DGtal::functions::Hull2D Namespace Reference

Hull2D namespace gathers useful functions to compute and return the convex hull or the alpha-shape of a range of 2D points. More...

Enumerations
enum	ThicknessDefinition { HorizontalVerticalThickness , EuclideanThickness }
	The 2 thickness definitions. More...

Functions
template<typename Stack , typename Point , typename Predicate >
void	updateHullWithStack (Stack &aStack, const Point &aNewPoint, const Predicate &aPredicate)
	Procedure that updates the hull when an extra point aNewPoint is considered: while the last three consecutive points, ie. aNewPoint and the last two points of the container are not oriented such that the predicate aPredicate returns 'true', the last point of the container is removed. More...
template<typename Stack , typename Point , typename Predicate >
void	updateHullWithAdaptedStack (Stack aStack, const Point &aNewPoint, const Predicate &aPredicate)
	Procedure that calls Hull2D::updateHullWithStack on a copy of the stack object used to retrieved the hull vertices. Useful when the first argument is a stack adapter returned by a function: it must be copied before being passed by reference in Hull2D::updateHullWithStack. More...
template<typename Stack , typename ForwardIterator , typename Predicate >
void	buildHullWithStack (Stack &aStack, const ForwardIterator &itb, const ForwardIterator &ite, const Predicate &aPredicate)
	Procedure that retrieves the vertices of the convex hull of a weakly externally visible polygon (WEVP) in linear-time. This technique is called Sklansky's scan, Graham's scan or 3-coins algorithm. It works for all WEVP [Toussaint and Avis, 1982 : [118]]. More...
template<typename Stack , typename ForwardIterator , typename Predicate >
void	buildHullWithAdaptedStack (Stack aStack, const ForwardIterator &itb, const ForwardIterator &ite, const Predicate &aPredicate)
	Procedure that calls Hull2D::buildHullWithStack on a copy of the stack object used to retrieved the hull vertices. Useful when the first argument is a stack adapter returned by a function: it must be copied before being passed by reference in Hull2D::buildHullWithStack. More...
template<typename ForwardIterator , typename OutputIterator , typename Predicate >
void	openGrahamScan (const ForwardIterator &itb, const ForwardIterator &ite, OutputIterator res, const Predicate &aPredicate)
	Procedure that retrieves the vertices of the convex hull of a weakly externally visible polygon (WEVP) in linear-time. More...
template<typename ForwardIterator , typename OutputIterator , typename Predicate >
void	closedGrahamScanFromVertex (const ForwardIterator &itb, const ForwardIterator &ite, OutputIterator res, const Predicate &aPredicate)
	Procedure that retrieves the vertices of the convex hull of a weakly externally visible polygon (WEVP) in linear-time. More...
template<typename ForwardIterator , typename OutputIterator , typename Predicate >
void	closedGrahamScanFromAnyPoint (const ForwardIterator &itb, const ForwardIterator &ite, OutputIterator res, const Predicate &aPredicate)
	Procedure that retrieves the vertices of the convex hull of a weakly externally visible polygon (WEVP) in linear-time. More...
template<typename ForwardIterator , typename OutputIterator , typename Predicate , typename PolarComparator >
void	grahamConvexHullAlgorithm (const ForwardIterator &itb, const ForwardIterator &ite, OutputIterator res, const Predicate &aPredicate, PolarComparator &aPolarComparator)
	Procedure that retrieves the vertices of the convex hull of a set of 2D points given by the range [ itb , ite ). This procedure follows the well-known Graham's algorithm [Graham, 1972 : [63]]. More...
template<typename ForwardIterator , typename OutputIterator , typename Predicate , typename PolarComparator >
void	grahamConvexHullAlgorithm (const ForwardIterator &itb, const ForwardIterator &ite, OutputIterator res, const Predicate &aPredicate)
	Procedure that retrieves the vertices of the convex hull of a set of 2D points given by the range [ itb , ite ). This procedure follows the well-known Graham's algorithm [Graham, 1972 : [63]]. More...
template<typename ForwardIterator , typename OutputIterator , typename Predicate >
void	andrewConvexHullAlgorithm (const ForwardIterator &itb, const ForwardIterator &ite, OutputIterator res, const Predicate &aPredicate)
	Procedure that retrieves the vertices of the hull of a set of 2D points given by the range [ itb , ite ). This procedure follows the well-known monotone-chain algorithm due to [Andrew, 1979 : [6]]. More...
template<typename ForwardIterator >
double	computeHullThickness (const ForwardIterator &itb, const ForwardIterator &ite, const ThicknessDefinition &def)
	Procedure to compute the convex hull thickness given from different definitions (Horizontal/vertical or Euclidean distances). It takes as input the vertices of the hull given by the range [itbn, ite). The procedure applies the classic rotating caliper to recover all anti-podal pairs. More...
template<typename ForwardIterator , typename TInputPoint >
double	computeHullThickness (const ForwardIterator &itb, const ForwardIterator &ite, const ThicknessDefinition &def, TInputPoint &antipodalEdgeP, TInputPoint &antipodalEdgeQ, TInputPoint &antipodalVertexR)
	Procedure to compute the convex hull thickness given from different definitions (Horizontal/vertical or Euclidean distances). It takes as input the vertices of the hull given by the range [itbn, ite). The procedure applies the classic rotating caliper to recover all anti-podal pairs. More...
template<typename TInputPoint >
double	getAngle (const TInputPoint &a, const TInputPoint &b, const TInputPoint &c, const TInputPoint &d)
template<typename TInputPoint >
bool	isCoLinearOpp (const TInputPoint &a, const TInputPoint &b, const TInputPoint &c, const TInputPoint &d)
template<typename TInputPoint >
double	getThicknessAntipodalPair (const TInputPoint &p, const TInputPoint &q, const TInputPoint &r, const ThicknessDefinition &def)
template<typename TInputPoint >
double	computeHProjDistance (const TInputPoint &a, const TInputPoint &b, const TInputPoint &c, bool &isInside)
template<typename TInputPoint >
double	computeVProjDistance (const TInputPoint &a, const TInputPoint &b, const TInputPoint &c, bool &isInside)
template<typename TInputPoint >
double	computeEuclideanDistance (const TInputPoint &a, const TInputPoint &b, const TInputPoint &c, bool &isInside)
template<typename ForwardIterator , typename OutputIterator , typename Functor >
void	melkmanConvexHullAlgorithm (const ForwardIterator &itb, const ForwardIterator &ite, OutputIterator res, Functor &aFunctor)
	Procedure that retrieves the vertices of the hull of a set of 2D points given by the range [ itb , ite ). This procedure follows the well-known Melkman algorithm [Melkman, 1979 : [90]]. More...

Variables
constexpr double	angleTolerance = 1e-6

Detailed Description

Hull2D namespace gathers useful functions to compute and return the convex hull or the alpha-shape of a range of 2D points.

Enumeration Type Documentation

ThicknessDefinition

enum DGtal::functions::Hull2D::ThicknessDefinition

The 2 thickness definitions.

Enumerator
HorizontalVerticalThickness
EuclideanThickness

Definition at line 78 of file Hull2DHelpers.h.

{HorizontalVerticalThickness, EuclideanThickness};

Function Documentation

andrewConvexHullAlgorithm()

template<typename ForwardIterator , typename OutputIterator , typename Predicate >

void DGtal::functions::Hull2D::andrewConvexHullAlgorithm	(	const ForwardIterator &	itb,
		const ForwardIterator &	ite,
		OutputIterator	res,
		const Predicate &	aPredicate
	)

Procedure that retrieves the vertices of the hull of a set of 2D points given by the range [ itb , ite ). This procedure follows the well-known monotone-chain algorithm due to [Andrew, 1979 : [6]].

• first, points are sorted along the horizontal axis.
• then, the lower and upper convex hull are computed by a simple Graham scan.

  See also

  Hull2D::openGrahamScan

  Postcondition

  The first point of the resulting list of extremal points follows the one with minimal x-coordinate and y-coordinate. Orientation depends on the predicate.

  Parameters

  itb	begin iterator
  ite	end iterator
  res	output iterator used to export the retrieved points
  aPredicate	any ternary predicate

  Template Parameters

  ForwardIterator	a model of forward and readable iterator
  OutputIterator	a model of incrementable and writable iterator
  Predicate	a model of ternary predicate

buildHullWithAdaptedStack()

template<typename Stack , typename ForwardIterator , typename Predicate >

void DGtal::functions::Hull2D::buildHullWithAdaptedStack	(	Stack	aStack,
		const ForwardIterator &	itb,
		const ForwardIterator &	ite,
		const Predicate &	aPredicate
	)

Procedure that calls Hull2D::buildHullWithStack on a copy of the stack object used to retrieved the hull vertices. Useful when the first argument is a stack adapter returned by a function: it must be copied before being passed by reference in Hull2D::buildHullWithStack.

Parameters

aStack	stack copy
itb	begin iterator
ite	end iterator
aPredicate	predicate

Template Parameters

Stack	a model of CStack
ForwardIterator	a model of forward and readable iterator
Predicate	a model of ternary predicate

See also

Hull2D::buildHullWithStack

buildHullWithStack()

template<typename Stack , typename ForwardIterator , typename Predicate >

void DGtal::functions::Hull2D::buildHullWithStack	(	Stack &	aStack,
		const ForwardIterator &	itb,
		const ForwardIterator &	ite,
		const Predicate &	aPredicate
	)

Procedure that retrieves the vertices of the convex hull of a weakly externally visible polygon (WEVP) in linear-time. This technique is called Sklansky's scan, Graham's scan or 3-coins algorithm.
It works for all WEVP [Toussaint and Avis, 1982 : [118]].

Parameters

aStack	reference to the stack of retrieved vertices
itb	begin iterator
ite	end iterator
aPredicate	predicate

Template Parameters

Stack	a model of CStack
ForwardIterator	a model of forward and readable iterator
Predicate	a model of ternary predicate

See also

Hull2D::updateHullWithStack

closedGrahamScanFromAnyPoint()

template<typename ForwardIterator , typename OutputIterator , typename Predicate >

void DGtal::functions::Hull2D::closedGrahamScanFromAnyPoint	(	const ForwardIterator &	itb,
		const ForwardIterator &	ite,
		OutputIterator	res,
		const Predicate &	aPredicate
	)

Procedure that retrieves the vertices of the convex hull of a weakly externally visible polygon (WEVP) in linear-time.

See also

Hull2D::buildHullWithStack Hull2D::closedGrahamScanFromVertex

NB: We do not assume that the starting point of the polygon is an extremal point like in Hull2D::closedGrahamScanFromVertex

Parameters

itb	begin iterator
ite	end iterator
res	output iterator used to export the retrieved points
aPredicate	any ternary predicate

Template Parameters

ForwardIterator	a model of forward and readable iterator
OutputIterator	a model of incremental and writable iterator
Predicate	a model of ternary predicate

closedGrahamScanFromVertex()

template<typename ForwardIterator , typename OutputIterator , typename Predicate >

void DGtal::functions::Hull2D::closedGrahamScanFromVertex	(	const ForwardIterator &	itb,
		const ForwardIterator &	ite,
		OutputIterator	res,
		const Predicate &	aPredicate
	)

Procedure that retrieves the vertices of the convex hull of a weakly externally visible polygon (WEVP) in linear-time.

See also

Hull2D::buildHullWithStack

Parameters

itb	begin iterator
ite	end iterator
res	output iterator used to export the retrieved points
aPredicate	any ternary predicate

Precondition

we assume that the starting point of the polygon is an extremal point.

Template Parameters

ForwardIterator	a model of forward and readable iterator
OutputIterator	a model of incremental and writable iterator
Predicate	a model of ternary predicate

computeEuclideanDistance()

template<typename TInputPoint >

double DGtal::functions::Hull2D::computeEuclideanDistance	(	const TInputPoint &	a,
		const TInputPoint &	b,
		const TInputPoint &	c,
		bool &	isInside
	)

Computes the euclidean distance a point c according to the segment [a, b]. (i.e the distance between c and its projected point on [a,b].

Parameters

[in]	a	one point of the segment.
[in]	b	a second point of the segment.
[in]	c	the point for which the vertical distance is computed.
[out]	isInside	indicates if the projected point is inside the segment or not.

computeHProjDistance()

template<typename TInputPoint >

double DGtal::functions::Hull2D::computeHProjDistance	(	const TInputPoint &	a,
		const TInputPoint &	b,
		const TInputPoint &	c,
		bool &	isInside
	)

Computes the horizontal distance a point according to the segment [ a , b ]. (i.e the horizontal projection distance of on [ a , b ]).

Note

if the segment [a, b] is horizontal (i.e a [1]==b[1]) then an infinite value (std::numerics<double>::max()) is returned.

Parameters

[in]	a	one point of the segment.
[in]	b	a second point of the segment.
[in]	c	the point for which the horizontal distance is computed.
[out]	isInside	indicates if the projected point is inside the segment or not.

computeHullThickness() [1/2]

template<typename ForwardIterator >

double DGtal::functions::Hull2D::computeHullThickness	(	const ForwardIterator &	itb,
		const ForwardIterator &	ite,
		const ThicknessDefinition &	def
	)

Procedure to compute the convex hull thickness given from different definitions (Horizontal/vertical or Euclidean distances). It takes as input the vertices of the hull given by the range [itbn, ite). The procedure applies the classic rotating caliper to recover all anti-podal pairs.

Typical use:

typedef PointVector<2,DGtal::int32_t> Point;
typedef InHalfPlaneBySimple3x3Matrix<Point, DGtal::int32_t> Functor;  
DGtal::MelkmanConvexHull<Point, Functor> ch; 
ch.add(Point(0,0));
ch.add(Point(11,1));
ch.add(Point(12,3));
ch.add(Point(8,3));
ch.add(Point(4,5));
ch.add(Point(2,6));
ch.add(Point(1,4));
double th = computeHullThickness(ch.begin(), ch.end(), 
                                 DGtal::functions::Hull2D::EuclideanThickness);

Parameters

[in]	itb	begin iterator on the convex hull points.
[in]	ite	end iterator on the convex hull points.
[in]	def	definition of the thickness used in the estimation (i.e HorizontalVerticalThickness or EuclideanThickness)

Note

If the convex hull contains 0, 1 or 2 points the thickness of 0 is returned.

Warning

The convex hull should be oriented in counter clockwise else it will return wrong result.

Examples

geometry/tools/exampleConvexHull2D.cpp.

Referenced by convexHull(), and testConvexHullCompThickness().

computeHullThickness() [2/2]

template<typename ForwardIterator , typename TInputPoint >

double DGtal::functions::Hull2D::computeHullThickness	(	const ForwardIterator &	itb,
		const ForwardIterator &	ite,
		const ThicknessDefinition &	def,
		TInputPoint &	antipodalEdgeP,
		TInputPoint &	antipodalEdgeQ,
		TInputPoint &	antipodalVertexR
	)

Procedure to compute the convex hull thickness given from different definitions (Horizontal/vertical or Euclidean distances). It takes as input the vertices of the hull given by the range [itbn, ite). The procedure applies the classic rotating caliper to recover all anti-podal pairs.

Typical use:

typedef PointVector<2,DGtal::int32_t> Point;
typedef InHalfPlaneBySimple3x3Matrix<Point, DGtal::int32_t> Functor;  
DGtal::MelkmanConvexHull<Point, Functor> ch; 
ch.add(Point(0,0));
ch.add(Point(11,1));
ch.add(Point(12,3));
ch.add(Point(8,3));
ch.add(Point(4,5));
ch.add(Point(2,6));
ch.add(Point(1,4));
Point p, q, s;
double th = computeHullThickness(ch.begin(), ch.end(), 
                                 DGtal::functions::Hull2D::EuclideanThickness, p, q, s);

Parameters

[in]	itb	begin iterator on the convex hull points.
[in]	ite	end iterator on the convex hull points.
[in]	def	definition of the thickness used in the estimation (i.e HorizontalVerticalThickness or EuclideanThickness)
[out]	antipodalEdgeP	one point of the antipodal edge associated to the minimal value of convex hull thickness.
[out]	antipodalEdgeQ	one point of the antipodal edge associated to the minimal value of convex hull thickness.
[out]	antipodalVertexR	the vertex of the antipodal pair associated to the minimal value of convex hull thickness.

Note

If the convex hull contains 0, 1 or 2 points the thickness of 0 is returned and the antipodal points are updated with the first points (if they exist).

Warning

The convex hull should be oriented in counter clockwise else it will return wrong result.

computeVProjDistance()

template<typename TInputPoint >

double DGtal::functions::Hull2D::computeVProjDistance	(	const TInputPoint &	a,
		const TInputPoint &	b,
		const TInputPoint &	c,
		bool &	isInside
	)

Computes the vertical distance a point according to the segment [a, b]. (i.e the vertical projection distance of on [a,b].

Note

if the segment [a, b] is vertical (i.e a [0]== b [0]) then an infinite value (std::numerics<double>::max()) is returned.

Parameters

[in]	a	one point of the segment.
[in]	b	a second point of the segment.
[in]	c	the point for which the vertical distance is computed.
[out]	isInside	indicates if the projected point is inside the segment or not.

getAngle()

template<typename TInputPoint >

double DGtal::functions::Hull2D::getAngle ( const TInputPoint &  a, const TInputPoint &  b, const TInputPoint &  c, const TInputPoint &  d  )

inline

Computes the angle between the line (a,b) and (c,d)

Parameters

[in]	a	one of point defining the first line.
[in]	b	a second point defining the first line.
[in]	c	a third point defining the second line.
[in]	d	a third point defining the second line.

Returns

the resulting angle is between 0 and 2M_PI.

getThicknessAntipodalPair()

template<typename TInputPoint >

double DGtal::functions::Hull2D::getThicknessAntipodalPair	(	const TInputPoint &	p,
		const TInputPoint &	q,
		const TInputPoint &	r,
		const ThicknessDefinition &	def
	)

Computes the thickness of an anti podal pair (represented by the segment [ p , q ] and vertex r) according to the given distance def definition.

If the distance definition is HorizontalVerticalThickness, it returns the minimal distance between the vertical/horizontal projection of r on ( p , q ).

If the distance definition is EuclideanThickness, it returns the distance between r and its projection on the line ( p ,q ).

Parameters

[in]	p	the first point of the edge anti podal pair.
[in]	q	the second point of the edge anti podal pair.
[in]	r	the vertex of the anti podal pair.
[in]	def	definition of the thickness used in the estimation (i.e HorizontalVerticalThickness or EuclideanThickness).

Referenced by testConvexHullCompThickness().

grahamConvexHullAlgorithm() [1/2]

template<typename ForwardIterator , typename OutputIterator , typename Predicate , typename PolarComparator >

void DGtal::functions::Hull2D::grahamConvexHullAlgorithm	(	const ForwardIterator &	itb,
		const ForwardIterator &	ite,
		OutputIterator	res,
		const Predicate &	aPredicate
	)

Procedure that retrieves the vertices of the convex hull of a set of 2D points given by the range [ itb , ite ). This procedure follows the well-known Graham's algorithm [Graham, 1972 : [63]].

• choose a pole and sort the points in order of increasing angle about the pole (with a counter-clockwise orientation).
• scan the sorted list of points and remove some points so that the given predicate returns 'true' for all sets of three consecutive points.

  See also

  Hull2D::closedGrahamScanFromVertex

  Postcondition

  The first point of the resulting list of extremal points is guaranteed to be the one with maximal x-coordinate and y-coordinate.

  Warning

  The predicate must be chosen so that is returns 'true' for counter-clockwise oriented 3-point sets. Otherwise, the procedure only returns the last convex hull edge.
  

  Parameters

  itb	begin iterator
  ite	end iterator
  res	output iterator used to export the retrieved points
  aPredicate	any ternary predicate

  Template Parameters

  ForwardIterator	a model of forward and readable iterator
  OutputIterator	a model of incremental and writable iterator
  Predicate	a model of ternary predicate

grahamConvexHullAlgorithm() [2/2]

template<typename ForwardIterator , typename OutputIterator , typename Predicate , typename PolarComparator >

void DGtal::functions::Hull2D::grahamConvexHullAlgorithm	(	const ForwardIterator &	itb,
		const ForwardIterator &	ite,
		OutputIterator	res,
		const Predicate &	aPredicate,
		PolarComparator &	aPolarComparator
	)

Procedure that retrieves the vertices of the convex hull of a set of 2D points given by the range [ itb , ite ). This procedure follows the well-known Graham's algorithm [Graham, 1972 : [63]].

• choose a pole and sort the points in order of increasing angle about the pole with the given comparator.
• scan the sorted list of points and remove some points so that the given predicate returns 'true' for all sets of three consecutive points.

  See also

  Hull2D::closedGrahamScanFromVertex

  Postcondition

  The first point of the resulting list of extremal points is guaranteed to be the one with maximal x-coordinate and y-coordinate.

  Warning

  The orientation of the predicate and of the polar comparator should be the same. Otherwise, the procedure only returns the last convex hull edge. For instance, you may use a predicate that returns 'true' for three points counter-clockwise oriented together with PolarPointComparatorBy2x2DetComputer.
  

  Parameters

  itb	begin iterator
  ite	end iterator
  res	output iterator used to export the retrieved points
  aPredicate	any ternary predicate
  aPolarComparator	any polar comparator

  Template Parameters

  ForwardIterator	a model of forward and readable iterator
  OutputIterator	a model of incremental and writable iterator
  Predicate	a model of ternary predicate
  PolarComparator	a model of CPolarPointComparator2D.

isCoLinearOpp()

template<typename TInputPoint >

bool DGtal::functions::Hull2D::isCoLinearOpp ( const TInputPoint &  a, const TInputPoint &  b, const TInputPoint &  c, const TInputPoint &  d  )

inline

Determine if the two vectors (a,b) and (c,d) are co linear from opposite directions

Parameters

[in]	a	one of point defining the first line.
[in]	b	a second point defining the first line.
[in]	c	a third point defining the second line.
[in]	d	a third point defining the second line.

Returns

true if the two vectors are co linear from opposite directions, else it returns false.

melkmanConvexHullAlgorithm()

template<typename ForwardIterator , typename OutputIterator , typename Functor >

void DGtal::functions::Hull2D::melkmanConvexHullAlgorithm	(	const ForwardIterator &	itb,
		const ForwardIterator &	ite,
		OutputIterator	res,
		Functor &	aFunctor
	)

Procedure that retrieves the vertices of the hull of a set of 2D points given by the range [ itb , ite ). This procedure follows the well-known Melkman algorithm [Melkman, 1979 : [90]].

See also

MelkmanConvexHull

Parameters

itb	begin iterator
ite	end iterator
res	output iterator used to export the retrieved points
aFunctor	aFunctor

Template Parameters

ForwardIterator	a model of forward and readable iterator
OutputIterator	a model of incrementable and writable iterator
Functor	a model of COrientationFunctor2

openGrahamScan()

template<typename ForwardIterator , typename OutputIterator , typename Predicate >

void DGtal::functions::Hull2D::openGrahamScan	(	const ForwardIterator &	itb,
		const ForwardIterator &	ite,
		OutputIterator	res,
		const Predicate &	aPredicate
	)

Procedure that retrieves the vertices of the convex hull of a weakly externally visible polygon (WEVP) in linear-time.

See also

Hull2D::buildHullWithStack

Parameters

itb	begin iterator
ite	end iterator
res	output iterator used to export the retrieved points
aPredicate	any ternary predicate

Template Parameters

ForwardIterator	a model of forward and readable iterator
OutputIterator	a model of incremental and writable iterator
Predicate	a model of ternary predicate

updateHullWithAdaptedStack()

template<typename Stack , typename Point , typename Predicate >

void DGtal::functions::Hull2D::updateHullWithAdaptedStack	(	Stack	aStack,
		const Point &	aNewPoint,
		const Predicate &	aPredicate
	)

Procedure that calls Hull2D::updateHullWithStack on a copy of the stack object used to retrieved the hull vertices. Useful when the first argument is a stack adapter returned by a function: it must be copied before being passed by reference in Hull2D::updateHullWithStack.

Parameters

aStack	stack copy
aNewPoint	new point
aPredicate	point predicate

Template Parameters

Stack	a model of CStack
Point	a model of point
Predicate	a model of ternary predicate

updateHullWithStack()

template<typename Stack , typename Point , typename Predicate >

void DGtal::functions::Hull2D::updateHullWithStack	(	Stack &	aStack,
		const Point &	aNewPoint,
		const Predicate &	aPredicate
	)

Procedure that updates the hull when an extra point aNewPoint is considered: while the last three consecutive points, ie. aNewPoint and the last two points of the container are not oriented such that the predicate aPredicate returns 'true',
the last point of the container is removed.

See also

Hull2D::buildHullWithStack

Parameters

aStack	reference to the stack of retrieved vertices
aNewPoint	new point
aPredicate	point predicate

Template Parameters

Stack	a model of CStack
Point	a model of point
Predicate	a model of ternary predicate

Variable Documentation

angleTolerance

constexpr double DGtal::functions::Hull2D::angleTolerance = 1e-6

constexpr

Definition at line 76 of file Hull2DHelpers.h.