From 9b879d294e2021f58b1dc1af6eb3e477aaf3d970 Mon Sep 17 00:00:00 2001 From: Clement Jamin Date: Fri, 9 Oct 2015 15:35:17 +0200 Subject: [PATCH] Fix doc according to Jean-Daniel's review --- .../doc/Triangulation/Triangulation.txt | 21 +++++++++++++------ 1 file changed, 15 insertions(+), 6 deletions(-) diff --git a/Triangulation/doc/Triangulation/Triangulation.txt b/Triangulation/doc/Triangulation/Triangulation.txt index 48a213b413e..4964b4c111b 100644 --- a/Triangulation/doc/Triangulation/Triangulation.txt +++ b/Triangulation/doc/Triangulation/Triangulation.txt @@ -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$. @@ -416,7 +415,10 @@ in the conflict zone are removed, leaving a hole that contains `p`. That hole is ``star shaped'' around `p` and thus is re-triangulated using `p` as a center vertex. -Delaunay triangulations also support vertex removal. +Delaunay triangulations support insertion of points, removal of vertices, +and localization of a query point inside the triangulation. +Note that inserting a set of points at once is much faster +than inserting the points one by one. ## Implementation ## @@ -451,7 +453,7 @@ The class `CGAL::Regular_triangulation` and represents regular triangulations. -A regular triangulation is similar to Delaunay triangulations, but +Regular triangulations are similar to Delaunay triangulations, but with weighted points. Let \f$ {S}^{(w)}\f$ be a set of weighted points in \f$ \mathbb{R}^D\f$. Let @@ -478,7 +480,12 @@ called the power sphere. A sphere \f$ {z}^{(w)}\f$ is said to be A triangulation of \f$ {S}^{(w)}\f$ is regular 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.