diff --git a/Polyline_simplification_2/examples/Polyline_simplification_2/points_and_vertices.cpp b/Polyline_simplification_2/examples/Polyline_simplification_2/points_and_vertices.cpp
index 34397eb1ddd..f582dde84c9 100644
--- a/Polyline_simplification_2/examples/Polyline_simplification_2/points_and_vertices.cpp
+++ b/Polyline_simplification_2/examples/Polyline_simplification_2/points_and_vertices.cpp
@@ -60,7 +60,7 @@ int main( )
   PS::simplify(ct, cid, Cost(), Stop(0.5), keep_points);
   print(ct, cid);
   PS::simplify(ct, cid, Cost(), Stop(0.5), keep_points);
-  ct.remove_points_from_constraint(cid);
+  ct.remove_points_without_corresponding_vertex(cid);
   print(ct, cid);  
 
   return 0;
diff --git a/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h b/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h
index 2179cb8a5ff..c1fec18c948 100644
--- a/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h
+++ b/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h
@@ -426,7 +426,7 @@ simplify(CGAL::Constrained_triangulation_plus_2<CDT>& ct,
 
   while(simplifier()){}
   if(! keep_points){
-    ct.remove_points_from_constraint(cid);
+    ct.remove_points_without_corresponding_vertex(cid);
   }
   return simplifier.number_of_removed_vertices();
 }
@@ -457,7 +457,7 @@ simplify(CGAL::Constrained_triangulation_plus_2<CDT>& ct,
 
   while(simplifier()){}
   if(! keep_points){
-    ct.remove_points_from_constraints();
+    ct.remove_points_without_corresponding_vertex();
   }
   return simplifier.number_of_removed_vertices();
 }
diff --git a/Triangulation_2/doc/Triangulation_2/CGAL/Constrained_triangulation_plus_2.h b/Triangulation_2/doc/Triangulation_2/CGAL/Constrained_triangulation_plus_2.h
index 5436daf9660..21107c1f9c5 100644
--- a/Triangulation_2/doc/Triangulation_2/CGAL/Constrained_triangulation_plus_2.h
+++ b/Triangulation_2/doc/Triangulation_2/CGAL/Constrained_triangulation_plus_2.h
@@ -427,8 +427,8 @@ Points_in_constraint_iterator points_in_constraint_end(Constraint_id cid) const
 \cgalAdvancedBegin
 Removes the vertex at `vicq` from the constraint and the triangulation.
 The point of that vertex remains stored in the sequence of original points
-of the constraint until `remove_points_without_vertex_from_constraint(Constraint_id)`
-or `remove_points_without_vertex_from_constraints()` is called.
+of the constraint until `remove_points_without_corresponding_vertex(Constraint_id)`
+or `remove_points_without_corresponding_vertex()` is called.
 
 The polyline simplification algorithm described in Chapter
 \ref Chapter_2D_Polyline_simplification 
diff --git a/Triangulation_2/include/CGAL/Polyline_constrained_triangulation_2.h b/Triangulation_2/include/CGAL/Polyline_constrained_triangulation_2.h
index a5545231f59..230594c0e71 100644
--- a/Triangulation_2/include/CGAL/Polyline_constrained_triangulation_2.h
+++ b/Triangulation_2/include/CGAL/Polyline_constrained_triangulation_2.h
@@ -623,13 +623,13 @@ public:
     Triangulation::insert_constraint(*u, *w);
   }
 
-  std::size_t remove_points_from_constraint(Constraint_id cid)
+  std::size_t remove_points_without_corresponding_vertex(Constraint_id cid)
   {
-    return hierarchy.remove_points_from_constraint(cid);
+    return hierarchy.remove_points_without_corresponding_vertex(cid);
   }
-  std::size_t remove_points_from_constraints()
+  std::size_t remove_points_without_corresponding_vertex()
   {
-    return hierarchy.remove_points_from_constraints();
+    return hierarchy.remove_points_without_corresponding_vertex();
   }
 
 
diff --git a/Triangulation_2/include/CGAL/Polyline_constraint_hierarchy_2.h b/Triangulation_2/include/CGAL/Polyline_constraint_hierarchy_2.h
index 14b637e96a7..ce000f840c6 100644
--- a/Triangulation_2/include/CGAL/Polyline_constraint_hierarchy_2.h
+++ b/Triangulation_2/include/CGAL/Polyline_constraint_hierarchy_2.h
@@ -253,8 +253,8 @@ public:
                 Vertex_it v,
                 Vertex_it w);
 
-  std::size_t remove_points_from_constraint(Constraint_id);
-  std::size_t remove_points_from_constraints();
+  std::size_t remove_points_without_corresponding_vertex(Constraint_id);
+  std::size_t remove_points_without_corresponding_vertex();
 
   Constraint_id concatenate(Constraint_id first, Constraint_id second);
   Constraint_id concatenate2(Constraint_id first, Constraint_id second);
@@ -648,7 +648,7 @@ void Polyline_constraint_hierarchy_2<T,Data>::simplify(Vertex_it uc,
 
 template <class T, class Data>
 std::size_t
-Polyline_constraint_hierarchy_2<T,Data>::remove_points_from_constraint(Constraint_id cid)
+Polyline_constraint_hierarchy_2<T,Data>::remove_points_without_corresponding_vertex(Constraint_id cid)
 {
   std::size_t n = 0;
   for(Point_it it = points_in_constraint_begin(cid); 
@@ -663,11 +663,11 @@ Polyline_constraint_hierarchy_2<T,Data>::remove_points_from_constraint(Constrain
 
 template <class T, class Data>
 std::size_t
-Polyline_constraint_hierarchy_2<T,Data>::remove_points_from_constraints()
+Polyline_constraint_hierarchy_2<T,Data>::remove_points_without_corresponding_vertex()
 {
   std::size_t n = 0;
   for(C_iterator it = constraint_set.begin(); it!= constraint_set.end(); ++it){
-    n+= remove_points_from_constraint(*it);
+    n+= remove_points_without_corresponding_vertex(*it);
   }
   return n;
 }