diff --git a/Polyhedron/demo/Polyhedron/Plugins/Surface_mesh/Parameterization_plugin.cpp b/Polyhedron/demo/Polyhedron/Plugins/Surface_mesh/Parameterization_plugin.cpp
index 9f0d3aed363..186c01e9dfd 100644
--- a/Polyhedron/demo/Polyhedron/Plugins/Surface_mesh/Parameterization_plugin.cpp
+++ b/Polyhedron/demo/Polyhedron/Plugins/Surface_mesh/Parameterization_plugin.cpp
@@ -618,7 +618,7 @@ void Polyhedron_demo_parameterization_plugin::parameterize(const Parameterizatio
 
   // Next is the gathering of the border halfedges of the connected component.
   // It is wrong to pass the underlying mesh tMesh: a sphere split in half does
-  // not have any border according if border_halfedges() is run with tMesh.
+  // not have any border if border_halfedges() is run with tMesh.
   //
   // The proper way would be to completely redesign the plugin to use Seam meshes
   // everywhere. But that's not worth it. Instead, we abuse the fact that faces
@@ -786,9 +786,10 @@ void Polyhedron_demo_parameterization_plugin::parameterize(const Parameterizatio
       typedef boost::unordered_map<vertex_descriptor, SMP::Cone_type>  Cones;
       Cones cmap;
 
-      SMP::internal::locate_unordered_cones<Seam_mesh,
-                                            boost::unordered_set<T_vertex_descriptor>,
-                                            Cones>(sMesh, unordered_cones, cmap);
+      if(!SMP::internal::locate_unordered_cones<Seam_mesh,
+                                                boost::unordered_set<T_vertex_descriptor>,
+                                                Cones>(sMesh, unordered_cones, cmap))
+        return;
 
       // vimap and uvmap
       typedef boost::unordered_map<vertex_descriptor, int> Indices;
diff --git a/Surface_mesh_parameterization/examples/Surface_mesh_parameterization/orbifold.cpp b/Surface_mesh_parameterization/examples/Surface_mesh_parameterization/orbifold.cpp
index 24ae85c1e28..29229b30ba9 100644
--- a/Surface_mesh_parameterization/examples/Surface_mesh_parameterization/orbifold.cpp
+++ b/Surface_mesh_parameterization/examples/Surface_mesh_parameterization/orbifold.cpp
@@ -108,9 +108,9 @@ int main(int argc, char * argv[])
   // Mark the cones in the seam mesh
   typedef boost::unordered_map<vertex_descriptor, SMP::Cone_type>  Cones;
   Cones cmap;
-  SMP::internal::locate_cones<Mesh,
-                              Cones_in_smesh_container,
-                              Cones>(mesh, cone_sm_vds, cmap);
+  bool locate_status = SMP::internal::locate_cones<Mesh,
+                                                   Cones_in_smesh_container,
+                                                   Cones>(mesh, cone_sm_vds, cmap);
 
   // The 2D points of the uv parametrisation will be written into this map
   // Note that this is a halfedge property map, and that uv values
diff --git a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/MVC_post_processor_3.h b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/MVC_post_processor_3.h
index 0bfb78d7f92..40636b8552b 100644
--- a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/MVC_post_processor_3.h
+++ b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/MVC_post_processor_3.h
@@ -461,7 +461,7 @@ private:
             typename VertexUVMap,
             typename VertexIndexMap,
             typename VertexParameterizedMap>
-  void fill_linear_system_matrix_mvc_from_ct_face(const CT& ct,
+  void fill_linear_system_matrix_mvc_from_ct_face(const CT& CGAL_precondition_code(ct),
                                                   typename CT::Finite_faces_iterator fh,
                                                   const VertexUVMap uvmap,
                                                   const VertexIndexMap vimap,
diff --git a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/internal/orbifold_cone_helper.h b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/internal/orbifold_cone_helper.h
index 9fd073d72fb..7417841398c 100644
--- a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/internal/orbifold_cone_helper.h
+++ b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/internal/orbifold_cone_helper.h
@@ -399,7 +399,7 @@ Error_code read_cones(const TriangleMesh& pm, const char* filename,
 template<typename SeamMesh,
          typename Cones_in_pmesh_vector,
          typename ConeMap>
-void locate_cones(const SeamMesh& mesh,
+bool locate_cones(const SeamMesh& mesh,
                   const Cones_in_pmesh_vector& cone_tm_vds,
                   ConeMap& cones)
 {
@@ -438,7 +438,7 @@ void locate_cones(const SeamMesh& mesh,
     }
   }
 
-  check_input_validity(mesh, cones, cone_tm_vds);
+  return check_input_validity(mesh, cones, cone_tm_vds);
 }
 
 /// Same as above, but the cones are NOT ordered and we thus use seam mesh
@@ -451,11 +451,12 @@ void locate_cones(const SeamMesh& mesh,
 template<typename SeamMesh,
          typename Cones_in_pmesh_set,
          typename ConeMap>
-void locate_unordered_cones(const SeamMesh& mesh,
+bool locate_unordered_cones(const SeamMesh& mesh,
                             const Cones_in_pmesh_set& cone_tm_vds,
                             ConeMap& cones)
 {
   CGAL_precondition(cones.empty());
+  CGAL_precondition(cone_tm_vds.size() == 3 || cone_tm_vds.size() == 4);
 
   typedef typename SeamMesh::TriangleMesh                                  TriangleMesh;
 
@@ -527,7 +528,7 @@ void locate_unordered_cones(const SeamMesh& mesh,
 
   } while(vertex_on_seam != end);
 
-  check_input_validity(mesh, cones, cone_tm_vds);
+  return check_input_validity(mesh, cones, cone_tm_vds);
 }
 
 } // namespace internal