mirror of https://github.com/CGAL/cgal
edits on the rest of the chapter - to be read and checked
some todos left
This commit is contained in:
Monique Teillaud
parent
b21dd31d5b
commit
129c79a711
|
|
@ -55,7 +55,7 @@ the action of \f$\mathcal G\f$ if there exists an element
|
|||
\f$\mathcal{M}\f$ is defined as the quotient of \f$\mathbb H^2\f$
|
||||
under the action of \f$\mathcal G\f$, named the fundamental group of
|
||||
\f$\mathcal M\f$: \f[ \mathcal{M} = \mathbb H^2 / \mathcal{G} \f]
|
||||
\f$pi\f$ denotes the natural projection map from \f$\mathbb H^2\f$
|
||||
\f$\pi\f$ denotes the natural projection map from \f$\mathbb H^2\f$
|
||||
onto \f$\mathcal M\f$.
|
||||
|
||||
\cgalFigureAnchor{P4HTriangulationOctagonId}
|
||||
|
|
@ -70,7 +70,7 @@ onto \f$\mathcal M\f$.
|
|||
<b>Center:</b> Illustration of periodicity in the hyperbolic
|
||||
plane. The figure shows a few periodic copies of the points in the central octagon.
|
||||
<b>Right:</b> The half-open octagon \f$\mathcal D\f$ is an original domain for \f$\mathcal{M}\f$. Note that
|
||||
only vertex of the octagon is included in the original domain.
|
||||
only one vertex of the octagon is included in the original domain.
|
||||
\cgalFigureCaptionEnd
|
||||
|
||||
By definition, all points of \f$\mathbb H^2\f$ that belong to the same
|
||||
|
|
@ -102,45 +102,43 @@ D\f$ under the action of \f$\mathcal G\f$.
|
|||
\cgalFigureCaptionEnd
|
||||
|
||||
|
||||
|
||||
\section P4HT2_representation Representation
|
||||
|
||||
TODO TODO TODO TODO TODO TODO
|
||||
TODO TODO TODO TODO TODO TODO
|
||||
|
||||
The construction of triangulations of \f$\mathcal{M}\f$ works in the hyperbolic plane \f$\mathbb H^2\f$.
|
||||
We require that all input points lie inside \f$\mathcal D\f$.
|
||||
\cgalModifBegin begin: ? \cgalModifEnd
|
||||
When there is no risk
|
||||
of confusion, the same notation will be used for a point on the
|
||||
surface \f$\mathcal M\f$ and its representative in \f$\mathcal
|
||||
D\f$. Similarly, \f$\mathcal{P}\f$ denotes both a set of points on the
|
||||
surface and the set of their representatives in \f$\mathcal
|
||||
D\f$.
|
||||
\cgalModifBegin end: ? \cgalModifEnd
|
||||
|
||||
Edges and faces, however, may
|
||||
cross the boundary of \f$\mathcal D\f$, i.e., they may be defined in terms of points
|
||||
both inside and outside \f$\mathcal D\f$. The points outside \f$\mathcal D\f$ are images
|
||||
of points inside \f$\mathcal D\f$ under the action of some translation in the group
|
||||
\f$\mathcal{G}\f$. So, to specify an edge or a face, we use two or three pairs of points
|
||||
and translations in \f$\mathcal{G}\f$; points in the original domain are paired with the
|
||||
identity translation \f$\mathbb 1\f$.
|
||||
\cgalModifBegin
|
||||
Note that equivalent translations of the group \f$\mathcal G\f$ can be reduced to a
|
||||
unique minimal representative, so a point-translation pair is uniquely defined.
|
||||
\cgalModifEnd
|
||||
The Delaunay triangulation of \f$\mathcal{M}\f$ defined by a point set
|
||||
\f$\mathcal{P}\f$ is defined as the projection by \f$\pi\f$ of the
|
||||
Delaunay triangulation in the plane \f$\mathbb H^2\f$ of the
|
||||
(infinite) set of points \f$\mathcal{G P}\f$ onto \f$\mathcal{M}\f$,
|
||||
under some validity conditions that will be detailed in ... .
|
||||
\cgalModifBegin TODO add ref to sec validity \cgalModifEnd
|
||||
|
||||
We require that all input points lie inside \f$\mathcal D\f$. As for
|
||||
orbits of points, all faces of the Delaunay triangulation of
|
||||
\f$\mathcal{G P}\f$ that are in the same orbit under the action of
|
||||
\f$\mathcal{G}\f$ project by \f$\pi\f$ onto the same face on the
|
||||
surface \f$\mathcal{M}\f$; the data structure stores a unique
|
||||
<i>canonical</i> representative of each orbit, which has at least one
|
||||
vertex in \f$\mathcal D\f$. Some faces have vertices both inside and
|
||||
outside \f$\mathcal D\f$. Such faces can be uniquely specified by
|
||||
three pairs of points in \f$\mathcal D\f$ and (reduced) translations
|
||||
of \f$\mathcal{G}\f$; points in the original domain are paired with
|
||||
the identity translation \f$\mathbb 1\f$. The underlying combinatorial
|
||||
triangulation is a \ref PkgTDS2Summary, enriched in each face by the
|
||||
three translations that are paired with the point in each vertex. See
|
||||
\cgalFigureRef{P4HTriangulationOrientationDS}.
|
||||
|
||||
A triangulation is a collection of vertices and faces that are linked together through incidence
|
||||
and adjacency relations. Each vertex stores a point, and gives access to one of its incident
|
||||
faces. Each face gives access to its three adjacent faces, to its three incident vertices, and
|
||||
to the translation paired with the point in each vertex.
|
||||
|
||||
The three vertices of a face are indexed with 0, 1, and 2 in positive (counter-clockwise)
|
||||
orientation. The orientation of a face in \f$\mathcal{M}\f$ is defined as the orientation of
|
||||
its representative in \f$\mathbb H^2\f$ defined by the translations it stores.
|
||||
|
||||
As in the underlying combinatorial triangulation (see the documentation for \ref PkgTDS2Summary),
|
||||
the neighbors of a face are indexed with 0, 1, 2 in such a
|
||||
way that the \f$i\f$-th neighbor is opposite to the vertex with the same index. Edges are not
|
||||
explicitly represented: an edge is given by a face and the index of its opposite vertex
|
||||
in that face.
|
||||
\cgalModifBegin
|
||||
See \cgalFigureRef{P4HTriangulationOrientationDS}.
|
||||
\cgalModifEnd
|
||||
\cgalModifBegin TODO make the translation more visible than the rest on the fig \cgalModifEnd
|
||||
|
||||
\cgalFigureAnchor{P4HTriangulationOrientationDS}
|
||||
<center>
|
||||
|
|
@ -148,146 +146,112 @@ See \cgalFigureRef{P4HTriangulationOrientationDS}.
|
|||
</center>
|
||||
\cgalFigureCaptionBegin{P4HTriangulationOrientationDS}
|
||||
Representation of a face \f$f\f$ stored in the triangulation data structure.
|
||||
Each vertex \f$v_i\f$ stores a point \f$p_i\f$ that is paired with a translation
|
||||
Each vertex \f$v_i\f$ stores a point \f$p_i\f$ paired with a translation
|
||||
\f$\tau_i\f$. Opposite to vertex \f$i\f$ is the \f$i\f$-th neighbor of \f$f\f$.
|
||||
\cgalFigureCaptionEnd
|
||||
|
||||
In the data structure each vertex stores the input point in \f$\mathcal D\f$ to which it
|
||||
corresponds. For every face on \f$\mathcal{M}\f$, the data structure stores the <i>canonical
|
||||
representative</i> of that face in \f$\mathbb H^2\f$.
|
||||
|
||||
We introduce some notation necessary for the definition of canonical representative.
|
||||
Let \f$\mathcal N\f$ denote the translations of \f$\mathcal G\f$ for which the image
|
||||
of \f$\mathcal D_O\f$ has at least one vertex in common with \f$\mathcal D_O\f$. Let
|
||||
\f$\mathcal D_{\mathcal N}\f$ be the union of \f$\mathcal D_O\f$ and these images. With the use
|
||||
of dummy points (see \ref P4HT2_Dummy_points), faces with at least one vertex in
|
||||
\f$\mathcal D\f$ are contained in \f$\mathcal D_{\mathcal N}\f$. We consider the images
|
||||
of \f$\mathcal D_O\f$ to be ordered counter-clockwise around \f$\mathcal D_O\f$,
|
||||
arbitrarily starting with the one labeled \f$abcd\f$.
|
||||
\cgalModifBegin
|
||||
See \cgalFigureRef{P4HTriangulationCanonicalRepExample}.
|
||||
\cgalModifEnd
|
||||
We then choose the canonical representative in \f$\mathbb H^2\f$ of a face on
|
||||
\f$\mathcal M\f$ so that:
|
||||
<ul>
|
||||
More precisely, the translations are elements of the subset
|
||||
\f$\mathcal N\f$ of \f$\mathcal G\f$ for which the image of
|
||||
\f$\mathcal D_O\f$ has at least one vertex in common with
|
||||
\f$\mathcal D_O\f$.
|
||||
See \cgalFigureRef{P4HTriangulationCanonicalRepExample}.
|
||||
We consider the images of \f$\mathcal D_O\f$ under translations of
|
||||
\f$\mathcal N\f$ to be ordered counterclockwise around
|
||||
\f$\mathcal D_O\f$, arbitrarily starting with the one corresponding to
|
||||
translation \f$abcd\f$. The canonical representative in
|
||||
\f$\mathbb H^2\f$ of a face on \f$\mathcal M\f$ is such that
|
||||
<ul>
|
||||
<li> either all vertices of the representative lie in \f$\mathcal D\f$, or
|
||||
<li> the representative has at least one vertex in \f$\mathcal D\f$ and is as close as
|
||||
possible to \f$abcd\f$ in the ordering defined above.
|
||||
</ul>
|
||||
|
||||
<li> the representative has at least one vertex in \f$\mathcal D\f$
|
||||
and is as close as possible to \f$abcd\f$ in the ordering defined above.
|
||||
</ul>
|
||||
\cgalModifBegin todo show p,q,r on the fig, otherwise
|
||||
the caption makes no sense... \cgalModifEnd
|
||||
|
||||
\cgalFigureAnchor{P4HTriangulationCanonicalRepExample}
|
||||
<center>
|
||||
<img src="periodic_face.svg" style="max-width:35%; width=35%; display: inline-block;"/>
|
||||
</center>
|
||||
\cgalFigureCaptionBegin{P4HTriangulationCanonicalRepExample}
|
||||
Illustration of the canonical representative in \f$\mathbb H^2\f$ of a face on
|
||||
\f$\mathcal M\f$. Among the three candidate faces, the canonical representative
|
||||
is the green one, which is closest to the region labeled \f$abcd\f$
|
||||
clockwise. We store in each face the translations to apply to obtain the
|
||||
face's canonical representative in
|
||||
\f$\mathbb H^2\f$, in this case \f$\mathbb 1, a\f$, and \f$\overline{b}\f$,
|
||||
corresponding to the points \f$p, q\f$, and \f$r\f$.
|
||||
Among the three faces in the orbit that have at least one vertex in
|
||||
\f$\mathcal D\f$, the canonical representative is the green one: it is
|
||||
closest, in the counterclockwise order, to the region labeled
|
||||
\f$abcd\f$. The translations \f$\mathbb 1, a\f$, and
|
||||
\f$\overline{b}\f$, corresponding to the points \f$p, q\f$, and
|
||||
\f$r\f$ are stored in the face in the data structure.
|
||||
\cgalFigureCaptionEnd
|
||||
|
||||
|
||||
<b>Validity</b> <br>
|
||||
A periodic hyperbolic triangulation of \f$\mathcal{M}\f$ is said to be `valid` if and only if
|
||||
\cgalModifBegin
|
||||
\section P4HT2_validity Validity
|
||||
|
||||
Let us now give details on the validity condition mentioned above.
|
||||
A periodic hyperbolic triangulation is said to be `valid` if and only if
|
||||
<ol>
|
||||
<li> Its underlying combinatorial graph, the triangulation data structure, is `valid`
|
||||
(see \ref PkgTDS2Summary).
|
||||
<li> Its underlying combinatorial graph, the triangulation data structure, is `valid`
|
||||
(see \ref PkgTDS2Summary) \cgalModifBegin todo give more
|
||||
precise link \cgalModifEnd and each face of the triangulation is positively oriented. See
|
||||
\cgalFigureRef{P4HTriangulationOrientationDS}.
|
||||
<li> All triangles are topological disks; \cgalModifBegin question
|
||||
bete : what else could it be? \cgalModifEnd
|
||||
this condition is satisfied when the Euler
|
||||
relation \f$V-E+F = -2\f$ is verified, where \f$V, E, F\f$ are the number
|
||||
of vertices, edges and faces of the triangulation, respectively.
|
||||
<li> The combinatorial graph of the triangulation does not contain cycles of length 1
|
||||
or 2; this condition is equivalent to requiring that the triangulation is a simplicial
|
||||
complex.
|
||||
<li> Each face of the triangulation is positively oriented. See
|
||||
\cgalFigureRef{P4HTriangulationOrientationDS}.
|
||||
<li> All triangles are topological disks; this condition is satisfied when the Euler
|
||||
relation \f$V-E+F = -2\f$ is verified, where \f$V, E, F\f$ are the number
|
||||
of vertices, edges and faces of the triangulation, respectively.
|
||||
</ol>
|
||||
\cgalModifEnd
|
||||
|
||||
|
||||
\section P4HT2_Delaunay Hyperbolic Delaunay triangulation
|
||||
|
||||
The class `Periodic_4_hyperbolic_Delaunay_triangulation_2` implements Delaunay triangulations of
|
||||
\f$\mathcal M\f$. The prefix "Periodic_4" emphasizes that the triangulation is periodic in the four
|
||||
directions defined by the hyperbolic translations \f$ a,b,c\f$, and \f$d\f$.
|
||||
|
||||
Delaunay triangulations have the <I>empty circle property</I>, that is, the circumscribing
|
||||
hyperbolic circle of each face does not contain any other vertex of the triangulation in its
|
||||
interior. These triangulations are uniquely defined if no more than three points are co-circular.
|
||||
Note however that the \cgal implementation computes a unique triangulation even in
|
||||
these cases \cgalCite{cgal:dt-pvr3d-03}. Note also that since hyperbolic circles coincide with
|
||||
Euclidean circles in the Poincaré disk model, the implementation uses Euclidean predicates to
|
||||
ensure the Delaunay property.
|
||||
|
||||
This implementation is fully dynamic: it supports both insertions of points and vertex removal.
|
||||
\cgalModifBegin
|
||||
Note, however, that the removal of a vertex is authorized only if the triangulation remains
|
||||
valid (as defined above). If this condition is not satisfied, then the vertex is <b>not</b>
|
||||
removed from the triangulation.
|
||||
\cgalModifEnd
|
||||
|
||||
|
||||
\section P4HT2_Dummy_points Dummy points
|
||||
|
||||
Some point sets do not admit a triangulation of \f$\mathcal M\f$.
|
||||
\cgalModifBegin
|
||||
For instance, a single point does not define a triangulation of \f$\mathcal M\f$, as the result
|
||||
would not be a simplicial complex.
|
||||
\cgalModifEnd
|
||||
Some point sets do not admit a triangulation of \f$\mathcal M\f$. For
|
||||
instance, a single point does not define a valid triangulation of
|
||||
\f$\mathcal M\f$, as the induced subdivision would not be a simplicial
|
||||
complex. Another example is shown in
|
||||
\cgalFigureRef{P4HNonSimplicialExample}.
|
||||
|
||||
\cgalFigureAnchor{P4HNonSimplicialExample}
|
||||
<center>
|
||||
<img src="non-triangulation.svg" style="max-width:45%; width=45%; display: inline-block;"/>
|
||||
</center>
|
||||
\cgalFigureCaptionBegin{P4HNonSimplicialExample}
|
||||
Example of a non-simplicial decomposition of the Bolza surface. Note that with the three
|
||||
points in the central octagon, we obtain a decomposition that is <b>not</b> a triangulation,
|
||||
since it is non-simplicial. On the figure a few cycles of length 2 are shown in color. Such
|
||||
cycles are double edges in the triangulation. Note also the pink edge between the two blue
|
||||
vertices: it corresponds to a loop on the surface.
|
||||
The three
|
||||
points in the central octagon define a subdivision of the Bolza surface that is <b>not</b> a triangulation,
|
||||
since it is non-simplicial. A few cycles of length 2 are shown in
|
||||
color, leading to double edges on the surface. The pink edge between the two blue
|
||||
vertices corresponds to a loop (cycle of length 1) on the surface.
|
||||
\cgalFigureCaptionEnd
|
||||
|
||||
For this reason, we initialize a triangulation of \f$\mathcal M\f$ with a predetermined set of
|
||||
14 points, called <i>dummy points</i>, whose triangulation in \f$\mathcal M\f$ is a simplicial
|
||||
complex and has the empty circle property. The set of dummy points has been proposed in
|
||||
We initialize a triangulation of \f$\mathcal M\f$ with a predetermined set of
|
||||
14 points, called <i>dummy points</i>, which ensures that the
|
||||
projection by $\f\pi\f$ of the Delaunay triangulation of \f$\mathcal{G P}\f$ onto
|
||||
\f$\mathcal M\f$ is a simplicial
|
||||
complex for any set of input points \f$\mathcal{P}\f$
|
||||
\cgalCite{cgal:btv-dtosl-16}.
|
||||
|
||||
\cgalModifBegin
|
||||
See \cgalFigureRef{P4HTriangulationDummyPoints}
|
||||
\cgalModifEnd
|
||||
|
||||
\cgalFigureAnchor{P4HTriangulationDummyPoints}
|
||||
<center>
|
||||
<img src="dummy-points.png" style="max-width:35%; width=35%; display: inline-block;"/>
|
||||
</center>
|
||||
\cgalFigureCaptionBegin{P4HTriangulationDummyPoints}
|
||||
Delaunay triangulation of \f$\mathcal M\f$ with the 14 dummy points. The set of dummy points
|
||||
Delaunay triangulation of \f$\mathcal M\f$ defined by the 14 dummy points. The set of dummy points
|
||||
contains the origin, the vertex of \f$\mathcal D\f$, the midpoints of the closed sides of
|
||||
\f$D\f$, and the hyperbolic midpoints of the segments between the origin and the vertices
|
||||
of the octagon.
|
||||
\cgalFigureCaptionEnd
|
||||
|
||||
\cgalModifBegin
|
||||
After the triangulation has been initialized with the dummy points, new input points can be
|
||||
inserted without ever losing the simplicial complex property. If sufficiently many well-distributed
|
||||
If sufficiently many well-distributed
|
||||
points are inserted, the dummy points become unnecessary, i.e., the triangulation remains a
|
||||
simplicial complex even if we remove them.
|
||||
If the set of input points is not sufficiently large, or if the points
|
||||
are not well-distributed on the surface, then the final triangulation will contain the
|
||||
dummy points necessary to guarantee that the triangulation is a simplicial complex.
|
||||
\cgalModifEnd
|
||||
|
||||
\cgalModifBegin tout ce qui concerne les experiences pourrait eventuellement
|
||||
descendre dans la sec Performance \cgalModifEnd
|
||||
Experiments have
|
||||
shown that, on average, all dummy points are removed from the triangulation with the insertion
|
||||
of less than 200 random points uniformly distributed (according to the Euclidean metric) in
|
||||
the unit disk.
|
||||
|
||||
\cgalModifBegin
|
||||
See \cgalFigureRef{P4HDummyPointsHistogram}.
|
||||
\cgalModifEnd
|
||||
the unit disk. See \cgalFigureRef{P4HDummyPointsHistogram}.
|
||||
\cgalModifBegin TODO cite socg IT \f$\mathcal{G P}\f$ \cgalModifEnd
|
||||
|
||||
\cgalFigureBegin{P4HDummyPointsHistogram, histogram-dummy-points.png}
|
||||
Histogram of the number of random input points needed to remove all dummy points in the
|
||||
|
|
@ -305,15 +269,39 @@ See \cgalFigureRef{P4HDummyPointsHistogram}.
|
|||
|
||||
\section P4HT2_design Software Design
|
||||
|
||||
\cgalModifBegin
|
||||
The two main classes `Periodic_4_hyperbolic_Delaunay_triangulation_2` and
|
||||
|
||||
The class `Periodic_4_hyperbolic_Delaunay_triangulation_2` implements
|
||||
Delaunay triangulations of the Bolza surface
|
||||
\f$\mathcal M\f$. The prefix "Periodic_4" emphasizes that the
|
||||
triangulation in the universal covering \f$\mathbb H^2\f$ is periodic in the four
|
||||
directions defined by the hyperbolic translations \f$ a,b,c\f$, and \f$d\f$.
|
||||
|
||||
Delaunay triangulations have the <I>empty circle property</I>, that is, the circumscribing
|
||||
hyperbolic circle of each face does not contain any other vertex of the triangulation in its
|
||||
interior. These triangulations are uniquely defined if no more than
|
||||
three points are co-circular; however that the \cgal implementation computes a unique triangulation even in
|
||||
these cases \cgalCite{cgal:dt-pvr3d-03}. Note also that since hyperbolic circles coincide with
|
||||
Euclidean circles in the Poincaré disk model, the implementation uses Euclidean predicates to
|
||||
ensure the Delaunay property.
|
||||
|
||||
This implementation is fully dynamic: it supports both insertions of points and vertex removal.
|
||||
Note, however, that the removal of a vertex is authorized only if the triangulation remains
|
||||
valid (as defined above). If this condition is not satisfied, then the vertex is <b>not</b>
|
||||
removed from the triangulation.
|
||||
|
||||
\cgalModifBegin (MT) J'arrete ici \cgalModifEnd
|
||||
|
||||
The two main classes `Periodic_4_hyperbolic_Delaunay_triangulation_2` and
|
||||
`Periodic_4_hyperbolic_triangulation_2` provide high-level geometric functionality and are
|
||||
responsible for the geometric validity of the triangulation.
|
||||
`Periodic_4_hyperbolic_Delaunay_triangulation_2` contains
|
||||
all the functionality that is specific to Delaunay triangulations, such as point insertion and
|
||||
vertex removal, the side-of-circle test, finding the conflicting region of a given point, and dual
|
||||
functions.
|
||||
\cgalModifEnd
|
||||
\cgalModifBegin mais c'est quoi ce truc ??? il n'y a PAS de classe
|
||||
`Periodic_4_hyperbolic_triangulation_2` !!!!!! a revoir completement (en prime c'est completement
|
||||
redondant pour Delaunay, ne pas repeter) \cgalModifEnd
|
||||
|
||||
`Periodic_4_hyperbolic_triangulation_2` contains all the functionality that is common
|
||||
to triangulations in general, such as location of a point in the triangulation
|
||||
\cgalCite{cgal:dpt-wt-02}, and access functions. Note that `Periodic_4_hyperbolic_triangulation_2`
|
||||
|
|
@ -327,36 +315,28 @@ in the software design by the fact that the triangulation classes take two templ
|
|||
necessary for the computation of Delaunay triangulations of the Bolza surface.
|
||||
It also provides the hyperbolic translations generating the group \f$\mathcal G\f$, as well as
|
||||
to the translations in the set \f$\mathcal N\f$.
|
||||
\cgalModifBegin
|
||||
This parameter must meet the requirements described in the concept
|
||||
`Periodic_4HyperbolicDelaunayTriangulationTraits_2`. For more details, see
|
||||
Section \ref P4HT2_traits.
|
||||
\cgalModifEnd
|
||||
|
||||
- the <B>triangulation data structure</B> class, which encodes the combinatorial structure of
|
||||
the triangulation.
|
||||
\cgalModifBegin
|
||||
This parameter must meet the requirements described in the concept
|
||||
`TriangulationDataStructure_2`, and for the purposes of this package is described in
|
||||
Section \ref P4HT2_datastructure.
|
||||
\cgalModifEnd
|
||||
|
||||
\subsection P4HT2_traits The Geometric Traits Parameter
|
||||
|
||||
The geometric traits class is the first template parameter of the periodic hyperbolic
|
||||
triangulation classes.
|
||||
\cgalModifBegin
|
||||
The requirements it must fulfill are described in the concept
|
||||
`Periodic_4HyperbolicDelaunayTriangulationTraits_2`, which extends the concept
|
||||
`HyperbolicDelaunayTriangulationTraits_2`, providing all necessary objects, predicates,
|
||||
and constructions, as well as the hyperbolic translations of the group \f$\mathcal G\f$.
|
||||
\cgalModifEnd
|
||||
|
||||
\cgalModifBegin
|
||||
This package offers a model for the concept `Periodic_4HyperbolicDelaunayTriangulationTraits_2`,
|
||||
which is the class `Periodic_4_hyperbolic_Delaunay_triangulation_traits_2`. The class requires
|
||||
two template parameters:
|
||||
\cgalModifEnd
|
||||
<ul>
|
||||
<li> A `Kernel` with a `number type` that guarantees exact computations on algebraic numbers
|
||||
with nested square roots, for example `CGAL::Cartesian<CORE::Expr>`. This parameter has
|
||||
|
|
@ -380,10 +360,8 @@ The triangulation data structure class is the second template parameter of the p
|
|||
triangulation classes. It is a container for the faces and vertices and maintains
|
||||
incidence and adjacency relations. For details, see Chapter
|
||||
\ref Chapter_2D_Triangulation_Data_Structure.
|
||||
\cgalModifBegin
|
||||
This parameter must meet the requirements described in the concept `TriangulationDataStructure_2`,
|
||||
for which \cgal offers the model `Triangulation_data_structure_2`.
|
||||
\cgalModifEnd
|
||||
This model is itself parameterized by a vertex
|
||||
base class and a face base class, which gives the possibility to customize the vertices and
|
||||
faces used by the triangulation data structure. To represent periodic hyperbolic triangulations,
|
||||
|
|
@ -391,16 +369,12 @@ the face base and vertex base classes must be models of the concepts
|
|||
`Periodic_4HyperbolicTriangulationDSFaceBase_2` and
|
||||
`Periodic_4HyperbolicTriangulationDSVertexBase_2`, respectively.
|
||||
|
||||
\cgalModifBegin
|
||||
The default value for the triangulation data structure parameter in the hyperbolic periodic
|
||||
triangulation classes is
|
||||
`Triangulation_data_structure_2< Periodic_4_hyperbolic_triangulation_face_base_2<Traits>, Periodic_4_hyperbolic_triangulation_vertex_base_2<Traits> >`,
|
||||
where `Traits` is a model of `Periodic_4HyperbolicDelaunayTriangulationTraits_2`.
|
||||
\cgalModifEnd
|
||||
|
||||
\cgalModifBegin
|
||||
\section P4HT2_examples Examples
|
||||
\cgalModifEnd
|
||||
\section P4HT2_examples Example
|
||||
|
||||
This example shows the initialization of a periodic hyperbolic 2D Delaunay triangulation, the
|
||||
insertion of random points uniformly distributed in the unit disk for the Euclidean metric, and the
|
||||
|
|
@ -409,9 +383,11 @@ properties of the triangulation after the insertion. It uses the default paramet
|
|||
|
||||
\cgalExample{Periodic_4_hyperbolic_triangulation_2/p4ht2_example_insertion.cpp}
|
||||
|
||||
\cgalModifBegin TODO lien vers exemples T2 pour fonctionnalites, dire
|
||||
que c'est similaire \cgalModifEnd
|
||||
|
||||
\section P4HT2_Performance Performance
|
||||
|
||||
\cgalModifBegin
|
||||
We perform tests to measure the insertion execution time of our implementation against existing
|
||||
\cgal code. We generate random points, uniformly distributed in the unit disk with respect to
|
||||
the Euclidean metric. From the generated points, we select 1 million points that lie in the original
|
||||
|
|
@ -438,7 +414,7 @@ have been executed on two machines:
|
|||
<tr><td> Non-periodic Euclidean (`double`) <td> 1 sec. <td> 1 sec.
|
||||
</table>
|
||||
</center>
|
||||
\cgalModifEnd
|
||||
|
||||
|
||||
\section P4HT2_Design Design and Implementation History
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue