diff --git a/Surface_mesh_parameterization/doc/Surface_mesh_parameterization/Concepts/BorderParameterizer_3.h b/Surface_mesh_parameterization/doc/Surface_mesh_parameterization/Concepts/BorderParameterizer_3.h
index 268e31fb274..c13f10fa124 100644
--- a/Surface_mesh_parameterization/doc/Surface_mesh_parameterization/Concepts/BorderParameterizer_3.h
+++ b/Surface_mesh_parameterization/doc/Surface_mesh_parameterization/Concepts/BorderParameterizer_3.h
@@ -45,7 +45,7 @@ Assign a 2D position (i.e.\ a `(u, v)` pair) on the shape  to (some of) the vert
 border of the mesh. Mark them as <I>parameterized</I>.
 
 */
-Error_code parameterize_border(const TriangleMesh& mesh);
+Error_code parameterize(const TriangleMesh& mesh);
 
 /*!
 
diff --git a/Surface_mesh_parameterization/include/CGAL/ARAP_parameterizer_3.h b/Surface_mesh_parameterization/include/CGAL/ARAP_parameterizer_3.h
index 2dbe3c8f22b..814a984279a 100644
--- a/Surface_mesh_parameterization/include/CGAL/ARAP_parameterizer_3.h
+++ b/Surface_mesh_parameterization/include/CGAL/ARAP_parameterizer_3.h
@@ -347,9 +347,7 @@ private:
 
       Useless_map useless_map;
       Useless_pmap useless_uvpmap(useless_map);
-      status = get_border_parameterizer().parameterize_border(mesh, bhd,
-                                                              useless_uvpmap,
-                                                              vpmap);
+      status = get_border_parameterizer().parameterize(mesh, bhd, useless_uvpmap, vpmap);
     }
 
     return status;
diff --git a/Surface_mesh_parameterization/include/CGAL/Circular_border_parameterizer_3.h b/Surface_mesh_parameterization/include/CGAL/Circular_border_parameterizer_3.h
index 403075b2c6e..801fda0efd1 100644
--- a/Surface_mesh_parameterization/include/CGAL/Circular_border_parameterizer_3.h
+++ b/Surface_mesh_parameterization/include/CGAL/Circular_border_parameterizer_3.h
@@ -106,10 +106,10 @@ public:
   /// on the circle. Mark them as <i>parameterized</i>.
   template <typename VertexUVmap, typename VertexParameterizedMap>
   Error_code
-  parameterize_border(const TriangleMesh& mesh,
-                      halfedge_descriptor bhd,
-                      VertexUVmap uvmap,
-                      VertexParameterizedMap vpmap)
+  parameterize(const TriangleMesh& mesh,
+               halfedge_descriptor bhd,
+               VertexUVmap uvmap,
+               VertexParameterizedMap vpmap)
   {
     // Nothing to do if no border
     if (bhd == halfedge_descriptor())
diff --git a/Surface_mesh_parameterization/include/CGAL/Fixed_border_parameterizer_3.h b/Surface_mesh_parameterization/include/CGAL/Fixed_border_parameterizer_3.h
index c95ed4857f5..e9e73bf173a 100644
--- a/Surface_mesh_parameterization/include/CGAL/Fixed_border_parameterizer_3.h
+++ b/Surface_mesh_parameterization/include/CGAL/Fixed_border_parameterizer_3.h
@@ -341,9 +341,8 @@ parameterize(TriangleMesh& mesh,
   // Count vertices
   int nbVertices = static_cast<int>(vertices.size());
 
-  // Compute (u,v) for border vertices
-  // and mark them as "parameterized"
-  status = get_border_parameterizer().parameterize_border(mesh, bhd, uvmap, vpm);
+  // Compute (u,v) for border vertices and mark them as "parameterized"
+  status = get_border_parameterizer().parameterize(mesh, bhd, uvmap, vpmap);
 
   if (status != Base::OK)
     return status;
diff --git a/Surface_mesh_parameterization/include/CGAL/LSCM_parameterizer_3.h b/Surface_mesh_parameterization/include/CGAL/LSCM_parameterizer_3.h
index 5929830566d..381440f9764 100644
--- a/Surface_mesh_parameterization/include/CGAL/LSCM_parameterizer_3.h
+++ b/Surface_mesh_parameterization/include/CGAL/LSCM_parameterizer_3.h
@@ -202,7 +202,7 @@ public:
 
     // Compute (u,v) for (at least two) border vertices
     // and mark them as "parameterized"
-    Error_code status = get_border_parameterizer().parameterize_border(mesh,bhd,uvmap,vpmap);
+    Error_code status = get_border_parameterizer().parameterize(mesh, bhd, uvmap, vpmap);
 
     if(status != Base::OK)
       return status;
diff --git a/Surface_mesh_parameterization/include/CGAL/Square_border_parameterizer_3.h b/Surface_mesh_parameterization/include/CGAL/Square_border_parameterizer_3.h
index a016ada5a10..dcc9fa80ac9 100644
--- a/Surface_mesh_parameterization/include/CGAL/Square_border_parameterizer_3.h
+++ b/Surface_mesh_parameterization/include/CGAL/Square_border_parameterizer_3.h
@@ -114,7 +114,7 @@ private:
     return len;
   }
 
-  /// Utility method for parameterize_border().
+  /// Utility method for parameterize().
   /// Compute the mesh iterator whose offset is closest to 'value'.
   halfedge_around_face_iterator closest_iterator(const TriangleMesh& mesh,
                                                  halfedge_descriptor bhd,
@@ -299,11 +299,11 @@ public:
   /// Assign to the vertices of the border of the mesh a 2D position
   /// (i.e.\ a (u,v) pair) on the border's shape. Mark them as <i>parameterized</i>.
   template<typename VertexUVMap, typename VertexParameterizedMap>
-  Error_code parameterize_border(const TriangleMesh& mesh,
-                                 halfedge_descriptor bhd,
-                                 VertexUVMap uvmap,
-                                 VertexParameterizedMap vpmap,
-                                 const char* filename = "../data/square_corners.selection.txt")
+  Error_code parameterize(const TriangleMesh& mesh,
+                          halfedge_descriptor bhd,
+                          VertexUVMap uvmap,
+                          VertexParameterizedMap vpmap,
+                          const char* filename = "../data/square_corners.selection.txt")
   {
   #ifdef DEBUG_TRACE
     std::cerr << "Map to a square" << std::endl;
diff --git a/Surface_mesh_parameterization/include/CGAL/Two_vertices_parameterizer_3.h b/Surface_mesh_parameterization/include/CGAL/Two_vertices_parameterizer_3.h
index 5838f965552..ca122e837ab 100644
--- a/Surface_mesh_parameterization/include/CGAL/Two_vertices_parameterizer_3.h
+++ b/Surface_mesh_parameterization/include/CGAL/Two_vertices_parameterizer_3.h
@@ -100,10 +100,10 @@ public:
   ///
   template <typename VertexUVmap, typename VertexParameterizedMap>
   typename Parameterizer_traits_3<TriangleMesh>::Error_code
-  parameterize_border(const TriangleMesh& mesh,
-                      halfedge_descriptor,
-                      VertexUVmap uvmap,
-                      VertexParameterizedMap vpmap)
+  parameterize(const TriangleMesh& mesh,
+               halfedge_descriptor,
+               VertexUVmap uvmap,
+               VertexParameterizedMap vpmap)
   {
     if(vertices_given){
       put(uvmap, vxmin, Point_2(0, 0.5));