#include <CGAL/Alpha_shape_3.h>

Inherits from

Dt.

Definition

The class Alpha_shape_3 represents the family of alpha shapes of points in the 3D space for all real \( \alpha\).

It maintains an underlying triangulation of the class Dt. Each k-dimensional face of Dt is associated with an interval that specifies for which values of alpha the face belongs to the alpha shape.

Note that this class is used for basic, weighted, and periodic Alpha Shapes.

The modifying functions insert and remove will overwrite the one inherited from the underlying triangulation class Dt. At the moment, only the static version is implemented.

Template Parameters

Dt must be either Delaunay_triangulation_3, Regular_triangulation_3, Periodic_3_Delaunay_triangulation_3 or Periodic_3_regular_triangulation_3. Note that Dt::Geom_traits, Dt::Vertex, and Dt::Face must be model the concepts AlphaShapeTraits_3, AlphaShapeVertex_3 and AlphaShapeCell_3, respectively.

The second template parameter ExactAlphaComparisonTag is a tag that, when set to Tag_true, triggers exact comparisons between alpha values. This is useful when the Delaunay triangulation is instantiated with an exact predicates inexact constructions kernel. By default the ExactAlphaComparisonTag is set to Tag_false as it induces a small overhead. Note that the tag ExactAlphaComparisonTag is currently ignored (meaning that the code will behave as if ExactAlphaComparisonTag were set to Tag_false) if Dt::Geom_traits::FT is not a floating point number type as this strategy does not make sense if the traits class already provides exact constructions.

Warning

When the tag ExactAlphaComparisonTag is set to Tag_true, the class Cartesian_converter is used internally to switch between the traits class and the CGAL kernel CGAL::Simple_cartesian<NT>, where NT can be either CGAL::Interval_nt or CGAL::Exact_rational. Cartesian_converter must thus offer the necessary functors to convert a three-dimensional point of the traits class to a three-dimensional point of CGAL::Simple_cartesian<NT>. However, these functors are not necessarily provided by the basic Cartesian_converter, for example when a custom point is used. In this case, a partial specialization of Cartesian_converter must be provided by the user. An example of such specialization is given in the two-dimensional Alpha Shapes example ex_alpha_projection_traits.cpp.
The tag ExactAlphaComparisonTag cannot be used in conjonction with periodic triangulations. When the tag ExactAlphaComparisonTag is set to Tag_true, the evaluations of predicates such as Side_of_oriented_sphere_3 are done lazily. Consequently, the predicates store pointers to the geometrical positions of the points passed as arguments of the predicates. It is thus important that these points are not temporary objects. Points of the triangulation are accessed using the function point(Cell_handle, int) of the underlying triangulation. In the case of periodic triangulations, the point(Cell_handle, int) function is actually a construction that returns a temporary, which thus cannot be used along with a lazy predicate evaluation.

I/O
The I/O operators are defined for iostream, and for the window stream provided by CGAL. The format for the iostream is an internal format.

Implementation

In GENERAL mode, the alpha intervals of each triangulation face is computed and stored at initialization time. In REGULARIZED mode, the alpha shape intervals of edges are not stored nor computed at initialization. Edges are simply classified on the fly upon request. This allows to have much faster building of alpha shapes in REGULARIZED mode.

Function Alpha_shape_3::alpha_find() uses linear search, while Alpha_shape_3::alpha_lower_bound() and Alpha_shape_3::alpha_upper_bound() use binary search. Alpha_shape_3::number_of_solid_components() performs a graph traversal and takes time linear in the number of cells of the underlying triangulation. Alpha_shape_3::find_optimal_alpha() uses binary search and takes time \( O(n \log n)\), where \( n\) is the number of points.

Examples:: Alpha_shapes_3/ex_alpha_shapes_3.cpp, Alpha_shapes_3/ex_alpha_shapes_exact_alpha.cpp, Alpha_shapes_3/ex_alpha_shapes_with_fast_location_3.cpp, Alpha_shapes_3/ex_periodic_alpha_shapes_3.cpp, Alpha_shapes_3/ex_weighted_alpha_shapes_3.cpp, and Alpha_shapes_3/ex_weighted_periodic_alpha_shapes_3.cpp.

Related Functions
(Note that these are not member functions.)
std::ostream &	operator<< (std::ostream &os, const Alpha_shape_3< Dt, ExactAlphaComparisonTag > &A)
	Inserts the alpha shape `A` for the current alpha value into the stream `os`. More...

Geomview_stream &	operator<< (Geomview_stream &W, const Alpha_shape_3< Dt, ExactAlphaComparisonTag > &A)
	Inserts the alpha shape `A` for the current alpha value into the Geomview stream `W`. More...

Types
enum	Mode { GENERAL, REGULARIZED }
	In GENERAL mode, In REGULARIZED mode,. More...

enum	Classification_type { EXTERIOR, SINGULAR, REGULAR, INTERIOR }
	Enum to classify the faces of the underlying triangulation with respect to the alpha shape. More...

typedef unspecified_type	Gt
	the alpha shape traits type. More...

typedef unspecified_type	FT
	the number type of alpha values. More...

typedef Dt::Point	Point
	The point type. More...

typedef unspecified_type	size_type
	The size type.

typedef unspecified_type	Alpha_iterator
	A bidirectional and non-mutable iterator that allow to traverse the increasing sequence of different alpha values. More...

Creation
	Alpha_shape_3 (FT alpha=0, Mode m=REGULARIZED)
	Introduces an empty alpha shape, sets the current alpha value to `alpha` and the mode to `m`.

	Alpha_shape_3 (Dt &dt, FT alpha=0, Mode m=REGULARIZED)
	Builds an alpha shape of mode `m` from the triangulation `dt`. More...

template<class InputIterator >
	Alpha_shape_3 (InputIterator first, InputIterator last, const FT &alpha=0, Mode m=REGULARIZED)
	Builds an alpha shape of mode `m` for the points in the range `[first,last)` and set the current alpha value to `alpha`. More...

Modifiers
template<class InputIterator >
std::ptrdiff_t	make_alpha_shape (InputIterator first, InputIterator last)
	Initialize the alpha shape data structure for points in the range `[first,last)`. More...

void	clear ()
	Clears the structure.

FT	set_alpha (const FT &alpha)
	Sets the \( \alpha\)-value to `alpha`. More...

Mode	set_mode (Mode m=REGULARIZED)
	Sets the mode of the alpha shape to `GENERAL` or `REGULARIZED`. More...

Query Functions
Mode	get_mode (void) const
	Returns whether the alpha shape is general or regularized.

const FT &	get_alpha (void) const
	Returns the current \( \alpha\)-value.

const FT &	get_nth_alpha (int n) const
	Returns the `n`-th `alpha`-value, sorted in an increasing order. More...

size_type	number_of_alphas () const
	Returns the number of different alpha-values.

Classification_type	classify (const Point &p, const FT &alpha=get_alpha()) const
	Locates a point `p` in the underlying triangulation and Classifies the associated k-face with respect to `alpha`.

Classification_type	classify (Cell_handle f, const FT &alpha=get_alpha()) const
	Classifies the cell `f` of the underlying triangulation with respect to `alpha`.

Classification_type	classify (Facet f, const FT &alpha=get_alpha()) const
	Classifies the facet `f` of the underlying triangulation with respect to `alpha`.

Classification_type	classify (Cell_handle f, int i, const FT &alpha=get_alpha()) const
	Classifies the facet of the cell `f` opposite to the vertex with index `i` of the underlying triangulation with respect to `alpha`.

Classification_type	classify (const Edge &e, const FT &alpha=get_alpha()) const
	Classifies the edge `e` with respect to `alpha` .

Classification_type	classify (Vertex_handle v, const FT &alpha=get_alpha()) const
	Classifies the vertex `v` of the underlying triangulation with respect to `alpha`.

Alpha_status< FT >	get_alpha_status (const Edge &e) const
	Returns the alpha-status of the edge `e`.

Alpha_status< FT >	get_alpha_status (const Facet &f) const
	Returns the alpha-status of the facet `f`.

template<class OutputIterator >
OutputIterator	get_alpha_shape_cells (OutputIterator it, Classification_type type, const FT &alpha=get_alpha())
	Write the cells which are of type `type` for the alpha value `alpha` to the sequence pointed to by the output iterator `it`. More...

template<class OutputIterator >
OutputIterator	get_alpha_shape_facets (OutputIterator it, Classification_type type, const FT &alpha=get_alpha())
	Write the facets which are of type `type` for the alpha value `alpha` to the sequence pointed to by the output iterator `it`. More...

template<class OutputIterator >
OutputIterator	get_alpha_shape_edges (OutputIterator it, Classification_type type, const FT &alpha=get_alpha())
	Write the edges which are of type `type` for the alpha value `alpha` to the sequence pointed to by the output iterator `it`. More...

template<class OutputIterator >
OutputIterator	get_alpha_shape_vertices (OutputIterator it, Classification_type type, const FT &alpha)
	Write the vertices which are of type `type` for the alpha value `alpha` to the sequence pointed to by the output iterator `it`. More...

template<class OutputIterator >
OutputIterator	filtration (OutputIterator it) const
	Output all the faces of the triangulation in increasing order of the alpha value for which they appear in the alpha complex. More...

template<class OutputIterator >
OutputIterator	filtration_with_alpha_values (OutputIterator it) const
	Output all the faces of the triangulation in increasing order of the alpha value for which they appear in the alpha complex. More...

Traversal of the alpha-Values
Alpha_iterator	alpha_begin () const
	Returns an iterator that allows to traverse the sorted sequence of \( \alpha\)-values of the family of alpha shapes.

Alpha_iterator	alpha_end () const
	Returns the corresponding past-the-end iterator.

Alpha_iterator	alpha_find (const FT &alpha) const
	Returns an iterator pointing to an element with \( \alpha\)-value `alpha`, or the corresponding past-the-end iterator if such an element is not found.

Alpha_iterator	alpha_lower_bound (const FT &alpha) const
	Returns an iterator pointing to the first element with \( \alpha\)-value not less than `alpha`.

Alpha_iterator	alpha_upper_bound (const FT &alpha) const
	Returns an iterator pointing to the first element with \( \alpha\)-value greater than `alpha`.

Operations
size_type	number_of_solid_components (const FT &alpha=get_alpha()) const
	Returns the number of solid components of the alpha shape, that is, the number of components of its regularized version.

Alpha_iterator	find_optimal_alpha (size_type nb_components) const
	Returns an iterator pointing to smallest \( \alpha\) value such that the alpha shape satisfies the following two properties: More...

Member Typedef Documentation

◆ Alpha_iterator

template<typename Dt , typename ExactAlphaComparisonTag >

typedef unspecified_type CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::Alpha_iterator

A bidirectional and non-mutable iterator that allow to traverse the increasing sequence of different alpha values.

Precondition: Its value_type is FT.

◆ FT

template<typename Dt , typename ExactAlphaComparisonTag >

typedef unspecified_type CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::FT

the number type of alpha values.

In case ExactAlphaComparisonTag is CGAL::Tag_false, it is Gt::FT.

In case ExactAlphaComparisonTag is CGAL::Tag_true, it is a number type allowing filtered exact comparisons (that is, interval arithmetic is first used before resorting to exact arithmetic). Access to the interval containing the exact value is provided through the function FT::Approximate_nt approx() const where FT::Approximate_nt is Interval_nt<Protected> with Protected=true. Access to the exact value is provided through the function FT::Exact_nt exact() const where FT::Exact_nt depends on the configuration of CGAL (it may be mpq_class, Gmpq, Quotient<CGAL::MP_Float>, etc). An overload for the function double to_double(FT) is also available. Its precision is controlled through FT::set_relative_precision_of_to_double() in exactly the same way as with Lazy_exact_nt<NT>, so a call to to_double may trigger an exact evaluation. It must be noted that an object of type FT is valid as long as the alpha shapes class that creates it is valid and has not been modified. For convenience, classical comparison operators are provided for the type FT.

◆ Gt

template<typename Dt , typename ExactAlphaComparisonTag >

typedef unspecified_type CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::Gt

the alpha shape traits type.

It has to derive from a triangulation traits class. For example Dt::Point is a point class.

◆ Point

template<typename Dt , typename ExactAlphaComparisonTag >

typedef Dt::Point CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::Point

The point type.

For basic alpha shapes, Point will be equal to Gt::Point_3. For weighted alpha shapes, Point will be equal to Gt::Weighted_point_3.

Member Enumeration Documentation

◆ Classification_type

template<typename Dt , typename ExactAlphaComparisonTag >

enum CGAL::Alpha_shape_3::Classification_type

Enum to classify the faces of the underlying triangulation with respect to the alpha shape.

In GENERAL mode, for \( k=(0,1,2)\), each k-dimensional simplex of the triangulation can be classified as EXTERIOR, SINGULAR, REGULAR or INTERIOR. In GENERAL mode a \( k\) simplex is REGULAR if it is on the boundary f the alpha complex and belongs to a \( k+1\) simplex in this complex and it is SINGULAR if it is a boundary simplex that is not included in a \( k+1\) simplex of the complex.

In REGULARIZED mode, for \( k=(0,1,2)\) each k-dimensional simplex of the triangulation can be classified as EXTERIOR, REGULAR or INTERIOR, i.e.\ there is no singular faces. A \( k\) simplex is REGULAR if it is on the boundary of alpha complex and belongs to a tetrahedral cell of the complex.

Enumerator
EXTERIOR
SINGULAR
REGULAR
INTERIOR

◆ Mode

template<typename Dt , typename ExactAlphaComparisonTag >

enum CGAL::Alpha_shape_3::Mode

In GENERAL mode, In REGULARIZED mode,.

Enumerator
GENERAL	the alpha complex can have singular faces, i.e., faces of dimension \( k\), for \( k=(0,1,2)\) that are not subfaces of a \( k+1\) face of the complex.
REGULARIZED	the complex is regularized, that is singular faces are dropped and the alpha complex includes only a subset of the tetrahedral cells of the triangulation and the subfaces of those cells.

Constructor & Destructor Documentation

◆ Alpha_shape_3() [1/2]

template<typename Dt , typename ExactAlphaComparisonTag >

CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::Alpha_shape_3	(	Dt &	dt,
		FT	alpha = `0`,
		Mode	m = `REGULARIZED`
	)

Builds an alpha shape of mode m from the triangulation dt.

Attention: This operation destroys the triangulation dt.

◆ Alpha_shape_3() [2/2]

template<typename Dt , typename ExactAlphaComparisonTag >

template<class InputIterator >

CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::Alpha_shape_3	(	InputIterator	first,
		InputIterator	last,
		const FT &	alpha = `0`,
		Mode	m = `REGULARIZED`
	)

Builds an alpha shape of mode m for the points in the range [first,last) and set the current alpha value to alpha.

Template Parameters

InputIterator must be an input iterator with value type Point (the point type of the underlying triangulation.)

Member Function Documentation

◆ filtration()

template<typename Dt , typename ExactAlphaComparisonTag >

template<class OutputIterator >

OutputIterator CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::filtration ( OutputIterator it ) const

Output all the faces of the triangulation in increasing order of the alpha value for which they appear in the alpha complex.

In case of equal alpha value lower dimensional faces are output first.

Template Parameters

OutputIterator must be an output iterator accepting variables of type Object.

Warning: The result of this function depends on the mode of the Alpha-shape. In most case, Alpha_shape_3::GENERAL is the most interesting one.

◆ filtration_with_alpha_values()

template<typename Dt , typename ExactAlphaComparisonTag >

template<class OutputIterator >

OutputIterator CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::filtration_with_alpha_values ( OutputIterator it ) const

Output all the faces of the triangulation in increasing order of the alpha value for which they appear in the alpha complex.

In case of equal alpha value lower dimensional faces are output first. In addition the value of alpha at which each face appears are also reported. Each face and its alpha value are reported successively.

Template Parameters

OutputIterator must be an output iterator accepting variables of type Object and FT. The class Dispatch_output_iterator can be used for this purpose.

Warning: The result of this function dependents on the mode of the Alpha-shape. In most case, Alpha_shape_3::GENERAL is the most interesting one.

◆ find_optimal_alpha()

template<typename Dt , typename ExactAlphaComparisonTag >

Alpha_iterator CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::find_optimal_alpha ( size_type nb_components ) const

Returns an iterator pointing to smallest \( \alpha\) value such that the alpha shape satisfies the following two properties:

All data points are either on the boundary or in the interior of the regularized version of the alpha shape.
The number of solid component of the alpha shape is equal to or smaller than nb_components.

◆ get_alpha_shape_cells()

template<typename Dt , typename ExactAlphaComparisonTag >

template<class OutputIterator >

OutputIterator CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::get_alpha_shape_cells	(	OutputIterator	it,
		Classification_type	type,
		const FT &	alpha = `get_alpha()`
	)

Write the cells which are of type type for the alpha value alpha to the sequence pointed to by the output iterator it.

Returns past the end of the output sequence.

◆ get_alpha_shape_edges()

template<typename Dt , typename ExactAlphaComparisonTag >

template<class OutputIterator >

OutputIterator CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::get_alpha_shape_edges	(	OutputIterator	it,
		Classification_type	type,
		const FT &	alpha = `get_alpha()`
	)

Write the edges which are of type type for the alpha value alpha to the sequence pointed to by the output iterator it.

Returns past the end of the output sequence.

◆ get_alpha_shape_facets()

template<typename Dt , typename ExactAlphaComparisonTag >

template<class OutputIterator >

OutputIterator CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::get_alpha_shape_facets	(	OutputIterator	it,
		Classification_type	type,
		const FT &	alpha = `get_alpha()`
	)

Write the facets which are of type type for the alpha value alpha to the sequence pointed to by the output iterator it.

Returns past the end of the output sequence.

◆ get_alpha_shape_vertices()

template<typename Dt , typename ExactAlphaComparisonTag >

template<class OutputIterator >

OutputIterator CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::get_alpha_shape_vertices	(	OutputIterator	it,
		Classification_type	type,
		const FT &	alpha
	)

Write the vertices which are of type type for the alpha value alpha to the sequence pointed to by the output iterator it.

Returns past the end of the output sequence.

◆ get_nth_alpha()

template<typename Dt , typename ExactAlphaComparisonTag >

const FT& CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::get_nth_alpha ( int n ) const

Returns the n-th alpha-value, sorted in an increasing order.

Precondition: n < number of alphas.

◆ make_alpha_shape()

template<typename Dt , typename ExactAlphaComparisonTag >

template<class InputIterator >

std::ptrdiff_t CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::make_alpha_shape	(	InputIterator	first,
		InputIterator	last
	)

Initialize the alpha shape data structure for points in the range [first,last).

Returns the number of data points inserted in the underlying triangulation.

If the function is applied to an non-empty alpha shape data structure, it is cleared before initialization.

Template Parameters

InputIterator must be an input iterator with value type Point.

◆ set_alpha()

template<typename Dt , typename ExactAlphaComparisonTag >

FT CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::set_alpha ( const FT & alpha )

Sets the \( \alpha\)-value to alpha.

Returns the previous \( \alpha\)-value.

Precondition: alpha \( \geq0\).

◆ set_mode()

template<typename Dt , typename ExactAlphaComparisonTag >

Mode CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag >::set_mode ( Mode m = REGULARIZED )

Sets the mode of the alpha shape to GENERAL or REGULARIZED.

Returns the previous mode. Changing the mode of an alpha shape entails a partial re-computation of the data structure.

Friends And Related Function Documentation

◆ operator<<() [1/2]

template<typename Dt , typename ExactAlphaComparisonTag >

std::ostream & operator<<	(	std::ostream &	os,
		const Alpha_shape_3< Dt, ExactAlphaComparisonTag > &	A
	)

Defined in CGAL/IO/io.h

Precondition: The insert operator must be defined for Point.

◆ operator<<() [2/2]

template<typename Dt , typename ExactAlphaComparisonTag >

Geomview_stream & operator<<	(	Geomview_stream &	W,
		const Alpha_shape_3< Dt, ExactAlphaComparisonTag > &	A
	)

Precondition: The insert operator must be defined for GT::Point and GT::Triangle.

Defined in CGAL/IO/Geomview_stream.h and CGAL/IO/alpha_shape_geomview_ostream_3.h

Inherits from

Definition

Related Functions

Types

Creation

Modifiers

Query Functions

Traversal of the alpha-Values

Operations

Member Typedef Documentation

◆ Alpha_iterator

◆ FT

◆ Gt

◆ Point

Member Enumeration Documentation

◆ Classification_type

◆ Mode

Constructor & Destructor Documentation

◆ Alpha_shape_3() [1/2]

◆ Alpha_shape_3() [2/2]

Member Function Documentation

◆ filtration()

◆ filtration_with_alpha_values()

◆ find_optimal_alpha()

◆ get_alpha_shape_cells()

◆ get_alpha_shape_edges()

◆ get_alpha_shape_facets()

◆ get_alpha_shape_vertices()

◆ get_nth_alpha()

◆ make_alpha_shape()

◆ set_alpha()

◆ set_mode()

Friends And Related Function Documentation

◆ operator<<() [1/2]

◆ operator<<() [2/2]