\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 4.12.2 - 2D Alpha Shapes
List of all members
CGAL::Alpha_shape_2< Dt, ExactAlphaComparisonTag > Class Template Reference
2D Alpha Shapes Reference

#include <CGAL/Alpha_shape_2.h>

Inherits from

Dt.

Definition

The class Alpha_shape_2 represents the family of \( \alpha\)-shapes of points in a plane for all positive \( \alpha\).

It maintains the underlying triangulation Dt which represents connectivity and order among its faces. Each \( k\)-dimensional face of the Dt is associated with an interval that specifies for which values of \( \alpha\) the face belongs to the \( \alpha\)-shape. There are links between the intervals and the \( k\)-dimensional faces of the triangulation.

Note that this class is at the same time used for basic and for weighted Alpha Shapes.

The modifying functions Alpha_shape_2::insert() and Alpha_shape_2::remove() will overwrite the one inherited from the underlying triangulation class Dt. At the moment, only the static version is implemented.

Template Parameters
Dtmust be either Delaunay_triangulation_2 or Regular_triangulation_2. Note that Dt::Geom_traits, Dt::Vertex, and Dt::Face must be model the concepts AlphaShapeTraits_2, AlphaShapeVertex_2 and AlphaShapeFace_2, respectively.
ExactAlphaComparisonTagis a tag that, when set to Tag_true, triggers exact comparisons between alpha values. This is useful when the underlying 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 two-dimensional point of the traits class to a two-dimensional point of CGAL::Simple_cartesian<NT>. However, these functors are not necessarily provided by the basic Cartesian_converter. For example when using the traits class CGAL::Projection_traits_xy_3, a CGAL::Point_3 is camouflaged as a Point_2 and the basic Cartesian_converter does not know how to convert from the camouflaged CGAL::Point_3 to the two-dimensional point of CGAL::Simple_cartesian<NT>. In this case, a partial specialization of Cartesian_converter must be provided by the user. An example of such specialization is given in the 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_circle_2 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(Face_handle, int) of the underlying triangulation. In the case of periodic triangulations, the point(Face_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 std::iostream. The format for the iostream is an internal format.

Implementation

The set of intervals associated with the \( k\)-dimensional faces of the underlying triangulation are stored in multimaps.

The cross links between the intervals and the \( k\)-dimensional faces of the triangulation are realized using methods in the \( k\)-dimensional faces themselves.

Alpha_shape_2::alpha_find() uses linear search, while Alpha_shape_2::alpha_lower_bound() and Alpha_shape_2::alpha_upper_bound() use binary search. Alpha_shape_2::number_of_solid_components() performs a graph traversal and takes time linear in the number of faces of the underlying triangulation. Alpha_shape_2::find_optimal_alpha() uses binary search and takes time \( O(n \log n)\), where \( n\) is the number of points.

Examples:
Alpha_shapes_2/ex_alpha_projection_traits.cpp, Alpha_shapes_2/ex_alpha_shapes_2.cpp, Alpha_shapes_2/ex_periodic_alpha_shapes_2.cpp, and Alpha_shapes_2/ex_weighted_alpha_shapes_2.cpp.

Types

enum  Classification_type { EXTERIOR, SINGULAR, REGULAR, INTERIOR }
 Distinguishes the different cases for classifying a \( k\)-dimensional face of the underlying triangulation of the \( \alpha\)-shape. More...
 
enum  Mode { GENERAL, REGULARIZED }
 In general, an alpha shape can be disconnected and contain many singular edges or vertices. More...
 
typedef unspecified_type Gt
 the alpha shape traits type. More...
 
typedef Gt::FT 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...
 
typedef unspecified_type Alpha_shape_vertices_iterator
 A bidirectional and non-mutable iterator that allow to traverse the vertices which belongs to the \( \alpha\)-shape for the current \( \alpha\). More...
 
typedef unspecified_type Alpha_shape_edges_iterator
 A bidirectional and non-mutable iterator that allow to traverse the edges which belongs to the \( \alpha\)-shape for the current \( \alpha\). More...
 

Creation

 Alpha_shape_2 (FT alpha=0, Mode m=GENERAL)
 Introduces an empty alpha-shape for a positive \( \alpha\)-value alpha. More...
 
 Alpha_shape_2 (Dt &dt, FT alpha=0, Mode m=GENERAL)
 Builds an alpha shape of mode m from the triangulation dt for a positive \( \alpha\)-value alpha. More...
 
template<class InputIterator >
 Alpha_shape_2 (InputIterator first, InputIterator last, const FT &alpha=0, Mode m=GENERAL)
 Initializes the family of alpha-shapes with the points in the range [first,last) and introduces an \( \alpha\)-shape for a positive \( \alpha\)-value alpha. More...
 

Operations

template<class InputIterator >
std::ptrdiff_t make_alpha_shape (InputIterator first, InputIterator last)
 Initialize the family of alpha-shapes with the 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...
 
const FT & get_alpha (void) const
 Returns the current \( \alpha\)-value.
 
const FT & get_nth_alpha (size_type 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.
 
Mode set_mode (Mode m=GENERAL)
 Sets the mode to its general or regularized version. More...
 
Mode get_mode (void) const
 Returns the mode, that is either GENERAL or REGULARIZED.
 
Alpha_shape_vertices_iterator alpha_shape_vertices_begin ()
 Starts at an arbitrary finite vertex which belongs to the \( \alpha\)-shape for the current \( \alpha\).
 
Alpha_shape_vertices_iterator alpha_shape_vertices_end ()
 Past-the-end iterator.
 
Alpha_shape_edges_iterator alpha_shape_edges_begin ()
 Starts at an arbitrary finite edge which belongs to the \( \alpha\)-shape for the current \( \alpha\). More...
 
Alpha_shape_edges_iterator alpha_shape_edges_end ()
 Past-the-end iterator.
 
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 the first element with \( \alpha\)-value such that the alpha shape satisfies the following two properties: More...
 
ostream & operator<< (std::ostream &os, const Alpha_shape_2< Dt > &A)
 Inserts the alpha shape for the current \( \alpha\)-value into the stream os. More...
 

Predicates

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 the alpha shape.
 
Classification_type classify (Face_handle f, const FT &alpha=get_alpha()) const
 Classifies the face f of the underlying triangulation with respect to the alpha shape.
 
Classification_type classify (Edge e, const FT &alpha=get_alpha()) const
 Classifies the edge e of the underlying triangulation with respect to the alpha shape.
 
Classification_type classify (Face_handle f, int i, const FT &alpha=get_alpha()) const
 Classifies the edge of the face f opposite to the vertex with index i of the underlying triangulation with respect to the alpha shape.
 
Classification_type classify (Vertex_handle v, const FT &alpha=get_alpha()) const
 Classifies the vertex v of the underlying triangulation with respect to the alpha shape.
 

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.
 

Member Typedef Documentation

◆ Alpha_iterator

template<typename Dt , typename ExactAlphaComparisonTag >
typedef unspecified_type CGAL::Alpha_shape_2< 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.

◆ Alpha_shape_edges_iterator

template<typename Dt , typename ExactAlphaComparisonTag >
typedef unspecified_type CGAL::Alpha_shape_2< Dt, ExactAlphaComparisonTag >::Alpha_shape_edges_iterator

A bidirectional and non-mutable iterator that allow to traverse the edges which belongs to the \( \alpha\)-shape for the current \( \alpha\).

Precondition
Its value_type is Dt::Edge.

◆ Alpha_shape_vertices_iterator

template<typename Dt , typename ExactAlphaComparisonTag >
typedef unspecified_type CGAL::Alpha_shape_2< Dt, ExactAlphaComparisonTag >::Alpha_shape_vertices_iterator

A bidirectional and non-mutable iterator that allow to traverse the vertices which belongs to the \( \alpha\)-shape for the current \( \alpha\).

Precondition
Its value_type is Dt::Vertex_handle.

◆ FT

template<typename Dt , typename ExactAlphaComparisonTag >
typedef Gt::FT CGAL::Alpha_shape_2< 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 CGAL::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 is Gmpq if gmp is available and Quotient<CGAL::MP_Float> otherwise). 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_2< 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_2< Dt, ExactAlphaComparisonTag >::Point

The point type.

For basic alpha shapes, Point will be equal to Gt::Point_2. For weighted alpha shapes, Point will be equal to Gt::Weighted_point_2.

Member Enumeration Documentation

◆ Classification_type

template<typename Dt , typename ExactAlphaComparisonTag >
enum CGAL::Alpha_shape_2::Classification_type

Distinguishes the different cases for classifying a \( k\)-dimensional face of the underlying triangulation of the \( \alpha\)-shape.

Enumerator
EXTERIOR 

if the face does not belong to the alpha-complex.

SINGULAR 

if the face belongs to the boundary of the alpha-shape, but is not incident to any 2-dimensional face of the alpha-complex

REGULAR 

if the face belongs to the boundary of the alpha-shape and is incident to a 2-dimensional face of the alpha-complex

INTERIOR 

if the face belongs to the alpha-complex, but does not belong to the boundary of the alpha-shape.

◆ Mode

template<typename Dt , typename ExactAlphaComparisonTag >
enum CGAL::Alpha_shape_2::Mode

In general, an alpha shape can be disconnected and contain many singular edges or vertices.

Its regularized version is formed by the set of regular edges and their vertices.

Enumerator
GENERAL 
REGULARIZED 

Constructor & Destructor Documentation

◆ Alpha_shape_2() [1/3]

template<typename Dt , typename ExactAlphaComparisonTag >
CGAL::Alpha_shape_2< Dt, ExactAlphaComparisonTag >::Alpha_shape_2 ( FT  alpha = 0,
Mode  m = GENERAL 
)

Introduces an empty alpha-shape for a positive \( \alpha\)-value alpha.

Precondition
alpha \( \geq~0\).

◆ Alpha_shape_2() [2/3]

template<typename Dt , typename ExactAlphaComparisonTag >
CGAL::Alpha_shape_2< Dt, ExactAlphaComparisonTag >::Alpha_shape_2 ( Dt &  dt,
FT  alpha = 0,
Mode  m = GENERAL 
)

Builds an alpha shape of mode m from the triangulation dt for a positive \( \alpha\)-value alpha.

Attention
This operation destroys the triangulation.
Precondition
alpha \( \geq~0\).

◆ Alpha_shape_2() [3/3]

template<typename Dt , typename ExactAlphaComparisonTag >
template<class InputIterator >
CGAL::Alpha_shape_2< Dt, ExactAlphaComparisonTag >::Alpha_shape_2 ( InputIterator  first,
InputIterator  last,
const FT &  alpha = 0,
Mode  m = GENERAL 
)

Initializes the family of alpha-shapes with the points in the range [first,last) and introduces an \( \alpha\)-shape for a positive \( \alpha\)-value alpha.

Template Parameters
InputIteratormust be an input iterator with the value type Point.
Precondition
alpha \( \geq0\).

Member Function Documentation

◆ alpha_shape_edges_begin()

template<typename Dt , typename ExactAlphaComparisonTag >
Alpha_shape_edges_iterator CGAL::Alpha_shape_2< Dt, ExactAlphaComparisonTag >::alpha_shape_edges_begin ( )

Starts at an arbitrary finite edge which belongs to the \( \alpha\)-shape for the current \( \alpha\).

In regularized mode, edges are represented as a pair (f,i), where f is an interior face of the \( \alpha\)-shape.

Examples:
Alpha_shapes_2/ex_alpha_shapes_2.cpp, Alpha_shapes_2/ex_periodic_alpha_shapes_2.cpp, and Alpha_shapes_2/ex_weighted_alpha_shapes_2.cpp.

◆ find_optimal_alpha()

template<typename Dt , typename ExactAlphaComparisonTag >
Alpha_iterator CGAL::Alpha_shape_2< Dt, ExactAlphaComparisonTag >::find_optimal_alpha ( size_type  nb_components) const

Returns an iterator pointing to the first element with \( \alpha\)-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.

If no such value is found, the iterator points to the first element with \( \alpha\)-value such that the alpha shape satisfies the second property.

◆ get_nth_alpha()

template<typename Dt , typename ExactAlphaComparisonTag >
const FT& CGAL::Alpha_shape_2< Dt, ExactAlphaComparisonTag >::get_nth_alpha ( size_type  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_2< Dt, ExactAlphaComparisonTag >::make_alpha_shape ( InputIterator  first,
InputIterator  last 
)

Initialize the family of alpha-shapes with the points in the range [first,last).

Returns the number of inserted points.

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

Template Parameters
InputIteratormust be an input iterator with the value type Point.

◆ set_alpha()

template<typename Dt , typename ExactAlphaComparisonTag >
FT CGAL::Alpha_shape_2< 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_2< Dt, ExactAlphaComparisonTag >::set_mode ( Mode  m = GENERAL)

Sets the mode to its general or regularized version.

Returns the previous mode.

Friends And Related Function Documentation

◆ operator
template<typename Dt , typename ExactAlphaComparisonTag >
ostream& operator<< ( std::ostream &  os,
const Alpha_shape_2< Dt > &  A 
)
related

Inserts the alpha shape for the current \( \alpha\)-value into the stream os.

Precondition
CGAL/IO/io.h must be included.
The insert operator must be defined for Point.