diff --git a/Triangulation_2/include/CGAL/Triangulation_2.h b/Triangulation_2/include/CGAL/Triangulation_2.h
index 3d2a6e73629..9c20817b5d6 100644
--- a/Triangulation_2/include/CGAL/Triangulation_2.h
+++ b/Triangulation_2/include/CGAL/Triangulation_2.h
@@ -895,49 +895,77 @@ is_valid(bool verbose, int level) const
   if (dimension() <= 0 || (dimension()==1 && number_of_vertices() == 2 ) )
     return result;
 
-  if (dimension() == 1) {
+  if (dimension() == 1)
+  {
     Finite_vertices_iterator it1 = finite_vertices_begin(),
                              it2(it1), it3(it1);
     ++it2;
     ++it3; ++it3;
     while( it3 != finite_vertices_end()) {
-     Orientation s = orientation(it1->point(),
-         it2->point(),
-         it3->point());
-     result = result && s == COLLINEAR ;
+     Orientation s = orientation(point(it1), point(it2), point(it3));
+     result = result && (s == COLLINEAR) ;
+     if(verbose && (s != COLLINEAR))
+     {
+       std::cerr << "Error: " << point(it1) << " "
+                              << point(it2) << " and "
+                              << point(it3) << " are not collinear" << std::endl;
+     }
+
      CGAL_triangulation_assertion(result);
      ++it1 ; ++it2; ++it3;
     }
   }
-  else { //dimension() == 2
-    for(Finite_faces_iterator it=finite_faces_begin();
-        it!=finite_faces_end(); it++) {
+  else //dimension() == 2
+  {
+    for(Finite_faces_iterator it=finite_faces_begin(); it!=finite_faces_end(); it++)
+    {
       CGAL_triangulation_assertion( ! is_infinite(it));
-      Orientation s = orientation(it->vertex(0)->point(),
-                                  it->vertex(1)->point(),
-                                  it->vertex(2)->point());
-      CGAL_triangulation_assertion( s == LEFT_TURN );
+      Orientation s = orientation(point(it, 0), point(it, 1), point(it, 2));
       result = result && ( s == LEFT_TURN );
+
+      if(verbose && (s != LEFT_TURN))
+      {
+        std::cerr << "Error: " << point(it, 0) << " "
+                               << point(it, 1) << " and "
+                               << point(it, 2) << " form a badly oriented face" << std::endl;
+      }
+
+      CGAL_triangulation_assertion(result);
     }
 
     Vertex_circulator start = incident_vertices(infinite_vertex());
     Vertex_circulator pc(start);
     Vertex_circulator qc(start); ++qc;
     Vertex_circulator rc(start); ++rc; ++rc;
-    do {
-      Orientation s = orientation(pc->point(),
-                                  qc->point(),
-                                  rc->point());
+    do
+    {
+      Orientation s = orientation(point(pc), point(qc), point(rc));
       CGAL_triangulation_assertion( s != LEFT_TURN );
       result = result && ( s != LEFT_TURN );
+
+      if(verbose && (s == LEFT_TURN))
+      {
+        std::cerr << "Error: " << point(pc) << " "
+                               << point(qc) << " and "
+                               << point(rc) << " form a badly oriented infinite face" << std::endl;
+      }
+
       ++pc ; ++qc ; ++rc;
     } while(pc != start);
 
     // check number of faces. This cannot be done by the Tds
     // which does not know the number of components nor the genus
-    result = result && (number_of_faces() == 2*(number_of_vertices()+1)
-                        - 4
-                        - degree(infinite_vertex()));
+    const bool genus_check = number_of_faces() == 2*(number_of_vertices()+1) - 4 - degree(infinite_vertex());
+    result = result && genus_check;
+    if(verbose && !genus_check)
+    {
+      std::cerr << "Error: Genus check fail " << number_of_faces()
+                << " vs " << 2*(number_of_vertices()+1) - 4 - degree(infinite_vertex())
+                << " (nv = " << number_of_vertices()
+                << " nf = " << number_of_faces()
+                << " and degree(infinite_vertex()) = " << degree(infinite_vertex()) << std::endl;
+    }
+
     CGAL_triangulation_assertion( result);
   }
   return result;
@@ -3789,6 +3817,219 @@ operator>>(std::istream& is, Triangulation_2<Gt, Tds> &tr)
   return is;
 }
 
+namespace internal {
+
+// Internal function used by operator==.
+template < class GT, class TDS1, class TDS2, typename FMAP, typename VMAP >
+bool
+test_next(const Triangulation_2<GT, TDS1>& t1,
+          const Triangulation_2<GT, TDS2>& t2,
+          typename Triangulation_2<GT, TDS1>::Face_handle f1,
+          typename Triangulation_2<GT, TDS2>::Face_handle f2,
+          FMAP& Fmap,
+          VMAP& Vmap)
+{
+  // This function tests and registers the 3 neighbors of f1/f2,
+  // and recursively calls itself over them.
+  // We don't use the call stack as it may overflow
+  // Returns false if an inequality has been found.
+
+  // Precondition: f1, f2 have been registered as well as their 3 vertices.
+  CGAL_triangulation_precondition(t1.dimension() >= 2);
+  CGAL_triangulation_precondition(Fmap[f1] == f2);
+  CGAL_triangulation_precondition(Vmap.find(f1->vertex(0)) != Vmap.end());
+  CGAL_triangulation_precondition(Vmap.find(f1->vertex(1)) != Vmap.end());
+  CGAL_triangulation_precondition(t1.dimension() == 1 || Vmap.find(f1->vertex(2)) != Vmap.end());
+
+  typedef Triangulation_2<GT, TDS1>               Tr1;
+  typedef Triangulation_2<GT, TDS2>               Tr2;
+  typedef typename Tr1::Vertex_handle             Vertex_handle1;
+  typedef typename Tr1::Face_handle               Face_handle1;
+  typedef typename Tr2::Vertex_handle             Vertex_handle2;
+  typedef typename Tr2::Face_handle               Face_handle2;
+
+  typedef typename VMAP::const_iterator           Vit;
+  typedef typename FMAP::const_iterator           Fit;
+
+  typedef typename Tr1::Geom_traits::Construct_point_2    Construct_point_2;
+  typedef typename Tr1::Geom_traits::Compare_xy_2         Compare_xy_2;
+
+  Compare_xy_2 cmp1 = t1.geom_traits().compare_xy_2_object();
+  Construct_point_2 cp = t1.geom_traits().construct_point_2_object();
+
+  std::vector<std::pair<Face_handle1, Face_handle2> > face_stack;
+  face_stack.emplace_back(f1, f2);
+
+  while(! face_stack.empty())
+  {
+    Face_handle1 f1 = face_stack.back().first;
+    Face_handle2 f2 = face_stack.back().second;
+    face_stack.pop_back();
+
+    for(int i=0; i <= t1.dimension(); ++i)
+    {
+      Face_handle1 n1 = f1->neighbor(i);
+      Fit fit = Fmap.find(n1);
+      Vertex_handle1 v1 = f1->vertex(i);
+      Vertex_handle2 v2 = Vmap[v1];
+      Face_handle2 n2 = f2->neighbor(f2->index(v2));
+      if(fit != Fmap.end())
+      {
+        // n1 was already registered.
+        if(fit->second != n2)
+          return false;
+
+        continue;
+      }
+
+      // n1 has not yet been registered.
+      // We check that the new vertices match geometrically.
+      // And we register them.
+      Vertex_handle1 vn1 = n1->vertex(n1->index(f1));
+      Vertex_handle2 vn2 = n2->vertex(n2->index(f2));
+      Vit vit = Vmap.find(vn1);
+      if(vit != Vmap.end())
+      {
+        // vn1 already registered
+        if(vit->second != vn2)
+          return false;
+      }
+      else
+      {
+        if(t2.is_infinite(vn2))
+          return false; // vn1 can't be infinite,
+
+        // since it would have been registered.
+        if(cmp1(cp(vn1->point()), cp(vn2->point())) != 0)
+          return false;
+
+        // We register vn1/vn2.
+        Vmap.emplace(vn1, vn2);
+      }
+
+      // We register n1/n2.
+      Fmap.emplace(n1, n2);
+      face_stack.emplace_back(n1, n2);
+    }
+  }
+
+  return true;
+}
+
+} // namespace internal
+
+template < class GT, class TDS1, class TDS2 >
+bool
+operator==(const Triangulation_2<GT, TDS1>& t1,
+           const Triangulation_2<GT, TDS2>& t2)
+{
+  typedef typename Triangulation_2<GT, TDS1>::Vertex_handle Vertex_handle1;
+  typedef typename Triangulation_2<GT, TDS1>::Face_handle   Face_handle1;
+  typedef typename Triangulation_2<GT, TDS2>::Vertex_handle Vertex_handle2;
+  typedef typename Triangulation_2<GT, TDS2>::Face_handle   Face_handle2;
+
+  typedef typename Triangulation_2<GT, TDS1>::Point                          Point;
+
+  typedef typename Triangulation_2<GT, TDS1>::Geom_traits::Equal_2           Equal_2;
+  typedef typename Triangulation_2<GT, TDS1>::Geom_traits::Compare_xy_2      Compare_xy_2;
+  typedef typename Triangulation_2<GT, TDS1>::Geom_traits::Construct_point_2 Construct_point_2;
+
+  Equal_2 equal = t1.geom_traits().equal_2_object();
+  Compare_xy_2 cmp1 = t1.geom_traits().compare_xy_2_object();
+  Compare_xy_2 cmp2 = t2.geom_traits().compare_xy_2_object();
+  Construct_point_2 cp = t1.geom_traits().construct_point_2_object();
+
+  // Some quick checks.
+  if(t1.dimension() != t2.dimension() ||
+     t1.number_of_vertices() != t2.number_of_vertices() ||
+     t1.number_of_faces() != t2.number_of_faces())
+    return false;
+
+  int dim = t1.dimension();
+  // Special case for dimension < 1.
+  // The triangulation is uniquely defined in these cases.
+  if(dim == -1)
+    return true;
+
+  // Special case for dimensions 0 and 1.
+  if(dim < 2)
+  {
+    // It's enough to test that the points are the same,
+    // since the triangulation is uniquely defined in this case.
+    std::vector<Point> V1 (t1.points_begin(), t1.points_end());
+    std::vector<Point> V2 (t2.points_begin(), t2.points_end());
+
+    std::sort(V1.begin(), V1.end(),
+              [&cmp1, &cp](const Point& p1, const Point& p2){ return cmp1(cp(p1), cp(p2))==SMALLER; });
+
+    std::sort(V2.begin(), V2.end(),
+              [&cmp2, &cp](const Point& p1, const Point& p2){ return cmp2(cp(p1), cp(p2))==SMALLER; });
+
+    return V1 == V2;
+  }
+
+  // We will store the mapping between the 2 triangulations vertices and faces in 2 maps.
+  std::unordered_map<Vertex_handle1, Vertex_handle2> Vmap;
+  std::unordered_map<Face_handle1, Face_handle2> Fmap;
+
+  // Handle the infinite vertex.
+  Vertex_handle1 v1 = t1.infinite_vertex();
+  Vertex_handle2 iv2 = t2.infinite_vertex();
+  Vmap.emplace(v1, iv2);
+
+  // We pick one infinite face of t1, and try to match it against the infinite faces of t2.
+  Face_handle1 f = v1->face();
+  Vertex_handle1 v2 = f->vertex((f->index(v1)+1)%(dim+1));
+  Vertex_handle1 v3 = f->vertex((f->index(v1)+2)%(dim+1));
+  const Point& p2 = v2->point();
+  const Point& p3 = v3->point();
+
+  std::vector<Face_handle2> ifs;
+  auto fc = t2.incident_faces(iv2), done(fc);
+  do {
+    ifs.push_back(fc);
+  } while(++fc != done);
+
+  for(typename std::vector<Face_handle2>::const_iterator fit = ifs.begin();
+                                                         fit != ifs.end(); ++fit)
+  {
+    int inf = (*fit)->index(iv2);
+
+    if(equal(cp(p2), cp((*fit)->vertex((inf+1)%(dim+1))->point())))
+      Vmap.emplace(v2, (*fit)->vertex((inf+1)%(dim+1)));
+    else if(dim == 2 && equal(cp(p2), cp((*fit)->vertex((inf+2)%(dim+1))->point())))
+      Vmap.emplace(v2, (*fit)->vertex((inf+2)%(dim+1)));
+    else
+      continue; // None matched v2.
+
+    if(equal(cp(p3), cp((*fit)->vertex((inf+1)%(dim+1))->point())))
+      Vmap.emplace(v3, (*fit)->vertex((inf+1)%(dim+1)));
+    else if(dim == 2 && equal(cp(p3), cp((*fit)->vertex((inf+2)%(dim+1))->point())))
+      Vmap.emplace(v3, (*fit)->vertex((inf+2)%(dim+1)));
+    else
+      continue; // None matched v3.
+
+    // Found it !
+    Fmap.emplace(f, *fit);
+    break;
+  }
+
+  if(Fmap.size() == 0)
+    return false;
+
+  // We now have one face, we need to propagate recursively.
+  return internal::test_next(t1, t2, Fmap.begin()->first, Fmap.begin()->second, Fmap, Vmap);
+}
+
+template < class GT, class Tds1, class Tds2 >
+inline
+bool
+operator!=(const Triangulation_2<GT, Tds1>& t1,
+           const Triangulation_2<GT, Tds2>& t2)
+{
+  return ! (t1 == t2);
+}
+
 } //namespace CGAL
 
 #include <CGAL/enable_warnings.h>
diff --git a/Triangulation_2/test/Triangulation_2/include/CGAL/_test_cls_triangulation_2.h b/Triangulation_2/test/Triangulation_2/include/CGAL/_test_cls_triangulation_2.h
index 15caad00e68..0aee5fb2e42 100644
--- a/Triangulation_2/test/Triangulation_2/include/CGAL/_test_cls_triangulation_2.h
+++ b/Triangulation_2/test/Triangulation_2/include/CGAL/_test_cls_triangulation_2.h
@@ -132,12 +132,10 @@ _test_cls_triangulation_2( const Triangul & )
   assert( T1.number_of_vertices() == 0 );
 
   Triangul T3(T1);
-  assert(T3.tds().vertices().size() == T1.tds().vertices().size());
-  assert(T3.tds().faces().size() == T1.tds().faces().size());
+  assert(T3 == T1);
 
   Triangul T4 = T1;
-  assert(T4.tds().vertices().size() == T1.tds().vertices().size());
-  assert(T4.tds().faces().size() == T1.tds().faces().size());
+  assert(T4 == T1);
 
   T3.swap(T1);
 
@@ -159,6 +157,7 @@ _test_cls_triangulation_2( const Triangul & )
   assert( T0_0.number_of_vertices() == 0 );
   assert( T0_0.number_of_faces() == 0);
   assert( T0_0.is_valid() );
+  assert( T0_0 == T0_0 );
 
   Triangul T0_1;
   Vertex_handle v0_1_0 = T0_1.insert(p0); assert( v0_1_0 != nullptr );
@@ -166,10 +165,18 @@ _test_cls_triangulation_2( const Triangul & )
   assert( T0_1.number_of_vertices() == 1 );
   assert( T0_1.number_of_faces() == 0);
   assert( T0_1.is_valid() );
+  assert( T0_0 != T0_1 );
 
   Triangul T0_1b(T0_1);
-  assert(T0_1b.tds().vertices().size() == T0_1.tds().vertices().size());
-  assert(T0_1b.tds().faces().size() == T0_1.tds().faces().size());
+  assert(T0_1b == T0_1);
+
+  Triangul T0_1c;
+  v0_1_0 = T0_1c.insert(p1); assert( v0_1_0 != nullptr );
+  assert( T0_1c.dimension() == 0 );
+  assert( T0_1c.number_of_vertices() == 1 );
+  assert( T0_1c.number_of_faces() == 0);
+  assert( T0_1c.is_valid() );
+  assert( T0_1 != T0_1c );
 
   // test insert_first()
   Triangul T0_2;
@@ -192,10 +199,10 @@ _test_cls_triangulation_2( const Triangul & )
   assert( T1_2.number_of_vertices() == 2 );
   assert( T1_2.number_of_faces() == 0 );
   assert( T1_2.is_valid() );
+  assert( T1_2 != T0_1 );
 
   Triangul T1_2b(T1_2);
-  assert(T1_2b.tds().vertices().size() == T1_2.tds().vertices().size());
-  assert(T1_2b.tds().faces().size() == T1_2.tds().faces().size());
+  assert(T1_2b == T1_2);
 
   // p1,p3,p2  [endpoints first]
   Triangul T1_3_0;
@@ -206,6 +213,7 @@ _test_cls_triangulation_2( const Triangul & )
   assert( T1_3_0.number_of_vertices() == 3 );
   assert( T1_3_0.number_of_faces() == 0 );
   assert( T1_3_0.is_valid() );
+  assert( T1_3_0 != T1_2 );
 
   // p1,p2,p3  [middle point first]
   Triangul T1_3_1;
@@ -216,6 +224,7 @@ _test_cls_triangulation_2( const Triangul & )
   assert( T1_3_1.number_of_vertices() == 3 );
   assert( T1_3_1.number_of_faces() == 0 );
   assert( T1_3_1.is_valid() );
+  assert( T1_3_1 == T1_3_0 );
 
   Triangul T1_5;
   Vertex_handle v1_5_1 = T1_5.insert(p1);
@@ -227,6 +236,7 @@ _test_cls_triangulation_2( const Triangul & )
   assert( T1_5.number_of_vertices() == 5 );
   assert( T1_5.number_of_faces() == 0 );
   assert( T1_5.is_valid() );
+  assert( T1_5 != T1_3_1 );
 
   // test insert_second()
   Triangul T1_6 = T0_2;
@@ -298,11 +308,13 @@ _test_cls_triangulation_2( const Triangul & )
 
   // test generic iterator insert
 #ifndef CGAL_CFG_NO_MEMBER_TEMPLATES
-  Triangul T2_4; T2_4.insert( Point_iterator(T2_1.finite_vertices_begin()),
-                         Point_iterator(T2_1.finite_vertices_end()) );
+  Triangul T2_4;
+  T2_4.insert( Point_iterator(T2_1.finite_vertices_begin()),
+               Point_iterator(T2_1.finite_vertices_end()) );
   assert( T2_4.dimension() == 2 );
   assert( T2_4.number_of_vertices() == 11 );
   assert( T2_4.is_valid() );
+  assert( T2_4 == T2_1 );
 #endif
 
   // test list iterator insert
@@ -320,6 +332,7 @@ _test_cls_triangulation_2( const Triangul & )
   assert( T2_6.dimension() == 2 );
   assert( T2_6.number_of_vertices() == 10 );
   assert( T2_6.is_valid() );
+  assert( T2_5 == T2_6 );
 
   // test grid insert
   Triangul T2_7;
@@ -496,21 +509,25 @@ _test_cls_triangulation_2( const Triangul & )
   assert( T1_5_2.dimension() == 1 );
   assert( T1_5_2.number_of_vertices() == 5 );
   assert( T1_5_2.is_valid() );
+  assert( T1_5_2 == T1_5 );
 
   // test copy_constructor with non-empty 2-triangulation
   Triangul T2_8_1(T2_8);
   assert( T2_8_1.is_valid());
+  assert( T2_8 == T2_8);
+
   Triangul T2_1_1( T2_1 );
   assert( T2_1_1.dimension() == 2 );
   assert( T2_1_1.number_of_vertices() == 11 );
   assert( T2_1_1.is_valid() );
+  assert( T2_1_1 == T2_1 );
 
   // test assignment operator
   Triangul T2_1_4 = T2_1;
   assert( T2_1_4.dimension() == 2 );
   assert( T2_1_4.number_of_vertices() == 11 );
   assert( T2_1_4.is_valid() );
-
+  assert( T2_1_4 == T2_1 );
 
   /*********************************************/
   /****** FINITE/INFINITE VERTICES/FACES *******/
diff --git a/Triangulation_3/include/CGAL/Triangulation_3.h b/Triangulation_3/include/CGAL/Triangulation_3.h
index 25d4f462201..f075bfdeb5e 100644
--- a/Triangulation_3/include/CGAL/Triangulation_3.h
+++ b/Triangulation_3/include/CGAL/Triangulation_3.h
@@ -24,14 +24,6 @@
 # include <CGAL/Profile_counter.h>
 #endif
 
-#include <iostream>
-#include <list>
-#include <set>
-#include <map>
-#include <unordered_map>
-#include <utility>
-#include <stack>
-
 #include <CGAL/Unique_hash_map.h>
 #include <CGAL/triangulation_assertions.h>
 #include <CGAL/Triangulation_utils_3.h>
@@ -77,6 +69,14 @@
 # include <tbb/scalable_allocator.h>
 #endif
 
+#include <iostream>
+#include <list>
+#include <set>
+#include <map>
+#include <unordered_map>
+#include <utility>
+#include <stack>
+
 #define CGAL_TRIANGULATION_3_USE_THE_4_POINTS_CONSTRUCTOR
 
 namespace CGAL {
@@ -7154,16 +7154,14 @@ is_valid_finite(Cell_handle c, bool verbose, int) const
 namespace internal {
 
 // Internal function used by operator==.
-template < class GT, class Tds1, class Tds2, class Lds >
+template < class GT, class Tds1, class Tds2, class Lds, typename CMAP, typename VMAP >
 bool
 test_next(const Triangulation_3<GT, Tds1, Lds>& t1,
           const Triangulation_3<GT, Tds2, Lds>& t2,
           typename Triangulation_3<GT, Tds1, Lds>::Cell_handle c1,
           typename Triangulation_3<GT, Tds2, Lds>::Cell_handle c2,
-          std::map<typename Triangulation_3<GT, Tds1, Lds>::Cell_handle,
-          typename Triangulation_3<GT, Tds2, Lds>::Cell_handle>& Cmap,
-          std::map<typename Triangulation_3<GT, Tds1, Lds>::Vertex_handle,
-          typename Triangulation_3<GT, Tds2, Lds>::Vertex_handle>& Vmap)
+          CMAP& Cmap,
+          VMAP& Vmap)
 {
   // This function tests and registers the 4 neighbors of c1/c2,
   // and recursively calls itself over them.
@@ -7186,8 +7184,8 @@ test_next(const Triangulation_3<GT, Tds1, Lds>& t1,
   typedef typename Tr2::Vertex_handle             Vertex_handle2;
   typedef typename Tr2::Cell_handle               Cell_handle2;
 
-  typedef typename std::map<Vertex_handle1, Vertex_handle2>::const_iterator Vit;
-  typedef typename std::map<Cell_handle1, Cell_handle2>::const_iterator     Cit;
+  typedef typename std::unordered_map<Vertex_handle1, Vertex_handle2>::const_iterator Vit;
+  typedef typename std::unordered_map<Cell_handle1, Cell_handle2>::const_iterator     Cit;
 
   typedef typename Tr1::Geom_traits::Construct_point_3    Construct_point_3;
   typedef typename Tr1::Geom_traits::Compare_xyz_3        Compare_xyz_3;
@@ -7196,7 +7194,7 @@ test_next(const Triangulation_3<GT, Tds1, Lds>& t1,
   Construct_point_3 cp = t1.geom_traits().construct_point_3_object();
 
   std::vector<std::pair<Cell_handle1, Cell_handle2> > cell_stack;
-  cell_stack.push_back(std::make_pair(c1, c2));
+  cell_stack.emplace_back(c1, c2);
 
   while(! cell_stack.empty())
   {
@@ -7242,12 +7240,12 @@ test_next(const Triangulation_3<GT, Tds1, Lds>& t1,
           return false;
 
         // We register vn1/vn2.
-        Vmap.insert(std::make_pair(vn1, vn2));
+        Vmap.emplace(vn1, vn2);
       }
 
       // We register n1/n2.
-      Cmap.insert(std::make_pair(n1, n2));
-      cell_stack.push_back(std::make_pair(n1, n2));
+      Cmap.emplace(n1, n2);
+      cell_stack.emplace_back(n1, n2);
     }
   }
 
@@ -7286,11 +7284,11 @@ operator==(const Triangulation_3<GT, Tds1, Lds>& t1,
   int dim = t1.dimension();
   // Special case for dimension < 1.
   // The triangulation is uniquely defined in these cases.
-  if(dim < 1)
+  if(dim == - 1)
     return true;
 
-  // Special case for dimension == 1.
-  if(dim == 1)
+  // Special case for dimensions 0 and 1.
+  if(dim < 2)
   {
     // It's enough to test that the points are the same,
     // since the triangulation is uniquely defined in this case.
@@ -7308,13 +7306,13 @@ operator==(const Triangulation_3<GT, Tds1, Lds>& t1,
 
   // We will store the mapping between the 2 triangulations vertices and
   // cells in 2 maps.
-  std::map<Vertex_handle1, Vertex_handle2> Vmap;
-  std::map<Cell_handle1, Cell_handle2> Cmap;
+  std::unordered_map<Vertex_handle1, Vertex_handle2> Vmap;
+  std::unordered_map<Cell_handle1, Cell_handle2> Cmap;
 
   // Handle the infinite vertex.
   Vertex_handle1 v1 = t1.infinite_vertex();
   Vertex_handle2 iv2 = t2.infinite_vertex();
-  Vmap.insert(std::make_pair(v1, iv2));
+  Vmap.emplace(v1, iv2);
 
   // We pick one infinite cell of t1, and try to match it against the
   // infinite cells of t2.
@@ -7364,7 +7362,7 @@ operator==(const Triangulation_3<GT, Tds1, Lds>& t1,
     }
 
     // Found it !
-    Cmap.insert(std::make_pair(c, *cit));
+    Cmap.emplace(c, *cit);
     break;
   }
 
diff --git a/Triangulation_3/test/Triangulation_3/include/CGAL/_test_cls_triangulation_3.h b/Triangulation_3/test/Triangulation_3/include/CGAL/_test_cls_triangulation_3.h
index fe5b2d041b2..9790692fd90 100644
--- a/Triangulation_3/test/Triangulation_3/include/CGAL/_test_cls_triangulation_3.h
+++ b/Triangulation_3/test/Triangulation_3/include/CGAL/_test_cls_triangulation_3.h
@@ -222,15 +222,23 @@ _test_cls_triangulation_3(const Triangulation &)
     }
 
   std::cout << "    Constructor1 " << std::endl;
-  Point p10(0,0,0);
-  Vertex_handle v0=T0.insert(p10);
+  Vertex_handle v0=T0.insert(p[0]);
   assert(T0.dimension() == 0);
   assert(T0.number_of_vertices() == 1);
   assert(T0.is_valid());
+  assert(T0 != Tm1);
 
   Cls T0d0(T0);
   assert(T0 == T0d0);
 
+  Cls T0d0b;
+  v0=T0d0b.insert(p[1]);
+  assert(T0d0b.dimension() == 0);
+  assert(T0d0b.number_of_vertices() == 1);
+  assert(T0d0b.is_valid());
+  assert(T0d0b != T0d0);
+  assert(T0d0b != Tm1);
+
   if (! del) // to avoid doing the following tests for both Delaunay
     // and non Delaunay triangulations
     {
@@ -244,6 +252,7 @@ _test_cls_triangulation_3(const Triangulation &)
   assert(T0.dimension() == 1);
   assert(T0.number_of_vertices() == 2);
   assert(T0.is_valid());
+  assert(T0 != T0d0);
 
   Cls T0d1(T0);
   assert(T0 == T0d1);
@@ -261,6 +270,7 @@ _test_cls_triangulation_3(const Triangulation &)
   assert(T0.dimension() == 2);
   assert(T0.number_of_vertices() == 3);
   assert(T0.is_valid());
+  assert(T0 != T0d1);
 
   Cls T0d2(T0);
   assert(T0 == T0d2);
@@ -278,6 +288,7 @@ _test_cls_triangulation_3(const Triangulation &)
   assert(T0.dimension() == 3);
   assert(T0.number_of_vertices() == 4);
   assert(T0.is_valid());
+  assert(T0 != T0d2);
 
   Cls T0d3(T0);
   assert(T0 == T0d3);