diff --git a/Polyhedron/demo/Polyhedron/Plugins/Surface_mesh/Parameterization_plugin.cpp b/Polyhedron/demo/Polyhedron/Plugins/Surface_mesh/Parameterization_plugin.cpp
index ee77431c72e..9f0d3aed363 100644
--- a/Polyhedron/demo/Polyhedron/Plugins/Surface_mesh/Parameterization_plugin.cpp
+++ b/Polyhedron/demo/Polyhedron/Plugins/Surface_mesh/Parameterization_plugin.cpp
@@ -33,8 +33,8 @@
 #include <CGAL/Surface_mesh_parameterization/Error_code.h>
 #include <CGAL/Surface_mesh_parameterization/LSCM_parameterizer_3.h>
 #include <CGAL/Surface_mesh_parameterization/Two_vertices_parameterizer_3.h>
-#include <CGAL/Surface_mesh_parameterization/internal/orbital_cone_helper.h>
-#include <CGAL/Surface_mesh_parameterization/Orbital_Tutte_parameterizer_3.h>
+#include <CGAL/Surface_mesh_parameterization/internal/orbifold_cone_helper.h>
+#include <CGAL/Surface_mesh_parameterization/Orbifold_Tutte_parameterizer_3.h>
 #include <CGAL/Surface_mesh_parameterization/parameterize.h>
 
 #include <CGAL/Textured_polyhedron_builder.h>
@@ -759,7 +759,7 @@ void Polyhedron_demo_parameterization_plugin::parameterize(const Parameterizatio
 
       CGAL_precondition(!unordered_cones.empty());
 
-      typedef SMP::Orbital_Tutte_parameterizer_3<Seam_mesh> Parameterizer;
+      typedef SMP::Orbifold_Tutte_parameterizer_3<Seam_mesh> Parameterizer;
 
       QDialog dialog(mw);
       Ui::OTE_dialog ui;
diff --git a/Surface_mesh_parameterization/examples/Surface_mesh_parameterization/CMakeLists.txt b/Surface_mesh_parameterization/examples/Surface_mesh_parameterization/CMakeLists.txt
index efb823ef2f4..fd1f36b6a81 100644
--- a/Surface_mesh_parameterization/examples/Surface_mesh_parameterization/CMakeLists.txt
+++ b/Surface_mesh_parameterization/examples/Surface_mesh_parameterization/CMakeLists.txt
@@ -44,7 +44,7 @@ if ( CGAL_FOUND )
       message(STATUS "SuiteSparse_LIBS: ${SuiteSparse_LIBRARIES}")
       add_definitions(-DCGAL_SMP_USE_SPARSESUITE_SOLVERS)
     else(SuiteSparse_FOUND)
-      message(STATUS "NOTICE: The example `orbital.cpp` will be compiled without the Sparsesuite library.")
+      message(STATUS "NOTICE: The example `orbifold.cpp` will be compiled without the Sparsesuite library.")
     endif(SuiteSparse_FOUND)
 
 # ------------------------------------------------------------------
@@ -53,13 +53,15 @@ if ( CGAL_FOUND )
 
     create_single_source_cgal_program( "discrete_authalic.cpp" )
     create_single_source_cgal_program( "lscm.cpp" )
-    create_single_source_cgal_program( "orbital.cpp" )
+    create_single_source_cgal_program( "orbifold.cpp" )
+    create_single_source_cgal_program( "orbifold_mapping.cpp" )
     create_single_source_cgal_program( "seam_Polyhedron_3.cpp" )
     create_single_source_cgal_program( "simple_parameterization.cpp" )
     create_single_source_cgal_program( "square_border_parameterizer.cpp" )
     
     if(SuiteSparse_FOUND)
-      target_link_libraries(orbital ${SuiteSparse_LIBRARIES})
+      target_link_libraries(orbifold ${SuiteSparse_LIBRARIES})
+      target_link_libraries(orbifold_mapping ${SuiteSparse_LIBRARIES})
     endif()
 
   else(EIGEN3_FOUND)
diff --git a/Surface_mesh_parameterization/examples/Surface_mesh_parameterization/orbifold.cpp b/Surface_mesh_parameterization/examples/Surface_mesh_parameterization/orbifold.cpp
index b399bbf5cc1..929c78be76a 100644
--- a/Surface_mesh_parameterization/examples/Surface_mesh_parameterization/orbifold.cpp
+++ b/Surface_mesh_parameterization/examples/Surface_mesh_parameterization/orbifold.cpp
@@ -10,7 +10,7 @@
 #include <CGAL/Surface_mesh_parameterization/internal/kernel_traits.h>
 #include <CGAL/Surface_mesh_parameterization/internal/shortest_path.h>
 
-#include <CGAL/Surface_mesh_parameterization/Orbital_Tutte_parameterizer_3.h>
+#include <CGAL/Surface_mesh_parameterization/Orbifold_Tutte_parameterizer_3.h>
 
 #include <CGAL/Polygon_mesh_processing/connected_components.h>
 #include <CGAL/Polygon_mesh_processing/measure.h>
@@ -119,8 +119,8 @@ int main(int argc, char * argv[])
   UV_pmap uvmap = sm.add_property_map<SM_halfedge_descriptor, Point_2>("h:uv").first;
 
   // Parameterizer
-  typedef SMP::Orbital_Tutte_parameterizer_3<Mesh>         Parameterizer;
-  Parameterizer parameterizer(SMP::Parallelogram, SMP::Cotangent);
+  typedef SMP::Orbifold_Tutte_parameterizer_3<Mesh>         Parameterizer;
+  Parameterizer parameterizer(SMP::Triangle, SMP::Cotangent);
 
   // a halfedge on the (possibly virtual) border
   // only used in output (will also be used to handle multiple connected components in the future)
diff --git a/Surface_mesh_parameterization/examples/Surface_mesh_parameterization/orbifold_mapping.cpp b/Surface_mesh_parameterization/examples/Surface_mesh_parameterization/orbifold_mapping.cpp
index cc97db45c4a..7ec62be4633 100644
--- a/Surface_mesh_parameterization/examples/Surface_mesh_parameterization/orbifold_mapping.cpp
+++ b/Surface_mesh_parameterization/examples/Surface_mesh_parameterization/orbifold_mapping.cpp
@@ -10,8 +10,8 @@
 #include <CGAL/Surface_mesh_parameterization/internal/Containers_filler.h>
 #include <CGAL/Surface_mesh_parameterization/internal/kernel_traits.h>
 #include <CGAL/Surface_mesh_parameterization/internal/shortest_path.h>
-#include <CGAL/Surface_mesh_parameterization/Orbital_Tutte_parameterizer_3.h>
-#include <CGAL/Surface_mesh_parameterization/Orbital_Tutte_sphere_mapping.h>
+#include <CGAL/Surface_mesh_parameterization/Orbifold_Tutte_parameterizer_3.h>
+#include <CGAL/Surface_mesh_parameterization/Orbifold_Tutte_sphere_mapping.h>
 
 #include <CGAL/Arr_non_caching_segment_traits_2.h>
 #include <CGAL/Arr_segment_traits_2.h>
@@ -84,7 +84,7 @@ int main(int argc, char * argv[])
   const SMP::Orbifold_type orb_type = SMP::Triangle;
 
   // Parameterizer
-  typedef SMP::Orbital_Tutte_parameterizer_3<Mesh>         Parameterizer;
+  typedef SMP::Orbifold_Tutte_parameterizer_3<Mesh>         Parameterizer;
   Parameterizer parameterizer(orb_type, SMP::Cotangent);
 
   // *-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
@@ -151,7 +151,7 @@ int main(int argc, char * argv[])
   parameterizer.parameterize(mesh_A, bhd_A, cmap_A, uvmap_A, vimap_A);
   std::cout << "Parameterized the first domain in " << task_timer.time() << " seconds" << std::endl;
 
-  std::ofstream out_A("orbital_source.off");
+  std::ofstream out_A("orbifold_source.off");
   SMP::IO::output_uvmap_to_off(mesh_A, bhd_A, uvmap_A, out_A);
 
   Embedded_mesh emesh_A(mesh_A, cmap_A, vimap_A, uvmap_A, orb_type);
@@ -221,7 +221,7 @@ int main(int argc, char * argv[])
   parameterizer.parameterize(mesh_B, bhd_B, cmap_B, uvmap_B, vimap_B);
   std::cout << "Parameterized the second domain in " << task_timer.time() << " seconds" << std::endl;
 
-  std::ofstream out_B("orbital_target.off");
+  std::ofstream out_B("orbifold_target.off");
   SMP::IO::output_uvmap_to_off(mesh_B, bhd_B, uvmap_B, out_B);
 
   Embedded_mesh emesh_B(mesh_B, cmap_B, vimap_B, uvmap_B, orb_type);
diff --git a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Orbifold_Tutte_parameterizer_3.h b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Orbifold_Tutte_parameterizer_3.h
index 1ebc973c828..1b27bd5af1e 100644
--- a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Orbifold_Tutte_parameterizer_3.h
+++ b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Orbifold_Tutte_parameterizer_3.h
@@ -18,12 +18,12 @@
 //
 // Author(s)     :
 
-#ifndef CGAL_SURFACE_MESH_PARAMETERIZATION_ORBITAL_TUTTE_PARAMETERIZER_3_H
-#define CGAL_SURFACE_MESH_PARAMETERIZATION_ORBITAL_TUTTE_PARAMETERIZER_3_H
+#ifndef CGAL_SURFACE_MESH_PARAMETERIZATION_ORBIFOLD_TUTTE_PARAMETERIZER_3_H
+#define CGAL_SURFACE_MESH_PARAMETERIZATION_ORBIFOLD_TUTTE_PARAMETERIZER_3_H
 
 #include <CGAL/Surface_mesh_parameterization/internal/angles.h>
 #include <CGAL/Surface_mesh_parameterization/internal/kernel_traits.h>
-#include <CGAL/Surface_mesh_parameterization/internal/orbital_cone_helper.h>
+#include <CGAL/Surface_mesh_parameterization/internal/orbifold_cone_helper.h>
 #include <CGAL/Surface_mesh_parameterization/IO/File_off.h>
 
 #include <CGAL/Surface_mesh_parameterization/Error_code.h>
@@ -55,9 +55,9 @@
 #include <vector>
 #include <utility>
 
-/// \file Orbital_Tutte_parameterizer_3.h
+/// \file Orbifold_Tutte_parameterizer_3.h
 
-// #define CGAL_SMP_OUTPUT_ORBITAL_MATRICES
+#define CGAL_SMP_OUTPUT_ORBIFOLD_MATRICES
 
 // @todo checks that cones are different, are on seams, seam is one connected
 //       component
@@ -84,7 +84,7 @@ template
     = Eigen_solver_traits<Eigen::SparseLU<Eigen_sparse_matrix<double>::EigenType> >
 #endif
 >
-class Orbital_Tutte_parameterizer_3
+class Orbifold_Tutte_parameterizer_3
 {
 private:
   typedef typename boost::graph_traits<SeamMesh>::vertex_descriptor    vertex_descriptor;
@@ -567,7 +567,7 @@ private:
 
     std::cout << "Filled M and Bf" << std::endl;
 
-#ifdef CGAL_SMP_OUTPUT_ORBITAL_MATRICES
+#ifdef CGAL_SMP_OUTPUT_ORBIFOLD_MATRICES
     std::ofstream outM("matrices/M.txt");
     outM.precision(20);
     outM << M.row_dimension() << " " << M.column_dimension() << std::endl;
@@ -586,7 +586,7 @@ private:
     outBf << Bf.size() << std::endl;
     outBf << Bf << std::endl;
 
-#ifdef CGAL_SMP_OUTPUT_ORBITAL_MATRICES_FOR_MATLAB
+#ifdef CGAL_SMP_OUTPUT_ORBIFOLD_MATRICES_FOR_MATLAB
     std::ofstream mat_outM("matrices/matrixM.dat");
     mat_outM.precision(20);
     for(int k=0; k<M.eigen_object().outerSize(); ++k) {
@@ -619,7 +619,7 @@ private:
       X[i] = Xf[i];
     }
 
-#ifdef CGAL_SMP_OUTPUT_ORBITAL_MATRICES
+#ifdef CGAL_SMP_OUTPUT_ORBIFOLD_MATRICES
     std::ofstream outf("matrices/X.txt");
     for(std::size_t i=0; i<n; ++i) {
       outf << X[i] << " ";
@@ -678,7 +678,7 @@ public:
     // add rotational constraints
     AddRotationalConstraint(mesh, cmap, vimap, A, B);
 
-#ifdef CGAL_SMP_OUTPUT_ORBITAL_MATRICES
+#ifdef CGAL_SMP_OUTPUT_ORBIFOLD_MATRICES
     std::cout << "A and B are filled" << std::endl;
     std::ofstream outA("matrices/A.txt"), outB("matrices/B.txt");
 
@@ -700,7 +700,7 @@ public:
     else // weight_type == Mean_value
       mean_value_laplacian(mesh, vimap, L);
 
-#ifdef CGAL_SMP_OUTPUT_ORBITAL_MATRICES
+#ifdef CGAL_SMP_OUTPUT_ORBIFOLD_MATRICES
     std::ofstream outL("matrices/L.txt");
     outL.precision(20);
     outL << L.row_dimension() << " " << L.column_dimension() << std::endl;
@@ -725,7 +725,7 @@ public:
     if(status != OK)
       return status;
 
-    std::ofstream out("orbital_result.off");
+    std::ofstream out("orbifold_result.off");
     IO::output_uvmap_to_off(mesh, bhd, uvmap, out);
 
     return OK;
@@ -733,8 +733,8 @@ public:
 
 /// Constructor
 public:
-  Orbital_Tutte_parameterizer_3(const Orbifold_type orb_type = Square,
-                                const Weight_type weight_type = Cotangent)
+  Orbifold_Tutte_parameterizer_3(const Orbifold_type orb_type = Square,
+                                 const Weight_type weight_type = Cotangent)
     :
       orb_type(orb_type),
       weight_type(weight_type)
@@ -745,4 +745,4 @@ public:
 
 } // namespace CGAL
 
-#endif // CGAL_SURFACE_MESH_PARAMETERIZATION_ORBITAL_TUTTE_PARAMETERIZER_3_H
+#endif // CGAL_SURFACE_MESH_PARAMETERIZATION_ORBIFOLD_TUTTE_PARAMETERIZER_3_H
diff --git a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Orbifold_Tutte_sphere_mapping.h b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Orbifold_Tutte_sphere_mapping.h
index f3aced86e52..b011f687e5b 100644
--- a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Orbifold_Tutte_sphere_mapping.h
+++ b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Orbifold_Tutte_sphere_mapping.h
@@ -18,10 +18,10 @@
 //
 // Author(s)     :
 
-#ifndef CGAL_SURFACE_MESH_PARAMETERIZATION_ORBITAL_TUTTE_SPHERE_MAPPING_H
-#define CGAL_SURFACE_MESH_PARAMETERIZATION_ORBITAL_TUTTE_SPHERE_MAPPING_H
+#ifndef CGAL_SURFACE_MESH_PARAMETERIZATION_ORBIFOLD_TUTTE_SPHERE_MAPPING_H
+#define CGAL_SURFACE_MESH_PARAMETERIZATION_ORBIFOLD_TUTTE_SPHERE_MAPPING_H
 
-#include <CGAL/Surface_mesh_parameterization/internal/orbital_cone_helper.h>
+#include <CGAL/Surface_mesh_parameterization/internal/orbifold_cone_helper.h>
 #include <CGAL/Surface_mesh_parameterization/internal/kernel_traits.h>
 
 #include <CGAL/Arr_default_overlay_traits.h>
@@ -914,4 +914,4 @@ public:
 
 } // namespace CGAL
 
-#endif // CGAL_SURFACE_MESH_PARAMETERIZATION_ORBITAL_TUTTE_SPHERE_MAPPING_H
+#endif // CGAL_SURFACE_MESH_PARAMETERIZATION_ORBIFOLD_TUTTE_SPHERE_MAPPING_H
diff --git a/Surface_mesh_parameterization/include/CGAL/surface_mesh_parameterization.h b/Surface_mesh_parameterization/include/CGAL/surface_mesh_parameterization.h
index 36d3f0d846f..1ec0d034cb7 100644
--- a/Surface_mesh_parameterization/include/CGAL/surface_mesh_parameterization.h
+++ b/Surface_mesh_parameterization/include/CGAL/surface_mesh_parameterization.h
@@ -35,7 +35,8 @@
 #include <CGAL/Surface_mesh_parameterization/LSCM_parameterizer_3.h>
 #include <CGAL/Surface_mesh_parameterization/Mean_value_coordinates_parameterizer_3.h>
 #include <CGAL/Surface_mesh_parameterization/MVC_post_processor_3.h>
-#include <CGAL/Surface_mesh_parameterization/Orbital_Tutte_parameterizer_3.h>
+#include <CGAL/Surface_mesh_parameterization/Orbifold_Tutte_parameterizer_3.h>
+#include <CGAL/Surface_mesh_parameterization/Orbifold_Tutte_sphere_mapping.h>
 
 #include <CGAL/Surface_mesh_parameterization/parameterize.h>