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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
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
vertex in each face. For the enumeration, we use a combination enumerator,
which is not documented, but provided in \cgal.
vertex in each face.
\cgalFigureBegin{triangulationfigbarycentric,barycentric-subdivision.png}
Barycentric subdivision in dimension \f$ d=2\f$.
@ -416,7 +415,10 @@ in the conflict zone are removed, leaving a hole that contains `p`. That
hole is ``star shaped'' around `p` and thus is re-triangulated using
`p` as a center vertex.
Delaunay triangulations also support vertex removal.
Delaunay triangulations support insertion of points, removal of vertices,
and localization of a query point inside the triangulation.
Note that inserting a set of points at once is much faster
than inserting the points one by one.
## Implementation ##
@ -451,7 +453,7 @@ The class `CGAL::Regular_triangulation<RegularTriangulationTraits, Triangulation
`CGAL::Triangulation<RegularTriangulationTraits, TriangulationDataStructure>`
and represents regular triangulations.
A regular triangulation is similar to Delaunay triangulations, but
Regular triangulations are similar to Delaunay triangulations, but
with weighted points.
Let \f$ {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
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 ##
@ -505,7 +512,9 @@ This simple example shows how to create a regular triangulation.
The current implementation locates points by walking in the
triangulation, and sorts the points with spatial sort to insert a
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
performance of the Delaunay triangulation on randomly distributed points.