Distinguished between the first cone and the second cone

Thus if the input cones are C_0 ... C_3, then the seam is going from C_0 to C_3.
This is important because the result is not always identical.

Plugin of the demo decides randomly for now, which is not satisfying.

Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Orbital_Tutte_parameterizer_3.h

@@ -149,12 +149,12 @@ private:
     if(orb_type == Parallelogram) {
       if(cmap.size() != 6) {
         std::cerr << "Using orb_type " << get_orbifold_type(orb_type)
-                  << " requires 6 vertices marked as cones in the seam mesh :" << std::endl;
+                  << " requires 4 vertices marked as cones (thus 6 in the seam mesh)" << std::endl;
         return ERROR_WRONG_PARAMETER;
       }
     } else if(cmap.size() != 4){ // orb_type == Square, Diamond, Triangle
       std::cerr << "Using orb_type " << get_orbifold_type(orb_type)
-                << " requires 4 vertices marked as cones in the seam mesh :" << std::endl;
+                << " requires 3 vertices marked as cones (thus 4 in the seam mesh)" << std::endl;
       return ERROR_WRONG_PARAMETER;
     }
 
@@ -224,7 +224,7 @@ private:
   {
     typename ConeMap::const_iterator cmit = cmap.begin(), cend = cmap.end();
     for(; cmit!=cend; ++cmit) {
-      if(cmit->second != Unique_cone)
+      if(cmit->second != First_unique_cone)
         continue;
 
       cone = cmit->first;
@@ -433,7 +433,7 @@ private:
         constrain_seam_segment(seam_segment, ang, current_line_id_in_A, A, B);
 
         // the last cone of the seam is constrained
-        if(is_in_map->second == Unique_cone) { // reached the end of the seam
+        if(is_in_map->second == Second_unique_cone) { // reached the end of the seam
           CGAL_assertion(hd1_target == hd2_target);
           addConstraint(A, B, current_line_id_in_A, hd1t_index,
                         1. /*entry in A*/, tcoords.back());

Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/internal/orbital_cone_helper.h

@@ -42,7 +42,8 @@ namespace Surface_mesh_parameterization {
 
 enum Cone_type
 {
-  Unique_cone = 0,
+  First_unique_cone = 0,
+  Second_unique_cone,
   Duplicated_cone
 };
 
@@ -127,11 +128,23 @@ void locate_cones(const Mesh& mesh,
 
   // the cones in the underlying mesh
   std::size_t cvdss = cone_vds_in_sm.size();
+
+  std::cout << cvdss << " cones to locate" << std::endl;
+
   for(std::size_t i=0; i<cvdss; ++i) {
     BM_vertex_descriptor smvd = cone_vds_in_sm[i];
     BOOST_FOREACH(vertex_descriptor vd, vertices(mesh)) {
       if(get(ppmap, vd) == get(pm_ppmap, smvd)) {
-        Cone_type ct = (i == 0 || i == cvdss-1) ? Unique_cone : Duplicated_cone;
+        Cone_type ct;
+        if(i == 0)
+          ct = First_unique_cone;
+        else if(i == cvdss-1)
+          ct = Second_unique_cone;
+        else
+          ct = Duplicated_cone;
+        std::cout << smvd << " and "
+                  << vd << " (" << target(halfedge(vd, mesh), mesh.mesh()) << ")"
+                  << " ct: " << ct << std::endl;
         cones.insert(std::make_pair(vd, ct));
       }
     }
@@ -184,6 +197,8 @@ void locate_unordered_cones(const Mesh& mesh,
   typedef typename Kernel_traits<Mesh>::PPM            PPM;
   const PPM ppmap = get(boost::vertex_point, mesh);
 
+  bool first_cone_met = false;
+
   // walk on the seam and mark if we encounter a cone
   vertex_descriptor end = vertex_on_seam;
   do {
@@ -205,8 +220,17 @@ void locate_unordered_cones(const Mesh& mesh,
         CGAL_assertion(other_hd.seam);
         other_hd.seam = false;
 
-        Cone_type ct = (target(hd, mesh) == source(other_hd, mesh)) ? Unique_cone
-                                                                    : Duplicated_cone;
+        Cone_type ct;
+        if(target(hd, mesh) == source(other_hd, mesh)) {
+          if(first_cone_met)
+            ct = Second_unique_cone;
+          else {
+            ct = First_unique_cone;
+            first_cone_met = true;
+          }
+        } else {
+          ct = Duplicated_cone;
+        }
 
         std::cout << "new cone with type: " << ct << std::endl;