diff --git a/Alpha_shapes_3/doc_tex/Alpha_shapes_3/alpha3.tex b/Alpha_shapes_3/doc_tex/Alpha_shapes_3/alpha3.tex
index 853e2a2e269..1a94ba02aa3 100644
--- a/Alpha_shapes_3/doc_tex/Alpha_shapes_3/alpha3.tex
+++ b/Alpha_shapes_3/doc_tex/Alpha_shapes_3/alpha3.tex
@@ -195,10 +195,9 @@ points cannot be inserted or removed.
 
 
 We currently do not specify concepts for the underlying triangulation
-type. Models that work for a basic alpha-shape are the classes
-\ccc{CGAL::Delaunay_triangulation_3} and 
-\ccc{CGAL::Triangulation_hierarchy_3} templated with a Delaunay
-triangulation. A model that works for a weighted alpha-shape is 
+type. Models that work for a basic alpha-shape are the instantiations
+of the class \ccc{CGAL::Delaunay_triangulation_3}.
+A model that works for a weighted alpha-shape is 
 the class \ccc{CGAL::Regular_triangulation_3}.
 
 The triangulation needs a geometric traits class as argument.
@@ -234,17 +233,18 @@ as underlying triangulation.
 
 \subsection{Building Basic Alpha Shapes for Many Points}
 When many points are input in the alpha shape, say more than 10 000,
-it pays off to use a triangulation hierarchy as underlying triangulation
-(cf. \ref{T3-concept-hierarchy}).
+it may pay off to use a Delaunay triangulation with \ccc{Fast_location}
+policy as underlying triangulation in order to speed up point location
+quesries (cf. \ref{Triangulation3-sec-locpol}).
 
-\ccIncludeExampleCode{Alpha_shapes_3/ex_alpha_shapes_with_hierarchy_3.cpp}
+\ccIncludeExampleCode{Alpha_shapes_3/ex_alpha_shapes_with_fast_location_3.cpp}
 %----------------------------------------------------------------------
 
 \subsection{Example for Weighted Alpha-Shapes\label{I1_SectWeightedAS3D}}
 
 The following examples build a weighted alpha shape requiring a
 regular triangulation as underlying triangulation.
-The alpha shape is build in \ccc{GENERAL} mode.
+The alpha shape is built in \ccc{GENERAL} mode.
 
 
 \ccIncludeExampleCode{Alpha_shapes_3/ex_weighted_alpha_shapes_3.cpp}
diff --git a/Alpha_shapes_3/examples/Alpha_shapes_3/README b/Alpha_shapes_3/examples/Alpha_shapes_3/README
index 79c00fd510d..e485ccd4535 100644
--- a/Alpha_shapes_3/examples/Alpha_shapes_3/README
+++ b/Alpha_shapes_3/examples/Alpha_shapes_3/README
@@ -1,12 +1,12 @@
-
-example_alpha : read input points, compute the alpha shape in 
+ex_alpha_shapes_3 : read input points, compute the alpha shape in 
 regularized mode and find the optimal value of $\alpha$, i. e. the smallest
-$\alpha$ such that all input point are interior or on the boundary of the
-alpha shape and the alpha shape has a single connected component.
+$\alpha$ such that all input points are in the interior or on the boundary of
+the alpha shape, and the alpha shape has a single connected component.
 
-example_big_alpha : buid the alpha shape for a huge set of points using
-a Delaunay hirarchy as underlying triangulation for efficiency.
+ex_alpha_shapes_with_fast_location_3 : build the alpha shape using an
+underlying Delaunay triangulation with Fast_location policy, for efficient
+point location.
 
-example_weight : build the weighted alpha shape of a small
-set of spheres  and explore the  boundary of the alpha complex for $\alpha =0$.
+ex_weighted_alpha_shapes_3 : build the weighted alpha shape of a small
+set of spheres and explore the boundary of the alpha complex for $\alpha=0$.
 This complex is the nerve of the union of the spheres.
diff --git a/Alpha_shapes_3/examples/Alpha_shapes_3/ex_alpha_shapes_with_hierarchy_3.cpp b/Alpha_shapes_3/examples/Alpha_shapes_3/ex_alpha_shapes_with_fast_location_3.cpp
similarity index 77%
rename from Alpha_shapes_3/examples/Alpha_shapes_3/ex_alpha_shapes_with_hierarchy_3.cpp
rename to Alpha_shapes_3/examples/Alpha_shapes_3/ex_alpha_shapes_with_fast_location_3.cpp
index 56de6cdd155..ee661186580 100644
--- a/Alpha_shapes_3/examples/Alpha_shapes_3/ex_alpha_shapes_with_hierarchy_3.cpp
+++ b/Alpha_shapes_3/examples/Alpha_shapes_3/ex_alpha_shapes_with_fast_location_3.cpp
@@ -1,6 +1,5 @@
 #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
 #include <CGAL/Delaunay_triangulation_3.h>
-#include <CGAL/Triangulation_hierarchy_3.h>
 #include <CGAL/Alpha_shape_3.h>
 
 #include <fstream>
@@ -10,12 +9,10 @@
 typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
 
 typedef CGAL::Alpha_shape_vertex_base_3<K>               Vb;
-typedef CGAL::Triangulation_hierarchy_vertex_base_3<Vb>  Vbh;
 typedef CGAL::Alpha_shape_cell_base_3<K>                 Fb;
-typedef CGAL::Triangulation_data_structure_3<Vbh,Fb>     Tds;
-typedef CGAL::Delaunay_triangulation_3<K,Tds>            Delaunay;
-typedef CGAL::Triangulation_hierarchy_3<Delaunay>        Delaunay_hierarchy;
-typedef CGAL::Alpha_shape_3<Delaunay_hierarchy>          Alpha_shape_3;
+typedef CGAL::Triangulation_data_structure_3<Vb,Fb>      Tds;
+typedef CGAL::Delaunay_triangulation_3<K,Tds,CGAL::Fast_location>  Delaunay;
+typedef CGAL::Alpha_shape_3<Delaunay>                    Alpha_shape_3;
 
 typedef K::Point_3                                  Point;
 typedef Alpha_shape_3::Alpha_iterator               Alpha_iterator;
@@ -23,7 +20,7 @@ typedef Alpha_shape_3::NT                           NT;
 
 int main()
 {
-  Delaunay_hierarchy dt;
+  Delaunay dt;
   std::ifstream is("./data/bunny_1000");
   int n;
   is >> n;