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Diffstat (limited to 'src/Alpha_complex/doc')
| -rw-r--r-- | src/Alpha_complex/doc/Intro_alpha_complex.h | 17 |
1 files changed, 14 insertions, 3 deletions
diff --git a/src/Alpha_complex/doc/Intro_alpha_complex.h b/src/Alpha_complex/doc/Intro_alpha_complex.h index b075d1fc..3c32a1e6 100644 --- a/src/Alpha_complex/doc/Intro_alpha_complex.h +++ b/src/Alpha_complex/doc/Intro_alpha_complex.h @@ -38,12 +38,12 @@ namespace alpha_complex { * * Alpha_complex is constructing a <a target="_blank" * href="http://doc.cgal.org/latest/Triangulation/index.html#Chapter_Triangulations">Delaunay Triangulation</a> - * \cite cgal:hdj-t-15b from <a target="_blank" href="http://www.cgal.org/">CGAL</a> (the Computational Geometry - * Algorithms Library \cite cgal:eb-15b) and is able to create a `SimplicialComplexForAlpha`. + * \cite cgal:hdj-t-19b from <a target="_blank" href="http://www.cgal.org/">CGAL</a> (the Computational Geometry + * Algorithms Library \cite cgal:eb-19b) and is able to create a `SimplicialComplexForAlpha`. * * The complex is a template class requiring an Epick_d <a target="_blank" * href="http://doc.cgal.org/latest/Kernel_d/index.html#Chapter_dD_Geometry_Kernel">dD Geometry Kernel</a> - * \cite cgal:s-gkd-15b from CGAL as template parameter. + * \cite cgal:s-gkd-19b from CGAL as template parameter. * * \remark * - When the simplicial complex is constructed with an infinite value of alpha, the complex is a Delaunay @@ -51,6 +51,17 @@ namespace alpha_complex { * - For people only interested in the topology of the \ref alpha_complex (for instance persistence), * \ref alpha_complex is equivalent to the \ref cech_complex and much smaller if you do not bound the radii. * \ref cech_complex can still make sense in higher dimension precisely because you can bound the radii. + * - Using the default `CGAL::Epeck_d` makes the construction safe. If you pass exact=true to create_complex, the + * filtration values are the exact ones converted to the filtration value type of the simplicial complex. This can be + * very slow. If you pass exact=false (the default), the filtration values are only guaranteed to have a small + * multiplicative error compared to the exact value, see <code><a class="el" target="_blank" + * href="https://doc.cgal.org/latest/Number_types/classCGAL_1_1Lazy__exact__nt.html"> + * CGAL::Lazy_exact_nt<NT>::set_relative_precision_of_to_double</a></code> for details. A drawback, when computing + * persistence, is that an empty exact interval [10^12,10^12] may become a non-empty approximate interval + * [10^12,10^12+10^6]. Using `CGAL::Epick_d` makes the computations slightly faster, and the combinatorics are still + * exact, but the computation of filtration values can exceptionally be arbitrarily bad. In all cases, we still + * guarantee that the output is a valid filtration (faces have a filtration value no larger than their cofaces). + * - For performances reasons, it is advised to use `Alpha_complex` with \ref cgal ≥ 5.0.0. * * \section pointsexample Example from points * |