From 7c3ac1bcd178c0bbb428fa9e079ee30ad21bb76a Mon Sep 17 00:00:00 2001 From: Laurent Rineau Date: Mon, 19 May 2025 15:45:56 +0200 Subject: [PATCH] use the introduction by Mael (rephrased a bit) --- .../Constrained_triangulation_3.txt | 26 +++++++++---------- 1 file changed, 13 insertions(+), 13 deletions(-) diff --git a/Constrained_triangulation_3/doc/Constrained_triangulation_3/Constrained_triangulation_3.txt b/Constrained_triangulation_3/doc/Constrained_triangulation_3/Constrained_triangulation_3.txt index ca17e75d8f7..334b8b9afc1 100644 --- a/Constrained_triangulation_3/doc/Constrained_triangulation_3/Constrained_triangulation_3.txt +++ b/Constrained_triangulation_3/doc/Constrained_triangulation_3/Constrained_triangulation_3.txt @@ -14,21 +14,20 @@ namespace CGAL { \section CT_3_CCDT_3 Conforming Delaunay Triangulations in 3D -This package implements the construction of a 3D constrained Delaunay triangulation, -conforming to the set of faces of a _Piecewise Linear Complex_, PLC for short. -The set of input polygonal constraints given as a PLC is meant to be represented as a -sub-complex of the triangulation. The produced triangulation has the type `Triangulation_3`, -described in the chapter \ref PkgTriangulation3. +3D triangulations are a way to partition space, which is useful in many applications. +Sometimes, we want specific faces to appear in the output, for example, to preserve sharp features. +When a triangulation exactly respects these constraints, it is called a _constrained_ triangulation. +In some cases, it is possible to honor the geometry of the constraints but not their combinatorics; +additional points must be inserted—this is called a _conforming_ triangulation. -For any PLC in 3D, -the algorithm builds a constrained Delaunay triangulation conforming to this PLC. -The constrained triangulation does not always exist, and it may be necessary -to add Steiner vertices to the PLC to make it tetrahedralizable. +This package implements an algorithm that constructs conforming triangulations of +3D polygonal constraints. In particular, it requires that these piecewise linear constraints +are part of a _Piecewise Linear Complex_ (PLC). The type of produced triangulations is +`Triangulation_3`, described in the chapter \ref PkgTriangulation3. The problem is described by Cohen-Steiner et al. \cgalCite{cgal:cohen2002conforming}, and by Si \cgalCite{cgal:si2008cdt3}. - \section CT_3_definitions Definitions In this section, we define the main concepts that have to be understood to use this package. @@ -52,7 +51,7 @@ The polygons can be non-convex, have holes, and arbitrarily many boundary segmen \cgalFigureCaptionBegin{CT_3_plc_fig} -A Piecewise Linear Complex, made of planar faces, connected by edges and vertices. +A Piecewise Linear Complex, made of planar faces connected by edges and vertices. \cgalFigureCaptionEnd @@ -60,10 +59,11 @@ A Piecewise Linear Complex, made of planar faces, connected by edges and vertice The goal of the algorithms developed in this package is to compute a constrained Delaunay mesh containing a given set of polygonal constraints in 3D as sub-complex. -See package \ref PkgTriangulation3 for more details on Delaunay triangulations. A triangulation is a _Delaunay triangulation_ if the circumscribing sphere of any simplex -of the triangulation contains no vertex in its interior. +of the triangulation contains no vertex in its interior (see package \ref PkgTriangulation3 +for more details on Delaunay triangulations). + A _constrained Delaunay triangulation_ of a PLC is a constrained triangulation which is as much Delaunay as possible, given that some facets are marked as _constrained_. More precisely, a triangulation is _constrained Delaunay_ if for any simplex \f$s\f$ of the triangulation,