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Fix doc according to Jean-Daniel's review

This commit is contained in:
Clement Jamin 2015-10-09 15:35:17 +02:00
parent 0cfa5d0a71
commit 9b879d294e
1 changed files with 15 additions and 6 deletions
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21
Triangulation/doc/Triangulation/Triangulation.txt
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@ -283,8 +283,7 @@ full cells adjacent to `c` are automatically subdivided to match the
subdivision of the full cell `c`. The barycentric subdivision of `c` is subdivision of the full cell `c`. The barycentric subdivision of `c` is
obtained by enumerating all the faces of `c` in order of decreasing obtained by enumerating all the faces of `c` in order of decreasing
dimension, from the dimension of `c` to dimension 1, and inserting a new dimension, from the dimension of `c` to dimension 1, and inserting a new
vertex in each face. For the enumeration, we use a combination enumerator, vertex in each face.
which is not documented, but provided in \cgal.
\cgalFigureBegin{triangulationfigbarycentric,barycentric-subdivision.png} \cgalFigureBegin{triangulationfigbarycentric,barycentric-subdivision.png}
Barycentric subdivision in dimension \f$ d=2\f$. Barycentric subdivision in dimension \f$ d=2\f$.
@ -416,7 +415,10 @@ in the conflict zone are removed, leaving a hole that contains `p`. That
hole is ``star shaped'' around `p` and thus is re-triangulated using hole is ``star shaped'' around `p` and thus is re-triangulated using
`p` as a center vertex. `p` as a center vertex.
Delaunay triangulations also support vertex removal. Delaunay triangulations support insertion of points, removal of vertices,
and localization of a query point inside the triangulation.
Note that inserting a set of points at once is much faster
than inserting the points one by one.
## Implementation ## ## Implementation ##
@ -451,7 +453,7 @@ The class `CGAL::Regular_triangulation<RegularTriangulationTraits, Triangulation
`CGAL::Triangulation<RegularTriangulationTraits, TriangulationDataStructure>` `CGAL::Triangulation<RegularTriangulationTraits, TriangulationDataStructure>`
and represents regular triangulations. and represents regular triangulations.
A regular triangulation is similar to Delaunay triangulations, but Regular triangulations are similar to Delaunay triangulations, but
with weighted points. with weighted points.
Let \f$ {S}^{(w)}\f$ be a set of weighted points in \f$ \mathbb{R}^D\f$. Let Let \f$ {S}^{(w)}\f$ be a set of weighted points in \f$ \mathbb{R}^D\f$. Let
@ -478,7 +480,12 @@ called the <I>power sphere</I>. A sphere \f$ {z}^{(w)}\f$ is said to be
A triangulation of \f$ {S}^{(w)}\f$ is <I>regular</I> if the power spheres A triangulation of \f$ {S}^{(w)}\f$ is <I>regular</I> if the power spheres
of all simplices are regular. of all simplices are regular.
\warning The removal of points is not supported yet. Regular triangulations support insertion of points,
and localization of a query point inside the triangulation.
Note that inserting a set of points at once is much faster
than inserting the points one by one.
\warning The removal of vertices is not supported yet.
## Implementation ## ## Implementation ##
@ -505,7 +512,9 @@ This simple example shows how to create a regular triangulation.
The current implementation locates points by walking in the The current implementation locates points by walking in the
triangulation, and sorts the points with spatial sort to insert a triangulation, and sorts the points with spatial sort to insert a
set of points. In the worst case, the expected complexity is set of points. In the worst case, the expected complexity is
\f$ O(n^{\lceil\frac{d}{2}\rceil}+n\log n)\f$. \f$ O(n^{\lceil\frac{d}{2}\rceil}+n\log n)\f$. When the algorithm is
run on \f$ n \f$ random points, the cost of inserting one point is
\f$ O(n^{1/d}) \f$.
We provide below (Figure \cgalFigureRef{Triangulationfigbenchmarks}) the We provide below (Figure \cgalFigureRef{Triangulationfigbenchmarks}) the
performance of the Delaunay triangulation on randomly distributed points. performance of the Delaunay triangulation on randomly distributed points.