\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - 3D Alpha Shapes

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CGAL::Alpha_shape_3< Dt, ExactAlphaComparisonTag > Class Template Reference
3D Alpha Shapes Reference

`#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. The second template parameter `ExactAlphaComparisonTag` is a tag that, when set to `CGAL::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 `CGAL::Tag_false` as it induces a small overhead. Note that since such a strategy does not make sense if used together with a traits class with exact constructions, the tag `ExactAlphaComparisonTag` is not taken into account if `Dt::Geom_traits::FT` is not a floating point number type.

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

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, and Alpha_shapes_3/ex_weighted_alpha_shapes_3.cpp.

## Related Functions
(Note that these are not member functions.)
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 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

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`.

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 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`.

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.

## Member Enumeration Documentation

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 |

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

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`.

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

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.

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.

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`.

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.

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.

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.

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.

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.

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`.

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` \( \geq0\).

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

template<typename Dt , typename ExactAlphaComparisonTag >

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

Inserts the alpha shape `A` for the current alpha value into the stream `os`.

Defined in `CGAL/IO/io.h`

Precondition
  The insert operator must be defined for `Point`.

template<typename Dt , typename ExactAlphaComparisonTag >

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

Inserts the alpha shape `A` for the current alpha value into the Geomview stream `W`.

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`