From a7decae3cdf47441cbd72c31e794176dbd3739c4 Mon Sep 17 00:00:00 2001 From: ROUVREAU Vincent Date: Mon, 25 May 2020 08:21:47 +0200 Subject: C++ version and documentation --- src/python/doc/alpha_complex_user.rst | 21 +++++++++++++++++++-- 1 file changed, 19 insertions(+), 2 deletions(-) (limited to 'src/python/doc/alpha_complex_user.rst') diff --git a/src/python/doc/alpha_complex_user.rst b/src/python/doc/alpha_complex_user.rst index d49f45b4..c1ed0eaa 100644 --- a/src/python/doc/alpha_complex_user.rst +++ b/src/python/doc/alpha_complex_user.rst @@ -16,8 +16,25 @@ Definition Remarks ^^^^^^^ -When an :math:`\alpha`-complex is constructed with an infinite value of :math:`\alpha^2`, -the complex is a Delaunay complex (with special filtration values). +* When an :math:`\alpha`-complex is constructed with an infinite value of :math:`\alpha^2`, the complex is a Delaunay + complex (with special filtration values). The Delaunay complex without filtration values is also available by + passing :code:`default_filtration_value = True` to :func:`~gudhi.AlphaComplex.create_simplex_tree`. +* For people only interested in the topology of the Alpha complex (for instance persistence), Alpha complex is + equivalent to the `Čech complex `_ and much smaller if + you do not bound the radii. `Čech complex `_ can still + make sense in higher dimension precisely because you can bound the radii. +* Using the default :code:`complexity = 'safe'` makes the construction safe. + If you pass :code:`complexity = 'exact'` to :func:`~gudhi.AlphaComplex.__init__`, 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 :code:`complexity = 'safe'` (the default) or :code:`complexity = 'fast'`, the filtration values are only + guaranteed to have a small multiplicative error compared to the exact value, see + `CGAL::Lazy_exact_nt::set_relative_precision_of_to_double `_ + 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 :code:`complexity = 'fast'` 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 `CGAL `_ :math:`\geq` 5.0.0. Example from points ------------------- -- cgit v1.2.3