Aim: a geometric kernel to compute the convex hull of floating points with integer-only arithmetic. Floating points are approximated with rational points with fixed precision (a given number of bits). All remaining computations are exact, as long as there is no overflow. More...

#include <DGtal/geometry/tools/QuickHullKernels.h>

Inheritance diagram for DGtal::ConvexHullRationalKernel< dim, TCoordinateInteger, TInternalInteger >:

[legend]

Public Types
typedef ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >	Base

typedef DGtal::PointVector< dim, CoordinateInteger >	CoordinatePoint

typedef DGtal::PointVector< dim, CoordinateInteger >	CoordinateVector

typedef CoordinateInteger	CoordinateScalar

typedef DGtal::PointVector< dim, InternalInteger >	InternalPoint

typedef DGtal::PointVector< dim, InternalInteger >	InternalVector

typedef InternalInteger	InternalScalar

typedef std::size_t	Size

typedef Size	Index

typedef std::vector< Index >	IndexRange

typedef std::array< Index, dim >	CombinatorialPlaneSimplex

Public Types inherited from DGtal::ConvexHullCommonKernel< dim, DGtal::int64_t, DGtal::int64_t >
typedef DGtal::int64_t	CoordinateInteger

typedef DGtal::int64_t	InternalInteger

typedef CoordinateInteger	CoordinateScalar

typedef InternalInteger	InternalScalar

typedef DGtal::PointVector< dim, CoordinateInteger >	CoordinatePoint

typedef DGtal::PointVector< dim, CoordinateInteger >	CoordinateVector

typedef DGtal::PointVector< dim, InternalInteger >	InternalPoint

typedef DGtal::PointVector< dim, InternalInteger >	InternalVector

typedef std::size_t	Size

typedef Size	Index

typedef std::vector< Index >	IndexRange

typedef std::array< Index, dim >	CombinatorialPlaneSimplex

typedef IntegerConverter< dim, CoordinateInteger >	Outer
	Converter to outer coordinate integers or lattice points / vector.

typedef IntegerConverter< dim, InternalInteger >	Inner
	Converter to inner internal integers or lattice points / vector.

Public Member Functions
	ConvexHullRationalKernel (double aPrecision=1024.)

bool	hasInfiniteFacets () const

bool	isHalfSpaceFacetInfinite (const HalfSpace &hs) const

template<typename InputPoint >
void	makeInput (std::vector< CoordinatePoint > &processed_points, IndexRange &input2comp, IndexRange &comp2input, const std::vector< InputPoint > &input_points, bool remove_duplicates)

template<typename OutputPoint >
void	convertPointTo (const CoordinatePoint &p, OutputPoint &out_p) const

HalfSpace	compute (const std::vector< CoordinatePoint > &vpoints, const CombinatorialPlaneSimplex &simplex, Index idx_below)

HalfSpace	compute (const std::vector< CoordinatePoint > &vpoints, const CombinatorialPlaneSimplex &simplex)

CoordinateVector	normal (const HalfSpace &H) const

CoordinateScalar	intercept (const HalfSpace &H) const

InternalScalar	dot (const HalfSpace &H1, const HalfSpace &H2) const

bool	equal (const HalfSpace &H1, const HalfSpace &H2) const

InternalScalar	height (const HalfSpace &H, const CoordinatePoint &p) const

InternalScalar	volume (const HalfSpace &H, const CoordinatePoint &p) const

bool	above (const HalfSpace &H, const CoordinatePoint &p) const

bool	aboveOrOn (const HalfSpace &H, const CoordinatePoint &p) const

bool	on (const HalfSpace &H, const CoordinatePoint &p) const

Public Member Functions inherited from DGtal::ConvexHullCommonKernel< dim, DGtal::int64_t, DGtal::int64_t >
	BOOST_CONCEPT_ASSERT ((concepts::CInteger< DGtal::int64_t >))

	BOOST_CONCEPT_ASSERT ((concepts::CInteger< DGtal::int64_t >))

	ConvexHullCommonKernel ()=default
	Default constructor.

HalfSpace	compute (const std::vector< CoordinatePoint > &vpoints, const CombinatorialPlaneSimplex &simplex, Index idx_below)

HalfSpace	compute (const std::vector< CoordinatePoint > &vpoints, const CombinatorialPlaneSimplex &simplex)

CoordinateVector	normal (const HalfSpace &H) const

CoordinateScalar	intercept (const HalfSpace &H) const

InternalScalar	dot (const HalfSpace &H1, const HalfSpace &H2) const

bool	equal (const HalfSpace &H1, const HalfSpace &H2) const

InternalScalar	height (const HalfSpace &H, const CoordinatePoint &p) const

InternalScalar	volume (const HalfSpace &H, const CoordinatePoint &p) const

bool	above (const HalfSpace &H, const CoordinatePoint &p) const

bool	aboveOrOn (const HalfSpace &H, const CoordinatePoint &p) const

bool	on (const HalfSpace &H, const CoordinatePoint &p) const

Data Fields
double	precision
	The precision as the common denominator for all rational points.

Static Public Attributes
static const Dimension	dimension

Static Public Attributes inherited from DGtal::ConvexHullCommonKernel< dim, DGtal::int64_t, DGtal::int64_t >
static const Dimension	dimension

Detailed Description

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>
struct DGtal::ConvexHullRationalKernel< dim, TCoordinateInteger, TInternalInteger >

Aim: a geometric kernel to compute the convex hull of floating points with integer-only arithmetic. Floating points are approximated with rational points with fixed precision (a given number of bits). All remaining computations are exact, as long as there is no overflow.

Description of template class 'ConvexHullRationalKernel'

Each floating point input coordinate x is converted to an integer through the following formula (Integer) round( x * precision ), where precision is the floating point value given at instanciation of the kernel.

Each output floating point coordinate is built from integer coordinates a through the formula ( (double) a ) / precision, where precision is the floating point value given at instanciation of the kernel.

See also: QuickHull algorithm in arbitrary dimension for convex hull and Delaunay cell complex computation

Template Parameters

dim	the dimension of the space of processed points.
TCoordinateInteger	the integer type that represents coordinates of lattice points, a model of concepts::CInteger.
TInternalInteger	the integer type that is used for internal computations of above/below plane tests, a model of concepts::CInteger. Must be at least as precise as TCoordinateInteger.

Examples: geometry/tools/exampleRationalBallQuickHull3D.cpp.

Definition at line 657 of file QuickHullKernels.h.

Member Typedef Documentation

◆ Base

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger > DGtal::ConvexHullRationalKernel< dim, TCoordinateInteger, TInternalInteger >::Base

Definition at line 660 of file QuickHullKernels.h.

◆ CombinatorialPlaneSimplex

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef std::array< Index, dim > DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::CombinatorialPlaneSimplex

Definition at line 195 of file QuickHullKernels.h.

◆ CoordinatePoint

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef DGtal::PointVector< dim, CoordinateInteger > DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::CoordinatePoint

Definition at line 188 of file QuickHullKernels.h.

◆ CoordinateScalar

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef CoordinateInteger DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::CoordinateScalar

Definition at line 184 of file QuickHullKernels.h.

◆ CoordinateVector

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef DGtal::PointVector< dim, CoordinateInteger > DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::CoordinateVector

Definition at line 189 of file QuickHullKernels.h.

◆ Index

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef Size DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::Index

Definition at line 193 of file QuickHullKernels.h.

◆ IndexRange

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef std::vector< Index > DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::IndexRange

Definition at line 194 of file QuickHullKernels.h.

◆ InternalPoint

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef DGtal::PointVector< dim, InternalInteger > DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::InternalPoint

Definition at line 190 of file QuickHullKernels.h.

◆ InternalScalar

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef InternalInteger DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::InternalScalar

Definition at line 185 of file QuickHullKernels.h.

◆ InternalVector

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef DGtal::PointVector< dim, InternalInteger > DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::InternalVector

Definition at line 191 of file QuickHullKernels.h.

◆ Size

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

typedef std::size_t DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::Size

Definition at line 192 of file QuickHullKernels.h.

Constructor & Destructor Documentation

◆ ConvexHullRationalKernel()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

DGtal::ConvexHullRationalKernel< dim, TCoordinateInteger, TInternalInteger >::ConvexHullRationalKernel ( double aPrecision = 1024. )

inline

Constructor with specified precision

Parameters

[in] aPrecision the chosen precision as the common denominator of all rationals (by defaut, 1024).

Definition at line 697 of file QuickHullKernels.h.

698 : precision( aPrecision ) {}

DGtal::ConvexHullRationalKernel::precision

double precision

The precision as the common denominator for all rational points.

Definition QuickHullKernels.h:691

Member Function Documentation

◆ above()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

bool DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::above	(	const HalfSpace &	H,
		const CoordinatePoint &	p ) const

inline

Parameters

H	the half-space
p	any point

Returns: 'true' iff p is strictly above this plane (so in direction N ).

Definition at line 333 of file QuickHullKernels.h.

334 { return height( H, p ) > 0; }

DGtal::ProbingMode::H

@ H

◆ aboveOrOn()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

bool DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::aboveOrOn	(	const HalfSpace &	H,
		const CoordinatePoint &	p ) const

inline

Parameters

H	the half-space
p	any point

Returns: 'true' iff p is above or lies on this plane (so in direction N ).

Definition at line 339 of file QuickHullKernels.h.

340 { return height( H, p ) >= 0; }

◆ compute() [1/2]

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

HalfSpace DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::compute	(	const std::vector< CoordinatePoint > &	vpoints,
		const CombinatorialPlaneSimplex &	simplex )

inline

Computes an halfspace from dimension points specified by simplex with vertices in a range vpoints of Point. Orientation is induced by the order of the points. If the simplex is degenrated, the half-space is invalid and has null normal.

Parameters

[in]	vpoints	a range of points over which the simplex is defined.
[in]	simplex	a range of dimension indices of points defining an hyperplane.

Returns: the corresponding halfspace (has null normal vector if simplex was not full dimensional)

Definition at line 257 of file QuickHullKernels.h.

    {
      typedef DGtal::SimpleMatrix< InternalScalar, dimension, dimension > Matrix;
      Matrix A;
      InternalVector N;
      for ( Dimension i = 1; i < dimension; i++ )
        for ( Dimension j = 0; j < dimension; j++ )
          A.setComponent( i-1, j,
                          Inner::cast( vpoints[ simplex[ i ] ][ j ]
                                       - vpoints[ simplex[ 0 ] ][ j ] ) );
      for ( Dimension j = 0; j < dimension; j++ )
        N[ j ] = A.cofactor( dimension-1, j );
      const InternalPoint ip = Inner::cast( vpoints[ simplex[ 0 ] ] );
      // c = N.dot( vpoints[ simplex[ 0 ] ] );
      return HalfSpace { N, N.dot( ip ) };
    }

◆ compute() [2/2]

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

HalfSpace DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::compute	(	const std::vector< CoordinatePoint > &	vpoints,
		const CombinatorialPlaneSimplex &	simplex,
		Index	idx_below )

inline

Computes an halfspace from dimension points specified by simplex with vertices in a range vpoints of Point. It is oriented such that the point of index idx_below is included in the half-space (i.e. below).

Parameters

[in]	vpoints	a range of points over which the simplex is defined.
[in]	simplex	a range of dimension indices of points defining an hyperplane.
[in]	idx_below	the index of a p-oint that is below the hyperplane.

Returns: the corresponding halfspace (has null normal vector if simplex was not full dimensional)

Definition at line 230 of file QuickHullKernels.h.

    {
      HalfSpace hs = compute( vpoints, simplex );
      if ( hs.N != InternalVector::zero ) 
        {
          const InternalPoint  ip = Inner::cast( vpoints[ idx_below ] );
          const InternalScalar nu = hs.N.dot( ip );
          //const Scalar nu = hs.N.dot( vpoints[ idx_below ] );
          if ( nu > hs.c ) { hs.N = -hs.N; hs.c = -hs.c; }
        }
      return hs;
    }

◆ convertPointTo()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

template<typename OutputPoint >

void DGtal::ConvexHullRationalKernel< dim, TCoordinateInteger, TInternalInteger >::convertPointTo	(	const CoordinatePoint &	p,
		OutputPoint &	out_p ) const

inline

Converts an integral point (as represented internally for QuickHull computations) to its corresponding output point representation.

Template Parameters

OutputPoint a model of point such that processing type Point is convertible to it.

Parameters

[in]	p	an integral point (as represented internally for QuickHull computations)
[out]	out_p	its corresponding output point representation.

Note: Each output floating point coordinate is built from integer coordinates a through the formula ( (double) a ) / precision, where precision is the floating point value given at instanciation of the kernel.

Definition at line 781 of file QuickHullKernels.h.

    {
      for ( Dimension k = 0; k < dimension; k++ )
        out_p[ k ] = ( (double) p[ k ] ) / precision;
    }

References DGtal::ConvexHullRationalKernel< dim, TCoordinateInteger, TInternalInteger >::dimension, and DGtal::ConvexHullRationalKernel< dim, TCoordinateInteger, TInternalInteger >::precision.

◆ dot()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

InternalScalar DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::dot	(	const HalfSpace &	H1,
		const HalfSpace &	H2 ) const

inline

Equivalent of the dot product of the normals of the half-spaces.

Parameters

H1	an half-space
H2	an half-space

Returns: a positive scalar if both half-spaces points to to the same hemisphere, a negative scalar if they point to opposite hemispheres, and zero if they are orthogonal.

Definition at line 297 of file QuickHullKernels.h.

    {
      return H1.N.dot( H2.N );
    }

◆ equal()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

bool DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::equal	(	const HalfSpace &	H1,
		const HalfSpace &	H2 ) const

inline

Parameters

H1	an half-space
H2	an half-space

Returns: 'true' if the half-spaces have the smae members.

Note: two half-spaces may be not equal but may represent the same set of points. For instance H1={{1,0},3} and H1={{2,0},6}.

Definition at line 310 of file QuickHullKernels.h.

    {
      return H1.c == H2.c && H1.N == H2.N;
    }

◆ hasInfiniteFacets()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

bool DGtal::ConvexHullRationalKernel< dim, TCoordinateInteger, TInternalInteger >::hasInfiniteFacets ( ) const

inline

Returns: 'true' if a kernel may induce infinite facets. This is typically the case of a Delaunay computation kernel, which casts points in higher dimension.

Definition at line 703 of file QuickHullKernels.h.

704 { return false; }

◆ height()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

InternalScalar DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::height	(	const HalfSpace &	H,
		const CoordinatePoint &	p ) const

inline

Parameters

H	the half-space
p	any point

Returns: the (signed) height of p wrt this plane.

Definition at line 318 of file QuickHullKernels.h.

319 { return H.N.dot( Inner::cast( p ) ) - H.c; }

◆ intercept()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

CoordinateScalar DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::intercept ( const HalfSpace & H ) const

inline

Parameters

H	the half-space

Returns: the intercept of this facet.

Definition at line 284 of file QuickHullKernels.h.

    {
      return Outer::cast( H.c );
    }

◆ isHalfSpaceFacetInfinite()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

bool DGtal::ConvexHullRationalKernel< dim, TCoordinateInteger, TInternalInteger >::isHalfSpaceFacetInfinite ( const HalfSpace & hs ) const

inline

Parameters

[in] hs an half-space corresponding to a facet.

Returns: 'true' if the facet associated to this half-space corresponds to an infinite facet.

Definition at line 710 of file QuickHullKernels.h.

    {
      (void) hs; // unused parameter      
      return false;
    }

◆ makeInput()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

template<typename InputPoint >

void DGtal::ConvexHullRationalKernel< dim, TCoordinateInteger, TInternalInteger >::makeInput	(	std::vector< CoordinatePoint > &	processed_points,
		IndexRange &	input2comp,
		IndexRange &	comp2input,
		const std::vector< InputPoint > &	input_points,
		bool	remove_duplicates )

inline

Transforms a range input_points of input points to a range processed_points of points adapted to a processing by QuickHull convex hull algorithm. Keep the mapping information between input and processed points.

Template Parameters

InputPoint any model of point whose components are convertible to Scalar.

Parameters

[out]	processed_points	the range of points prepared for a process by QuickHull.
[out]	input2comp	the surjective mapping between the input_points range and the processed_points range used for computation.
[out]	comp2input	the injective mapping between the processed_points range used for computation and the input_points range.
[in]	input_points	the range of input points.
[in]	remove_duplicates	when 'true', this method removes possible duplicates in input_points and processed_points may thus be of smaller size, otherwise, when 'false', it means that there are no duplicates in input_points.

Note: Each floating point input coordinate x is converted to an integer through the following formula (Integer) round( x * precision ), where precision is the floating point value given at instanciation of the kernel.

Definition at line 747 of file QuickHullKernels.h.

    {
      const auto F = [&] ( InputPoint input ) -> CoordinatePoint
        {
          CoordinatePoint p;
          for ( Dimension i = 0; i < dimension; i++ )
            p[ i ] = CoordinateScalar( round( input[ i ] * precision ) );
          return p;
        };
      DGtal::detail::transform( processed_points, input2comp, comp2input,
                                input_points.cbegin(), input_points.cend(),
                                F, remove_duplicates );
    }

References DGtal::ConvexHullRationalKernel< dim, TCoordinateInteger, TInternalInteger >::dimension, and DGtal::detail::transform().

◆ normal()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

CoordinateVector DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::normal ( const HalfSpace & H ) const

inline

Parameters

H	the half-space

Returns: the normal to this facet.

Definition at line 277 of file QuickHullKernels.h.

    {
      return Outer::cast( H.N );
    }

◆ on()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

bool DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::on	(	const HalfSpace &	H,
		const CoordinatePoint &	p ) const

inline

Parameters

H	the half-space
p	any point

Returns: 'true' iff p lies on this plane.

Definition at line 345 of file QuickHullKernels.h.

346 { return height( H, p ) == 0; }

◆ volume()

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

InternalScalar DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::volume	(	const HalfSpace &	H,
		const CoordinatePoint &	p ) const

inline

Parameters

H	the half-space
p	any point

Returns: the volume of the vectors spanned by the simplex and this point.

Definition at line 324 of file QuickHullKernels.h.

    {
      InternalScalar v = height( H, p );
      return v < InternalScalar( 0 ) ? -v : v;
    }

Field Documentation

◆ dimension

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

const Dimension DGtal::ConvexHullCommonKernel< dim, TCoordinateInteger, TInternalInteger >::dimension

static

Definition at line 196 of file QuickHullKernels.h.

Referenced by DGtal::ConvexHullRationalKernel< dim, TCoordinateInteger, TInternalInteger >::convertPointTo(), and DGtal::ConvexHullRationalKernel< dim, TCoordinateInteger, TInternalInteger >::makeInput().

◆ precision

template<Dimension dim, typename TCoordinateInteger = DGtal::int64_t, typename TInternalInteger = DGtal::int64_t>

double DGtal::ConvexHullRationalKernel< dim, TCoordinateInteger, TInternalInteger >::precision

The precision as the common denominator for all rational points.

Definition at line 691 of file QuickHullKernels.h.

Referenced by DGtal::ConvexHullRationalKernel< dim, TCoordinateInteger, TInternalInteger >::convertPointTo().

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

QuickHullKernels.h

Public Types

Public Member Functions

Data Fields

Static Public Attributes

Detailed Description

Member Typedef Documentation

◆ Base

◆ CombinatorialPlaneSimplex

◆ CoordinatePoint

◆ CoordinateScalar

◆ CoordinateVector

◆ Index

◆ IndexRange

◆ InternalPoint

◆ InternalScalar

◆ InternalVector

◆ Size

Constructor & Destructor Documentation

◆ ConvexHullRationalKernel()

Member Function Documentation

◆ above()

◆ aboveOrOn()

◆ compute() [1/2]

◆ compute() [2/2]

◆ convertPointTo()

◆ dot()

◆ equal()

◆ hasInfiniteFacets()

◆ height()

◆ intercept()

◆ isHalfSpaceFacetInfinite()

◆ makeInput()

◆ normal()

◆ on()

◆ volume()

Field Documentation

◆ dimension

◆ precision