer_list< Point > l)

template<typename PointIterator >
static RationalPolytope	makeRationalSimplex (Integer d, PointIterator itB, PointIterator itE)

static RationalPolytope	makeRationalSimplex (std::initializer_list< Point > l)

template<typename PointIterator >
static bool	isSimplexFullDimensional (PointIterator itB, PointIterator itE)

static bool	isSimplexFullDimensional (std::initializer_list< Point > l)

template<typename PointIterator >
static SimplexType	simplexType (PointIterator itB, PointIterator itE)

static SimplexType	simplexType (std::initializer_list< Point > l)

template<typename PointIterator >
static void	displaySimplex (std::ostream &out, PointIterator itB, PointIterator itE)

static void	displaySimplex (std::ostream &out, std::initializer_list< Point > l)

Convex envelope services
enum class	EnvelopeAlgorithm { DIRECT , LATTICE_SET }

PointRange	FC (const PointRange &Z, EnvelopeAlgorithm algo=EnvelopeAlgorithm::DIRECT) const

PointRange	envelope (const PointRange &Z, EnvelopeAlgorithm algo=EnvelopeAlgorithm::DIRECT) const

PointRange	relativeEnvelope (const PointRange &Z, const PointRange &Y, EnvelopeAlgorithm algo=EnvelopeAlgorithm::DIRECT) const

template<typename Predicate >
PointRange	relativeEnvelope (const PointRange &Z, const Predicate &PredY, EnvelopeAlgorithm algo=EnvelopeAlgorithm::DIRECT) const

Size	depthLastEnvelope () const

Detailed Description

template<typename TKSpace>
class DGtal::DigitalConvexity< TKSpace >

Aim: A helper class to build polytopes from digital sets and to check digital k-convexity and full convexity.

Description of template class 'DigitalConvexity'

See also: moduleDigitalConvexity

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

Template Parameters

TKSpace an arbitrary model of CCellularGridSpaceND.

Examples: geometry/curves/exampleDigitalConvexity.cpp, geometry/curves/exampleRationalConvexity.cpp, geometry/volumes/checkFullConvexityTheorems.cpp, geometry/volumes/digitalPolyhedronBuilder3D.cpp, and geometry/volumes/standardDigitalPolyhedronBuilder3D.cpp.

Definition at line 77 of file DigitalConvexity.h.

Member Typedef Documentation

◆ Cell

template<typename TKSpace >

typedef KSpace::Cell DGtal::DigitalConvexity< TKSpace >::Cell

Definition at line 87 of file DigitalConvexity.h.

◆ CellGeometry

template<typename TKSpace >

typedef DGtal::CellGeometry< KSpace > DGtal::DigitalConvexity< TKSpace >::CellGeometry

Definition at line 93 of file DigitalConvexity.h.

◆ Counter

template<typename TKSpace >

typedef DGtal::BoundedLatticePolytopeCounter< Space > DGtal::DigitalConvexity< TKSpace >::Counter

Definition at line 96 of file DigitalConvexity.h.

◆ Integer

template<typename TKSpace >

typedef KSpace::Integer DGtal::DigitalConvexity< TKSpace >::Integer

Definition at line 84 of file DigitalConvexity.h.

◆ Interval

template<typename TKSpace >

typedef Counter::Interval DGtal::DigitalConvexity< TKSpace >::Interval

Definition at line 97 of file DigitalConvexity.h.

◆ KSpace

template<typename TKSpace >

typedef TKSpace DGtal::DigitalConvexity< TKSpace >::KSpace

Definition at line 83 of file DigitalConvexity.h.

◆ LatticePolytope

template<typename TKSpace >

typedef DGtal::BoundedLatticePolytope< Space > DGtal::DigitalConvexity< TKSpace >::LatticePolytope

Definition at line 91 of file DigitalConvexity.h.

◆ LatticeSet

template<typename TKSpace >

typedef DGtal::LatticeSetByIntervals< Space > DGtal::DigitalConvexity< TKSpace >::LatticeSet

Definition at line 98 of file DigitalConvexity.h.

◆ Point

template<typename TKSpace >

typedef KSpace::Point DGtal::DigitalConvexity< TKSpace >::Point

Definition at line 85 of file DigitalConvexity.h.

◆ PointRange

template<typename TKSpace >

typedef std::vector<Point> DGtal::DigitalConvexity< TKSpace >::PointRange

Definition at line 94 of file DigitalConvexity.h.

◆ PointSet

template<typename TKSpace >

typedef std::unordered_set<Point> DGtal::DigitalConvexity< TKSpace >::PointSet

Definition at line 95 of file DigitalConvexity.h.

◆ Polytope

template<typename TKSpace >

typedef DGtal::BoundedLatticePolytope< Space > DGtal::DigitalConvexity< TKSpace >::Polytope

Definition at line 90 of file DigitalConvexity.h.

◆ RationalPolytope

template<typename TKSpace >

typedef DGtal::BoundedRationalPolytope< Space > DGtal::DigitalConvexity< TKSpace >::RationalPolytope

Definition at line 92 of file DigitalConvexity.h.

◆ Self

template<typename TKSpace >

typedef DigitalConvexity<TKSpace> DGtal::DigitalConvexity< TKSpace >::Self

Definition at line 82 of file DigitalConvexity.h.

◆ Size

template<typename TKSpace >

typedef std::size_t DGtal::DigitalConvexity< TKSpace >::Size

Definition at line 89 of file DigitalConvexity.h.

◆ Space

template<typename TKSpace >

typedef KSpace::Space DGtal::DigitalConvexity< TKSpace >::Space

Definition at line 88 of file DigitalConvexity.h.

◆ Vector

template<typename TKSpace >

typedef KSpace::Vector DGtal::DigitalConvexity< TKSpace >::Vector

Definition at line 86 of file DigitalConvexity.h.

Member Enumeration Documentation

◆ EnvelopeAlgorithm

template<typename TKSpace >

enum class DGtal::DigitalConvexity::EnvelopeAlgorithm

strong

Choice of algorithm for computing the fully convex envelope of a digital set.

Enumerator
DIRECT	Slightly faster but quite ugly big function
LATTICE_SET	Slightly slower function but decomposes well the algorithm

Definition at line 642 of file DigitalConvexity.h.

      { DIRECT ,
        LATTICE_SET 
      };

◆ SimplexType

template<typename TKSpace >

enum class DGtal::DigitalConvexity::SimplexType

strong

The possible types for simplices.

Enumerator
INVALID	When there are not the right number of vertices.
DEGENERATED	When the points of the simplex are not in general position.
UNITARY	When its edges form a unit parallelotope (det = +/- 1)
COMMON	Common simplex.

Definition at line 237 of file DigitalConvexity.h.

                          {
      INVALID,     
      DEGENERATED, 
      UNITARY,     
      COMMON       
    };

Constructor & Destructor Documentation

◆ ~DigitalConvexity()

template<typename TKSpace >

DGtal::DigitalConvexity< TKSpace >::~DigitalConvexity ( )

default

Destructor.

◆ DigitalConvexity() [1/4]

template<typename TKSpace >

DGtal::DigitalConvexity< TKSpace >::DigitalConvexity ( )

default

Constructor.

◆ DigitalConvexity() [2/4]

template<typename TKSpace >

DGtal::DigitalConvexity< TKSpace >::DigitalConvexity ( const Self & other )

default

Copy constructor.

Parameters

other the object to clone.

◆ DigitalConvexity() [3/4]

template<typename TKSpace >

DGtal::DigitalConvexity< TKSpace >::DigitalConvexity	(	Clone< KSpace >	K,
		bool	safe = `false`
	)

Constructor from cellular space.

Parameters

K	any cellular grid space.
safe	when 'true' performs convex hull computations with arbitrary precision integer (if available), otherwise chooses a compromise between speed and precision (int64_t).

◆ DigitalConvexity() [4/4]

template<typename TKSpace >

DGtal::DigitalConvexity< TKSpace >::DigitalConvexity	(	Point	lo,
		Point	hi,
		bool	safe = `false`
	)

Constructor from lower and upper points.

Parameters

lo	the lowest point of the domain (bounding box for computations).
hi	the highest point of the domain (bounding box for computations).
safe	when 'true' performs convex hull computations with arbitrary precision integer (if available), otherwise chooses a compromise between speed and precision (int64_t).

Member Function Documentation

◆ BOOST_CONCEPT_ASSERT()

template<typename TKSpace >

DGtal::DigitalConvexity< TKSpace >::BOOST_CONCEPT_ASSERT ( (concepts::CCellularGridSpaceND< TKSpace >) )

private

◆ CvxH()

template<typename TKSpace >

LatticePolytope DGtal::DigitalConvexity< TKSpace >::CvxH ( const PointRange & X ) const

inline

Given a range of distinct points X, computes the tightiest polytope that enclosed it. Note that this polytope may contain more lattice points than the given input points.

Parameters

X	any range of pairwise distinct points

Returns: the corresponding lattice polytope.

Note: alias for DigitalConvexity::makePolytope

Definition at line 504 of file DigitalConvexity.h.

    {
      return makePolytope( X );
    }

References DGtal::DigitalConvexity< TKSpace >::makePolytope().

◆ depthLastEnvelope()

template<typename TKSpace >

Size DGtal::DigitalConvexity< TKSpace >::depthLastEnvelope ( ) const

Returns: the number of iterations of the last process FC^*(Z):=FC(FC(....FC(Z)...)), i.e. the last call to DigitalConvexity::envelope or DigitalConvexity::relativeEnvelope .

Referenced by SCENARIO().

◆ displaySimplex() [1/2]

template<typename TKSpace >

template<typename PointIterator >

static void DGtal::DigitalConvexity< TKSpace >::displaySimplex	(	std::ostream &	out,
		PointIterator	itB,
		PointIterator	itE
	)

static

Displays information about the simplex formed by the given range [itB,itE) of lattice points.

Template Parameters

PointIterator any model of forward iterator on Point.

Parameters

[in,out]	out	the output stream where information is outputed.
	itB	the start of the range of n+1 points defining the simplex.
	itE	past the end the range of n+1 points defining the simplex.

◆ displaySimplex() [2/2]

template<typename TKSpace >

static void DGtal::DigitalConvexity< TKSpace >::displaySimplex	(	std::ostream &	out,
		std::initializer_list< Point >	l
	)

static

Displays information about simplex formed by the given list l of lattice points.

Parameters

[in,out]	out	the output stream where information is outputed.
	l	any list of lattice points.

◆ envelope()

template<typename TKSpace >

PointRange DGtal::DigitalConvexity< TKSpace >::envelope	(	const PointRange &	Z,
		EnvelopeAlgorithm	algo = `EnvelopeAlgorithm::DIRECT`
	)		const

Computes the fully convex envelope of Z, i.e. \( FC^*(Z):=FC(FC( \ldots FC(Z) \ldots )) \), for Z a range of points, until stabilization of the iterative process.

Parameters

Z	any range of points (must be sorted).
algo	the chosen method of computation.

Returns: \( FC^*( Z ) \)

Note: If Z is fully convex, then the output is Z itself. Otherwise, the returned set of points includes Z and is fully convex.

Referenced by checkCvxHPlusHypercubeFullConvexity(), checkProjectionFullConvexity(), checkSkelStarCvxHFullConvexity(), main(), and SCENARIO().

◆ eraseInterval()

template<typename TKSpace >

static void DGtal::DigitalConvexity< TKSpace >::eraseInterval	(	Interval	I,
		std::vector< Interval > &	V
	)

staticprivate

Erase the interval I from the intervals in V such that the integer in I are not part of V anymore.

Parameters

[in]	I	is a closed interval
[in,out]	V	is a sorted list of closed intervals

◆ Extr() [1/2]

template<typename TKSpace >

PointRange DGtal::DigitalConvexity< TKSpace >::Extr ( const LatticeSet & C ) const

Parameters

C	a range of cells represented as a lattice set.

Returns: the range of digital points that are the extremal vertices to the cells in C.

◆ Extr() [2/2]

template<typename TKSpace >

PointRange DGtal::DigitalConvexity< TKSpace >::Extr ( const PointRange & C ) const

Parameters

C	a range of cells represented with points in Khalimsky coordinates.

Returns: the range of digital points that are the extremal vertices to the cells in C.

◆ ExtrCvxH()

template<typename TKSpace >

PointRange DGtal::DigitalConvexity< TKSpace >::ExtrCvxH ( const PointRange & X ) const

Given a range of distinct points X, computes the vertices of the tightiest polytope that enclosed it.

Parameters

X	any range of pairwise distinct points

Returns: the vertices or extrema of CvxH(X).

Note: The method works in nD for full dimensional convex hulls. It can handle not full dimensional convex hull up to dimension 3 included.

◆ ExtrSkel()

template<typename TKSpace >

PointRange DGtal::DigitalConvexity< TKSpace >::ExtrSkel ( const LatticeSet & C ) const

Parameters

C	a range of cells represented as a lattice set.

Returns: the range of digital points that are the extremal vertices to the skeleton of the cells in C.

◆ FC()

template<typename TKSpace >

PointRange DGtal::DigitalConvexity< TKSpace >::FC	(	const PointRange &	Z,
		EnvelopeAlgorithm	algo = `EnvelopeAlgorithm::DIRECT`
	)		const

Computes FC(Z):=Extr(Skel(Star(CvxH(Z)))), for Z a range of points

Parameters

Z	any range of points (must be sorted).
algo	the chosen method of computation.

Returns: FC( Z )

◆ FC_direct()

template<typename TKSpace >

PointRange DGtal::DigitalConvexity< TKSpace >::FC_direct ( const PointRange & Z ) const

private

Computes FC(Z):=Extr(Skel(Star(CvxH(Z)))), for Z a range of points

Parameters

Z	any range of points (must be sorted).

Returns: FC( Z )

◆ FC_LatticeSet()

template<typename TKSpace >

PointRange DGtal::DigitalConvexity< TKSpace >::FC_LatticeSet ( const PointRange & Z ) const

private

Computes FC(Z):=Extr(Skel(Star(CvxH(Z)))), for Z a range of points

Parameters

Z	any range of points (must be sorted).

Returns: FC( Z )

◆ filter()

template<typename TKSpace >

template<typename Predicate >

static PointRange DGtal::DigitalConvexity< TKSpace >::filter	(	const PointRange &	E,
		const Predicate &	Pred
	)

staticprivate

Filters the points of E and outputs only the ones that satisfies the given predicate Pred.

Template Parameters

Predicate the type of a predicate Point -> boolean

Parameters

[in]	E	any range of point
[in]	Pred	the predicate Point -> boolean

Returns: the subset of E whose elements satisfy the predicate Pred.

◆ insidePoints() [1/2]

template<typename TKSpace >

static PointRange DGtal::DigitalConvexity< TKSpace >::insidePoints ( const LatticePolytope & polytope )

static

Parameters

polytope any lattice polytope.

Returns: the range of digital points that belongs to the polytope.

Referenced by SCENARIO().

◆ insidePoints() [2/2]

template<typename TKSpace >

static PointRange DGtal::DigitalConvexity< TKSpace >::insidePoints ( const RationalPolytope & polytope )

static

Parameters

polytope any rational polytope.

Returns: the range of digital points that belongs to the polytope.

◆ interiorPoints() [1/2]

template<typename TKSpace >

static PointRange DGtal::DigitalConvexity< TKSpace >::interiorPoints ( const LatticePolytope & polytope )

static

Parameters

polytope any lattice polytope.

Returns: the range of digital points that belongs to the interior of the polytope.

◆ interiorPoints() [2/2]

template<typename TKSpace >

static PointRange DGtal::DigitalConvexity< TKSpace >::interiorPoints ( const RationalPolytope & polytope )

static

Parameters

polytope any rational polytope.

Returns: the range of digital points that belongs to the interior of the polytope.

◆ is0Convex()

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::is0Convex ( const PointRange & X ) const

Tells if a given point range X is digitally 0-convex, i.e. \( \mathrm{Cvxh}(X) \cap \mathbb{Z}^d = X \). It works for arbitrary set of points in arbitrary dimenion.

Parameters

X	any range of pairwise distinct points

Returns: 'true' iff X is fully digitally convex.

Referenced by checkFullConvexityCharacterization(), DGtal::NeighborhoodConvexityAnalyzer< TKSpace, K >::is0Convex(), DGtal::NeighborhoodConvexityAnalyzer< TKSpace, K >::isComplementary0Convex(), and SCENARIO().

◆ isFullyConvex() [1/3]

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::isFullyConvex ( const LatticePolytope & P ) const

Tells if a given polytope P is fully digitally convex. The digital 0-convexity is the usual property \( Conv( P \cap Z^d ) = P \cap Z^d) \). Otherwise the property asks that the points inside P touch as many k-cells that the convex hull of P, for any valid dimension k.

Parameters

P	any lattice polytope such that `P.canBeSummed() == true`.

Returns: 'true' iff the polytope P is fully digitally convex.

Note: A polytope is always digitally 0-convex. Furthermore, if it is not digitally d-1-convex then it is digitally d-convex (d := KSpace::dimension). Hence, we only check k-convexity for 1 <= k <= d-1.

◆ isFullyConvex() [2/3]

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::isFullyConvex	(	const PointRange &	X,
		bool	convex0 = `false`
	)		const

Tells if a given point range X is fully digitally convex. The test uses the morphological characterization of full convexity. It is slightly slower than testing full convexity on simplices, but it works for arbitrary set of points in arbitrary dimenion.

Parameters

X	any range of pairwise distinct points
convex0	when 'true' indicates that X is known to be digitally 0-convex, otherwise the method will check it also.

Returns: 'true' iff X is fully digitally convex.

Referenced by checkCvxHPlusHypercubeFullConvexity(), checkFullConvexityCharacterization(), checkSkelStarCvxHFullConvexity(), DGtal::NeighborhoodConvexityAnalyzer< TKSpace, K >::isComplementaryFullyConvex(), DGtal::NeighborhoodConvexityAnalyzer< TKSpace, K >::isFullyConvex(), and SCENARIO().

◆ isFullyConvex() [3/3]

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::isFullyConvex ( const RationalPolytope & P ) const

Tells if a given polytope P is fully digitally convex. The digital 0-convexity is the usual property \( Conv( P \cap Z^d ) = P \cap Z^d) \). Otherwise the property asks that the points inside P touch as many k-cells that the convex hull of P, for any valid dimension k.

Parameters

P	any rational polytope such that `P.canBeSummed() == true`.

Returns: 'true' iff the polytope P is fully digitally convex.

Note: A polytope is always digitally 0-convex. Furthermore, if it is not digitally d-1-convex then it is digitally d-convex (d := KSpace::dimension). Hence, we only check k-convexity for 1 <= k <= d-1.

◆ isFullyConvexFast()

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::isFullyConvexFast ( const PointRange & X ) const

Tells if a given point range X is fully digitally convex. The test uses the morphological characterization of full convexity and a fast way to compute lattice points within a polytope. It works for arbitrary set of points in arbitrary dimenion.

Parameters

X	any range of pairwise distinct points

Returns: 'true' iff X is fully digitally convex.

Note: This method is generally faster than DigitalConvexity::isFullyConvex if (1) the set is indeed is fully convex, (2) the dimension is high (>= 3 or 4).

Referenced by SCENARIO().

◆ isFullySubconvex() [1/6]

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::isFullySubconvex	(	const LatticePolytope &	P,
		const CellGeometry &	C
	)		const

Tells if a given polytope P is digitally fully subconvex to some cell cover C. The digital 0-subconvexity is the usual property \( Conv( P \cap Z^d ) \subset C \cap Z^d) \). Otherwise the property asks that the k-cells intersected by the convex hull of P is a subset of the k-cells of C.

Parameters

P	any lattice polytope such that `P.canBeSummed() == true`.
C	any cell cover geometry (i.e. a cubical complex).

Returns: 'true' iff the polytope P is digitally fully subconvex to C.

Note: This method only checks the k-subconvexity for valid dimensions stored in C.

◆ isFullySubconvex() [2/6]

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::isFullySubconvex	(	const LatticePolytope &	P,
		const LatticeSet &	StarX
	)		const

Tells if a given polytope P is digitally fully subconvex to some lattice set Star_X, i.e. the cell cover of some set X represented by lattice points.

Parameters

P	any lattice polytope such that `P.canBeSummed() == true`.
StarX	any lattice set representing an open cubical complex.

Returns: 'true' iff Y is digitally fully subconvex to X.

◆ isFullySubconvex() [3/6]

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::isFullySubconvex	(	const Point &	a,
		const Point &	b,
		const CellGeometry &	C
	)		const

Tells if a given segment from a to b is digitally fully subconvex (i.e. tangent) to some cell cover C. The digital 0-subconvexity is the usual property \( Conv( P \cap Z^d ) \subset C \cap Z^d) \). Otherwise the property asks that the k-cells intersected by the convex hull of the segment is a subset of the k-cells of C.

Parameters

a	any point
b	any point
C	any cell cover geometry (i.e. a cubical complex).

Returns: 'true' iff the segment is a digitally fully subconvex of C, i.e. the two points are cotangent.

Note: Three times faster than building a (degenerated) lattice polytope and then checking if it subconvex.

◆ isFullySubconvex() [4/6]

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::isFullySubconvex	(	const Point &	a,
		const Point &	b,
		const LatticeSet &	StarX
	)		const

Tells if a given segment from a to b is digitally fully subconvex (i.e. tangent) to some open complex StarX.

Parameters

a	any point
b	any point
StarX	any lattice set representing an open cubical complex.

Returns: 'true' iff the segment is a digitally fully subconvex of C, i.e. the two points are cotangent.

Note: Three times faster than building a (degenerated) lattice polytope and then checking if it subconvex.

◆ isFullySubconvex() [5/6]

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::isFullySubconvex	(	const PointRange &	Y,
		const LatticeSet &	StarX
	)		const

Tells if a given set of points Y is digitally fully subconvex to some lattice set Star_X, i.e. the cell cover of some set X represented by lattice points.

Parameters

Y	any set of points
StarX	any lattice set representing an open cubical complex.

Returns: 'true' iff Y is digitally fully subconvex to X.

Note: This method is slower than the two others, since it builds the polytope embracing Y. However it is much more generic since the two other methods require a Minkowski summable polytope, i.e. P.canBeSummed() == true.

Referenced by main(), and SCENARIO().

◆ isFullySubconvex() [6/6]

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::isFullySubconvex	(	const RationalPolytope &	P,
		const CellGeometry &	C
	)		const

Tells if a given polytope P is digitally fully subconvex to some cell cover C. The digital 0-subconvexity is the usual property \( Conv( P \cap Z^d ) \subset C \cap Z^d) \). Otherwise the property asks that the k-cells intersected by the convex hull of P is a subset of the k-cells of C.

Parameters

P	any rational polytope such that `P.canBeSummed() == true`.
C	any cell cover geometry (i.e. a cubical complex).

Returns: 'true' iff the polytope P is digitally fully subconvex to C.

Note: This method only checks the k-subconvexity for valid dimensions stored in C.

◆ isKConvex() [1/2]

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::isKConvex	(	const LatticePolytope &	P,
		const Dimension	k
	)		const

Tells if a given polytope P is digitally k-convex. The digital 0-convexity is the usual property \( Conv( P \cap Z^d ) = P \cap Z^d) \). Otherwise the property asks that the points inside P touch as many k-cells that the convex hull of P.

Parameters

P	any lattice polytope such that `P.canBeSummed() == true`.
k	the dimension for which the digital k-convexity is checked, 0 <= k <= KSpace::dimension.

Returns: 'true' iff the polytope P is digitally k-convex.

Note: A polytope is always digitally 0-convex. Furthermore, if it is not digitally d-1-convex then it is digitally not d-convex (d := KSpace::dimension).

Referenced by SCENARIO().

◆ isKConvex() [2/2]

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::isKConvex	(	const RationalPolytope &	P,
		const Dimension	k
	)		const

Tells if a given polytope P is digitally k-convex. The digital 0-convexity is the usual property \( Conv( P \cap Z^d ) = P \cap Z^d) \). Otherwise the property asks that the points inside P touch as many k-cells that the convex hull of P.

Parameters

P	any rational polytope such that `P.canBeSummed() == true`.
k	the dimension for which the digital k-convexity is checked, 0 <= k <= KSpace::dimension.

Returns: 'true' iff the polytope P is digitally k-convex.

Note: A polytope is always digitally 0-convex. Furthermore, if it is not digitally d-1-convex then it is digitally not d-convex (d := KSpace::dimension).

◆ isKSubconvex() [1/3]

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::isKSubconvex	(	const LatticePolytope &	P,
		const CellGeometry &	C,
		const Dimension	k
	)		const

Tells if a given polytope P is digitally k-subconvex of some cell cover C. The digital 0-subconvexity is the usual property \( Conv( P \cap Z^d ) \subset C \cap Z^d) \). Otherwise the property asks that the k-cells intersected by the convex hull of P is a subset of the k-cells of C.

Parameters

P	any lattice polytope such that `P.canBeSummed() == true`.
C	any cell cover geometry (i.e. a cubical complex).
k	the dimension for which the digital k-convexity is checked, 0 <= k <= KSpace::dimension.

Returns: 'true' iff the polytope P is a digitally k-subconvex of C.

◆ isKSubconvex() [2/3]

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::isKSubconvex	(	const Point &	a,
		const Point &	b,
		const CellGeometry &	C,
		const Dimension	k
	)		const

Tells if a given segment from a to b is digitally k-subconvex (i.e. k-tangent) to some cell cover C. The digital 0-subconvexity is the usual property \( Conv( P \cap Z^d ) \subset C \cap Z^d) \). Otherwise the property asks that the k-cells intersected by the convex hull of the segment is a subset of the k-cells of C.

Parameters

a	any point
b	any point
C	any cell cover geometry (i.e. a cubical complex).
k	the dimension for which the digital k-convexity is checked, 0 <= k <= KSpace::dimension.

Returns: 'true' iff the segment is a digitally k-subconvex of C, i.e. the two points are k-cotangent.

Note: Three times faster than building a (degenerated) lattice polytope and then checking if it subconvex.

◆ isKSubconvex() [3/3]

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::isKSubconvex	(	const RationalPolytope &	P,
		const CellGeometry &	C,
		const Dimension	k
	)		const

Tells if a given polytope P is digitally k-subconvex of some cell cover C. The digital 0-subconvexity is the usual property \( Conv( P \cap Z^d ) \subset C \cap Z^d) \). Otherwise the property asks that the k-cells intersected by the convex hull of P is a subset of the k-cells of C.

Parameters

P	any rational polytope such that `P.canBeSummed() == true`.
C	any cell cover geometry (i.e. a cubical complex).
k	the dimension for which the digital k-convexity is checked, 0 <= k <= KSpace::dimension.

Returns: 'true' iff the polytope P is a digitally k-subconvex of C.

◆ isSimplexFullDimensional() [1/2]

template<typename TKSpace >

template<typename PointIterator >

static bool DGtal::DigitalConvexity< TKSpace >::isSimplexFullDimensional	(	PointIterator	itB,
		PointIterator	itE
	)

static

Checks if the given range [itB,itE) of lattice points form a full dimensional simplex, i.e. it must contain Space::dimension+1 points in general position.

Template Parameters

PointIterator any model of forward iterator on Point.

Parameters

itB	the start of the range of n+1 points defining the simplex.
itE	past the end the range of n+1 points defining the simplex.

Referenced by main(), and SCENARIO().

◆ isSimplexFullDimensional() [2/2]

template<typename TKSpace >

static bool DGtal::DigitalConvexity< TKSpace >::isSimplexFullDimensional ( std::initializer_list< Point > l )

static

Checks if the given list of lattice points l form a full dimensional simplex, i.e. it must contain Space::dimension+1 points in general position.

Parameters

l	any list of d+1 points in general positions.

◆ isValid()

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::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.

◆ makeCellCover() [1/3]

template<typename TKSpace >

CellGeometry DGtal::DigitalConvexity< TKSpace >::makeCellCover	(	const LatticePolytope &	P,
		Dimension	i = `0`,
		Dimension	k = `KSpace::dimension`
	)		const

Builds the cell geometry containing all the j-cells touching the lattice polytope P, for i <= j <= k. It conbains thus all the j-cells intersecting the convex hull of P.

Parameters

P	any lattice polytope such that `P.canBeSummed() == true`.
i	the first dimension for which the cell cover is computed.
k	the last dimension for which the cell cover is computed.

◆ makeCellCover() [2/3]

template<typename TKSpace >

CellGeometry DGtal::DigitalConvexity< TKSpace >::makeCellCover	(	const RationalPolytope &	P,
		Dimension	i = `0`,
		Dimension	k = `KSpace::dimension`
	)		const

Builds the cell geometry containing all the j-cells touching the rational polytope P, for i <= j <= k. It conbains thus all the j-cells intersecting the convex hull of P.

Parameters

P	any rational polytope such that `P.canBeSummed() == true`.
i	the first dimension for which the cell cover is computed.
k	the last dimension for which the cell cover is computed.

◆ makeCellCover() [3/3]

template<typename TKSpace >

template<typename PointIterator >

CellGeometry DGtal::DigitalConvexity< TKSpace >::makeCellCover	(	PointIterator	itB,
		PointIterator	itE,
		Dimension	i = `0`,
		Dimension	k = `KSpace::dimension`
	)		const

Builds the cell geometry containing all the j-cells touching a point of [itB,itE), for i <= j <= k.

Template Parameters

PointIterator any model of input iterator on Points.

Parameters

itB	start of a range of arbitrary points.
itE	past the end of a range of arbitrary points.
i	the first dimension for which the cell cover is computed.
k	the last dimension for which the cell cover is computed.

Referenced by main(), and SCENARIO().

◆ makePolytope()

template<typename TKSpace >

LatticePolytope DGtal::DigitalConvexity< TKSpace >::makePolytope	(	const PointRange &	X,
		bool	make_minkowski_summable = `false`
	)		const

Given a range of distinct points X, computes the tightiest polytope that enclosed it. Note that this polytope may contain more lattice points than the given input points.

Parameters

	X	any range of pairwise distinct points
[in]	make_minkowski_summable	Other constraints are added so that we can perform axis aligned Minkowski sums on this polytope. Useful in 2D/3D for checking digital k-convexity (see moduleDigitalConvexity).

Returns: the corresponding lattice polytope.

Referenced by checkCvxHPlusHypercubeFullConvexity(), checkFullConvexityCharacterization(), DGtal::DigitalConvexity< TKSpace >::CvxH(), and SCENARIO().

◆ makeRationalSimplex() [1/2]

template<typename TKSpace >

template<typename PointIterator >

static RationalPolytope DGtal::DigitalConvexity< TKSpace >::makeRationalSimplex	(	Integer	d,
		PointIterator	itB,
		PointIterator	itE
	)

static

Constructs a rational polytope from a rational simplex given as a range [itB,itE) of lattice points. Note that the range must contain Space::dimension+1 points or less in general position.

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) }.

Referenced by SCENARIO().

◆ makeRationalSimplex() [2/2]

template<typename TKSpace >

static RationalPolytope DGtal::DigitalConvexity< TKSpace >::makeRationalSimplex ( std::initializer_list< Point > l )

static

Constructs a rational 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) }.

◆ makeSimplex() [1/2]

template<typename TKSpace >

template<typename PointIterator >

static LatticePolytope DGtal::DigitalConvexity< TKSpace >::makeSimplex	(	PointIterator	itB,
		PointIterator	itE
	)

static

Constructs a lattice polytope from a simplex given as a range [itB,itE) of lattice points. Note that the range must contain Space::dimension+1 points or less in general position.

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.

Referenced by main(), and SCENARIO().

◆ makeSimplex() [2/2]

template<typename TKSpace >

static LatticePolytope DGtal::DigitalConvexity< TKSpace >::makeSimplex ( std::initializer_list< Point > l )

static

Constructs a lattice polytope from a simplex given as an initializer_list.

Parameters

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

Precondition: Note that the list must contain no more than Space::dimension+1 points in general position.

◆ operator=()

template<typename TKSpace >

Self & DGtal::DigitalConvexity< TKSpace >::operator= ( const Self & other )

default

Assignment.

Parameters

other the object to copy.

Returns: a reference on 'this'.

◆ relativeEnvelope() [1/2]

template<typename TKSpace >

PointRange DGtal::DigitalConvexity< TKSpace >::relativeEnvelope	(	const PointRange &	Z,
		const PointRange &	Y,
		EnvelopeAlgorithm	algo = `EnvelopeAlgorithm::DIRECT`
	)		const

Computes the fully convex envelope of Z relative to fully convex digital set Y, i.e. \( FC^*_Y(Z):=FC_Y(FC_Y( \ldots FC_Y(Z) \ldots )) \) for Z a range of points, until stabilization of the iterative process.

Parameters

Z	any range of points (must be sorted).
Y	any range of points (must be sorted) that is fully convex.
algo	the chosen method of computation.

Returns: \( FC^*_Y( Z ) \)

Note: If Z is fully convex, then the output is Z itself. Otherwise, the returned set of points includes Z and is fully convex.

Referenced by main(), and SCENARIO().

◆ relativeEnvelope() [2/2]

template<typename TKSpace >

template<typename Predicate >

PointRange DGtal::DigitalConvexity< TKSpace >::relativeEnvelope	(	const PointRange &	Z,
		const Predicate &	PredY,
		EnvelopeAlgorithm	algo = `EnvelopeAlgorithm::DIRECT`
	)		const

Computes the fully convex envelope of Z relative to fully convex digital set Y defined by a corresponding predicate PredY. It computes \( FC^*_Y(Z):=FC_Y(FC_Y( \ldots FC_Y(Z) \ldots )) \) for Z a range of points, until stabilization of the iterative process.

Template Parameters

Predicate the type of a predicate Point -> boolean

Parameters

Z	any range of points (must be sorted).
PredY	a Point predicate such that `PredY(p)==true` iff p belongs to Y.
algo	the chosen method of computation.

Returns: \( FC^*_Y( Z ) \)

Note: If Z is fully convex, then the output is Z itself. Otherwise, the returned set of points includes Z and is fully convex.

◆ selfDisplay()

template<typename TKSpace >

void DGtal::DigitalConvexity< TKSpace >::selfDisplay ( std::ostream & out ) const

Writes/Displays the object on an output stream.

Parameters

out	the output stream where the object is written.

◆ simplexType() [1/2]

template<typename TKSpace >

template<typename PointIterator >

static SimplexType DGtal::DigitalConvexity< TKSpace >::simplexType	(	PointIterator	itB,
		PointIterator	itE
	)

static

Returns the type of simplex formed by the given range [itB,itE) of lattice points.

Template Parameters

PointIterator any model of forward iterator on Point.

Parameters

itB	the start of the range of n+1 points defining the simplex.
itE	past the end the range of n+1 points defining the simplex.

Returns: the type of simplex formed by the given range [itB,itE) of lattice points.

Referenced by SCENARIO().

◆ simplexType() [2/2]

template<typename TKSpace >

static SimplexType DGtal::DigitalConvexity< TKSpace >::simplexType ( std::initializer_list< Point > l )

static

Returns the type of simplex formed by the given list l of lattice points.

Parameters

l	any list of lattice points.

Returns: the type of simplex formed by the given list of lattice points.

◆ sizeStarCvxH()

template<typename TKSpace >

Integer DGtal::DigitalConvexity< TKSpace >::sizeStarCvxH ( const PointRange & X ) const

Computes the number of cells in Star(CvxH(X)) for X a digital set.

Parameters

X	any range of lattice points

Returns: the number of cells touching the convex hull of X, represented as lattice points with Khalimsky coordinates.

◆ Skel()

template<typename TKSpace >

LatticeSet DGtal::DigitalConvexity< TKSpace >::Skel ( const LatticeSet & C ) const

Parameters

C	a range of cells represented as a lattice set.

Returns: the set of cells, represented as a lattice set, that form the skeleton of the given range of cells C.

◆ space()

template<typename TKSpace >

const KSpace & DGtal::DigitalConvexity< TKSpace >::space ( ) const

Returns: a const reference to the cellular grid space used by this object.

Referenced by DGtal::NeighborhoodConvexityAnalyzer< TKSpace, K >::NeighborhoodConvexityAnalyzer(), and DGtal::NeighborhoodConvexityAnalyzer< TKSpace, K >::space().

◆ Star()

template<typename TKSpace >

LatticeSet DGtal::DigitalConvexity< TKSpace >::Star	(	const PointRange &	X,
		Dimension	axis = `dimension`
	)		const

Builds the cell complex Star(X) for X a digital set, represented as a lattice set (stacked row representation).

Parameters

X	any range of lattice points
axis	specifies the projection axis for the row representation if below space dimension, otherwise chooses the axis that minimizes memory/computations.

Returns: the set of cells, represented as a lattice set, that touches points of X, i.e. Star(X).

Note: It is useful to specify an axis if you wish later to compare or make operations with several lattice sets. They must indeed have the same axis.

◆ StarCells()

template<typename TKSpace >

LatticeSet DGtal::DigitalConvexity< TKSpace >::StarCells	(	const PointRange &	C,
		Dimension	axis = `dimension`
	)		const

Builds the cell complex Star(C) for C a range of cells, represented as a lattice set (stacked row representation).

Parameters

C	a range of cells represented with points in Khalimsky coordinates.
axis	specifies the projection axis for the row representation if below space dimension, otherwise chooses the axis that minimizes memory/computations.

Returns: the set of cells, represented as a lattice set, that touches cells of C, i.e. Star(C).

Note: It is useful to specify an axis if you wish later to compare or make operations with several lattice sets. They must indeed have the same axis.

◆ StarCvxH()

template<typename TKSpace >

LatticeSet DGtal::DigitalConvexity< TKSpace >::StarCvxH	(	const PointRange &	X,
		Dimension	axis = `dimension`
	)		const

Builds the cell complex Star(CvxH(X)) for X a digital set, represented as a lattice set (stacked row representation).

Parameters

X	any range of lattice points
axis	specifies the projection axis for the row representation if below space dimension, otherwise chooses the axis that minimizes memory/computations.

Returns: the range of cells touching the convex hull of X, represented as a lattice set (cells are represented with Khalimsky coordinates).

Note: It is useful to specify an axis if you wish later to compare or make operations with several lattice sets. They must indeed have the same axis.

Referenced by checkSkelStarCvxHFullConvexity(), and SCENARIO().

◆ toLatticeSet()

template<typename TKSpace >

LatticeSet DGtal::DigitalConvexity< TKSpace >::toLatticeSet	(	const PointRange &	X,
		Dimension	axis = `dimension`
	)		const

Builds the lattice set (stacked row representation) associated to the given range of points.

Parameters

X	any range of lattice points
axis	specifies the projection axis for the row representation if below space dimension, otherwise chooses the axis that minimizes memory/computations.

Returns: the lattice set that represents the exact same points as X

Note: It is useful to specify an axis if you wish later to compare or make operations with several lattice sets. They must indeed have the same axis.

◆ toPointRange()

template<typename TKSpace >

PointRange DGtal::DigitalConvexity< TKSpace >::toPointRange ( const LatticeSet & L ) const

Builds the range of lattice points associated to the given lattice set.

Parameters

L	any lattice set.

Returns: the point range that represents the exact same points as L

◆ U()

template<typename TKSpace >

PointRange DGtal::DigitalConvexity< TKSpace >::U	(	Dimension	i,
		const PointRange &	X
	)		const

Performs the digital Minkowski sum of X along direction i

Parameters

i	any valid dimension
X	any sorted range of digital points

Returns: the sorted range of digital points X union the translation of X of one along direction i.

Referenced by checkCvxHPlusHypercubeFullConvexity().

Field Documentation

◆ dimension

template<typename TKSpace >

const Dimension DGtal::DigitalConvexity< TKSpace >::dimension = KSpace::dimension

static

Definition at line 100 of file DigitalConvexity.h.

◆ myDepthLastFCE

template<typename TKSpace >

Size DGtal::DigitalConvexity< TKSpace >::myDepthLastFCE

mutableprotected

The number of iterations of the last FullyConvexEnvelope operation.

Definition at line 880 of file DigitalConvexity.h.

◆ myK

template<typename TKSpace >

KSpace DGtal::DigitalConvexity< TKSpace >::myK

protected

The cellular grid space where computations are done.

Definition at line 872 of file DigitalConvexity.h.

◆ mySafe

template<typename TKSpace >

bool DGtal::DigitalConvexity< TKSpace >::mySafe

protected

when 'true' performs convex hull computations with arbitrary precision integer (if available), otherwise chooses a compromise between speed and precision (int64_t).

Definition at line 877 of file DigitalConvexity.h.

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

DigitalConvexity.h

Public Types

Public Member Functions

Static Public Member Functions

Static Public Attributes

Protected Attributes

Private Member Functions

Static Private Member Functions

Simplex services

Convex envelope services

Detailed Description

Member Typedef Documentation

◆ Cell

◆ CellGeometry

◆ Counter

◆ Integer

◆ Interval

◆ KSpace

◆ LatticePolytope

◆ LatticeSet

◆ Point

◆ PointRange

◆ PointSet

◆ Polytope

◆ RationalPolytope

◆ Self

◆ Size

◆ Space

◆ Vector

Member Enumeration Documentation

◆ EnvelopeAlgorithm

◆ SimplexType

Constructor & Destructor Documentation

◆ ~DigitalConvexity()

◆ DigitalConvexity() [1/4]

◆ DigitalConvexity() [2/4]

◆ DigitalConvexity() [3/4]

◆ DigitalConvexity() [4/4]

Member Function Documentation

◆ BOOST_CONCEPT_ASSERT()

◆ CvxH()

◆ depthLastEnvelope()

◆ displaySimplex() [1/2]

◆ displaySimplex() [2/2]

◆ envelope()

◆ eraseInterval()

◆ Extr() [1/2]

◆ Extr() [2/2]

◆ ExtrCvxH()

◆ ExtrSkel()

◆ FC()

◆ FC_direct()

◆ FC_LatticeSet()

◆ filter()

◆ insidePoints() [1/2]

◆ insidePoints() [2/2]

◆ interiorPoints() [1/2]

◆ interiorPoints() [2/2]

◆ is0Convex()

◆ isFullyConvex() [1/3]

◆ isFullyConvex() [2/3]

◆ isFullyConvex() [3/3]

◆ isFullyConvexFast()

◆ isFullySubconvex() [1/6]

◆ isFullySubconvex() [2/6]

◆ isFullySubconvex() [3/6]

◆ isFullySubconvex() [4/6]

◆ isFullySubconvex() [5/6]

◆ isFullySubconvex() [6/6]

◆ isKConvex() [1/2]

◆ isKConvex() [2/2]

◆ isKSubconvex() [1/3]

◆ isKSubconvex() [2/3]

◆ isKSubconvex() [3/3]

◆ isSimplexFullDimensional() [1/2]

◆ isSimplexFullDimensional() [2/2]

◆ isValid()

◆ makeCellCover() [1/3]

◆ makeCellCover() [2/3]

◆ makeCellCover() [3/3]

◆ makePolytope()