Doc review: Add the starting point of the documentation

1 files changed, 10 insertions, 0 deletions

diff --git a/src/Alpha_complex/doc/Intro_alpha_complex.h b/src/Alpha_complex/doc/Intro_alpha_complex.h
index fd6f4081..adc1378f 100644
--- a/src/Alpha_complex/doc/Intro_alpha_complex.h
+++ b/src/Alpha_complex/doc/Intro_alpha_complex.h
@@ -51,6 +51,16 @@ 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 &ge; 5.0.0.
  *
  * \section pointsexample Example from points