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Andreas Fabri
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@ -40,7 +40,7 @@ that the \f$ \alpha\f$-shape degenerates to the point-set \f$ S\f$ for
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will prevent us even from moving the spoon between two points since
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it's way too large. So we will never spoon up ice-cream lying in the
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inside of the convex hull of \f$ S\f$, and hence the \f$ \alpha\f$-shape for
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\f$ \alpha \rightarrow \infty\f$ is the convex hull of \f$ S\f$.\footnote{ice cream, ice cream!!! The wording of this introductory paragraphs is borrowed from Kaspar Fischer's " Introduction to Alpha Shapes" which can be found at http://people.inf.ethz.ch/fischerk/pubs/as.pdf. The picture has been taken from Walter Luh's homepage at http://www.stanford.edu/&wtilde;luh/cs448b/alphashapes.html.}
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\f$ \alpha \rightarrow \infty\f$ is the convex hull of \f$ S\f$.
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# Definitions # {#Alpha_shapes_2Definitions}
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@ -77,7 +77,7 @@ finite number of different \f$ \alpha\f$-shapes and corresponding
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\section I1_SectAlpha_Shape_2 Functionality
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The class `CGAL::Alpha_shape_2<Dt>` represents the family of
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The class `Alpha_shape_2<Dt>` represents the family of
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\f$ \alpha\f$-shapes of points in a plane for <I>all</I> positive
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\f$ \alpha\f$. It maintains the underlying triangulation `Dt` which
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represents connectivity and order among squared radius of its faces. Each
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@ -86,7 +86,7 @@ that specifies for which values of \f$ \alpha\f$ the face belongs to the
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\f$ \alpha\f$-shape. There are links between the intervals and the
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\f$ k\f$-dimensional faces of the triangulation.
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The class `CGAL::Alpha_shape_2<Dt>` provides functions to set and
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The class `Alpha_shape_2<Dt>` provides functions to set and
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get the current \f$ \alpha\f$-value, as well as an iterator that enumerates
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the \f$ \alpha\f$-values where the \f$ \alpha\f$-shape changes.
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@ -114,23 +114,23 @@ points cannot be inserted or removed.
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We currently do not specify concepts for the underlying triangulation
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type. Models that work for a basic alpha-shape are the classes
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`CGAL::Delaunay_triangulation_2` and
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`CGAL::Triangulation_hierarchy_2` templated with a Delaunay
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`Delaunay_triangulation_2` and
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`Triangulation_hierarchy_2` templated with a Delaunay
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triangulation. A model that works for a weighted alpha-shape is
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the class `CGAL::Regular_triangulation_2`.
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the class `Regular_triangulation_2`.
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The triangulation needs a geometric traits class as argument.
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The requirements of this class are described in the
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concept `CGAL::AlphaShapeTraits_2` for which
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concept `AlphaShapeTraits_2` for which
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the \cgal kernels
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and `CGAL::Weighted_alpha_shape_euclidean_traits_2` are models.
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and `Weighted_alpha_shape_euclidean_traits_2` are models.
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There are no requirements on the triangulation data structure.
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However it must be parameterized with
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vertex and face classes, which are model of the concepts
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`AlphaShapeVertex_2` and `AlphaShapeFace_2`,
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by default the classes `CGAL::Alpha_shape_vertex_base_2<Gt>`
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and `CGAL::Alpha_shape_face_base_2<Gt>`.
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by default the classes `Alpha_shape_vertex_base_2<Gt>`
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and `Alpha_shape_face_base_2<Gt>`.
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# Examples # {#Alpha_shapes_2Examples}
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@ -144,9 +144,9 @@ For the geometric traits class we can use a \cgal kernel.
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For the triangulation data structure traits, we have to
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choose the vertex and face classes needed for alpha shapes,
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namely `CGAL::Alpha_shape_vertex_base_2<Gt, Dv>` and `CGAL::Alpha_shape_face_base_2<Gt,Df>`.
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As default vertex and face type they use `CGAL::Triangulation_vertex_base_2<Gt>`
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and `CGAL::Triangulation_face_base_2<Gt>` respectively.
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namely `Alpha_shape_vertex_base_2<Gt, Dv>` and `Alpha_shape_face_base_2<Gt,Df>`.
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As default vertex and face type they use `Triangulation_vertex_base_2<Gt>`
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and `Triangulation_face_base_2<Gt>` respectively.
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The following code snippet shows how to obtain a basic
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alpha shape type.
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@ -190,7 +190,7 @@ typedef CGAL::Alpha_shape_2<Ht> Alpha_shape_2;
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A weighted alpha shape, needs a regular triangulation as
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underlying triangulation `Dt`, and it needs a particular
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face class, namely `CGAL::Regular_triangulation_face_base_2<Gt>`.
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face class, namely `Regular_triangulation_face_base_2<Gt>`.
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Note that there is no special weighted alpha shape class.
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\code{.cpp}
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@ -19,13 +19,13 @@ for <I>weighted</I> Alpha Shapes.
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### Parameters ###
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The template parameter `Dt` has to be either `Delaunay_triangulation_2` or `Regular_triangulation_2`.
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Note that `DT::Geom_traits`, `DT::Vertex` and `DT::Face` must model the concepts `AlphaShapeTraits_2`,
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`AlphaShapeVertex_2` and `AlphaShapeFace_2` respectively.
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Note that `Dt::Geom_traits`, `Dt::Vertex` and `Dt::Face` must be model the concepts `AlphaShapeTraits_2`,
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`AlphaShapeVertex_2` and `AlphaShapeFace_2`, respectively.
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The template parameter `ExactAlphaComparisonTag` is a tag that, when set to
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`CGAL::Tag_true`, triggers exact comparisons between alpha values. This is useful
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\link Tag_true `Tag_true`\endlink, triggers exact comparisons between alpha values. This is useful
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when the underlying triangulation is instantiated with an exact predicates inexact constructions
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kernel. By default the `ExactAlphaComparisonTag` is set to `CGAL::Tag_false` as it induces a small
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kernel. By default the `ExactAlphaComparisonTag` is set to \link Tag_false `Tag_false`\endlink as it induces a small
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overhead. Note that since such a strategy does not make sense if used together with a traits class with exact constructions,
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the tag `ExactAlphaComparisonTag` is not taken into account if `Dt::Geom_traits::FT` is not a floating point number type.
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@ -35,7 +35,7 @@ Inherits from `Dt`.
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This class is the underlying triangulation class.
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The modifying functions `insert` and `remove` will overwrite
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The modifying functions `Alpha_shape_2::insert()` and `Alpha_shape_2::remove()` will overwrite
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the inherited functions. At the moment, only the static version is implemented.
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### I/O ###
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@ -53,12 +53,12 @@ The cross links between the intervals and the \f$ k\f$-dimensional faces of the
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triangulation are realized using methods in the \f$ k\f$-dimensional faces
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themselves.
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`A.alpha_find` uses linear search, while
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`A.alpha_lower_bound` and `A.alpha_upper_bound`
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`Alpha_shape_2::alpha_find()` uses linear search, while
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`Alpha_shape_2::alpha_lower_bound()` and `Alpha_shape_2::alpha_upper_bound()`
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use binary search.
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`A.number_of_solid_components` performs a graph traversal and takes time
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`Alpha_shape_2::number_of_solid_components` performs a graph traversal and takes time
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linear in the number of faces of the underlying triangulation.
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`A.find_optimal_alpha` uses binary search and takes time
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`Alpha_shape_2::find_optimal_alpha` uses binary search and takes time
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\f$ O(n \log n)\f$, where \f$ n\f$ is the number of points.
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*/
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@ -6,17 +6,15 @@ namespace CGAL {
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The class `Alpha_shape_face_base_2` is the default model for the concept `AlphaShapeFace_2`.
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### Parameters ###
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\tparam Traits has to be a model of `AlphaShapeTraits_2`.
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The template parameter `Traits` has to be a model of `AlphaShapeTraits_2`.
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The template parameter `Fb` has to be a model of `TriangulationFaceBase_2` (or `RegularTriangulationFaceBase_2`)
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\tparam Fb has to be a model of `TriangulationFaceBase_2` (or `RegularTriangulationFaceBase_2`)
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if `Alpha_shape_face_base_2` is intended to be used with an alpha-shape class based on a
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`Delaunay_triangulation_2` (or a `Regular_triangulation_2`).
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The template parameter `ExactAlphaComparisonTag` is a tag that, when set to
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`CGAL::Tag_true`, triggers exact comparisons between alpha values. See the description
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provided in the documentation of `Alpha_shape_2` for more details. The default value is `CGAL::Tag_false`.
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\tparam ExactAlphaComparisonTag is a tag that, when set to
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\link Tag_true `Tag_true`\endlink, triggers exact comparisons between alpha values. See the description
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provided in the documentation of `Alpha_shape_2` for more details. The default value is \link Tag_false `Tag_false`\endlink.
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\models ::AlphaShapeFace_2
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*/
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@ -7,17 +7,15 @@ namespace CGAL {
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The class `Alpha_shape_vertex_base_2` is the default model for the concept
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`AlphaShapeVertex_2`.
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### Parameters ###
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\tparam Traits has to be a model of `AlphaShapeTraits_2`.
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The template parameter `Traits` has to be a model of `AlphaShapeTraits_2`.
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The template parameter `Vb` has to be a model of `TriangulationVertexBase_2` (or `RegularTriangulationVertexBase_2`)
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\tparam Vb has to be a model of `TriangulationVertexBase_2` (or `RegularTriangulationVertexBase_2`)
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if `Alpha_shape_vertex_base_2` is intended to be used with an alpha-shape class based on a
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`Delaunay_triangulation_2` (or a `Regular_triangulation_2`).
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The template parameter `ExactAlphaComparisonTag` is a tag that, when set to
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`CGAL::Tag_true`, triggers exact comparisons between alpha values. See the description
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provided in the documentation of `Alpha_shape_2` for more details. The default value is `CGAL::Tag_false`.
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\tparam ExactAlphaComparisonTag is a tag that, when set to
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\link Tag_true `Tag_true`\endlink, triggers exact comparisons between alpha values. See the description
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provided in the documentation of `Alpha_shape_2` for more details. The default value is \link Tag_false `Tag_false`\endlink.
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\models ::AlphaShapeVertex_2
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*/
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@ -4,7 +4,7 @@
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/*!
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\addtogroup PkgAlphaShape2
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\todo check generated documentation
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\PkgDescriptionBegin{2D Alpha Shapes,PkgAlphaShape2Summary}
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\PkgPicture{alpha-detail.png}
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\PkgAuthors{Tran Kai Frank Da}
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@ -31,10 +31,10 @@ triangulation and the other on its generalization, the regular triangulation,
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replacing the natural distance by the power to weighted points. The metric used
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determines an underlying triangulation of the alpha shape and thus, the version
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computed.
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The <I>basic alpha shape</I> (cf. \ref I1_SectClassicAS2D ) is associated with the Delaunay triangulation
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(cf. \ref Section_2D_Triangulations_Delaunay ).
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The <I>basic alpha shape</I> (cf. \ref I1_SectClassicAS2D) is associated with the Delaunay triangulation
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(cf. Section \ref Section_2D_Triangulations_Delaunay "Delaunay Triangulations").
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The <I>weighted alpha shape</I> (cf. \ref I1_SectWeightedAS2D ) is associated with the regular triangulation
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(cf. \ref Section_2D_Triangulations_Regular ).
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(cf. Section \ref Section_2D_Triangulations_Regular "Regular Triangulations").
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There is a close connection between alpha shapes and the underlying
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triangulations. More precisely, the \f$ \alpha\f$-complex of \f$ S\f$ is a
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