diff --git a/Triangulation_2/include/CGAL/Triangulation_2/internal/Triangulation_2_projection_traits_base_3.h b/Triangulation_2/include/CGAL/Triangulation_2/internal/Triangulation_2_projection_traits_base_3.h
index 9a160041dda..9a728577a4b 100644
--- a/Triangulation_2/include/CGAL/Triangulation_2/internal/Triangulation_2_projection_traits_base_3.h
+++ b/Triangulation_2/include/CGAL/Triangulation_2/internal/Triangulation_2_projection_traits_base_3.h
@@ -190,6 +190,7 @@ public:
   {
     CGAL_PROFILER("Projected_intersect_3::operator()")
     CGAL_TIME_PROFILER("Projected_intersect_3::operator()")
+    typedef boost::variant<Point, Segment> variant_type;
     const Vector_3 u1 = cross_product(s1.to_vector(), normal);
     if(u1 == NULL_VECTOR)
       return K().intersect_3_object()(s1.supporting_line(), s2);
@@ -203,21 +204,29 @@ public:
 
     auto planes_intersection = intersection(plane_1, plane_2);
     if(! planes_intersection) {
+#ifdef CGAL_T2_PTB_3_DEBUG
       std::cerr << "planes_intersection is empty\n";
+#endif
       return boost::none;
     }
     if(const Line* line = boost::get<Line>(&*planes_intersection))
     {
+      // check if the intersection line intersects both segments by
+      // checking if a point on the intersection line is between
+      // the segments endpoints
       const Point& pi = line->point(0);
-      if(cross_product(normal, pi - s1.source())
+      if(  cross_product(normal, pi - s1.source())
          * cross_product(normal, pi - s1.target()) > FT(0)
          ||
-         cross_product(normal, pi - s2.source())
+           cross_product(normal, pi - s2.source())
          * cross_product(normal, pi - s2.target()) > FT(0) )
       {
-        // the intersection of the lines is not inside the segments
+        // the intersection of the supporting lines is not inside both segments
+#ifdef CGAL_T2_PTB_3_DEBUG
         std::cerr << "intersection not inside\n";
         return boost::none;
+#endif
+        return boost::none;
       }
       else
       {
@@ -230,16 +239,67 @@ public:
         if(! inter){
           return boost::none;
         }
-        if(const Point* point = boost::get<Point>(&*inter)){
-          typedef  boost::variant<Point, Segment> variant_type;
+        if(const Point* point = boost::get<Point>(&*inter)){        
           return boost::make_optional(variant_type(*point));
         }
       }
     }
     if(boost::get<Plane_3>(&*planes_intersection))
     {
-      std::cerr << "coplanar lines\n";
-      CGAL_error();
+#ifdef CGAL_T2_PTB_3_DEBUG
+      std::cerr << "coplanar supporting lines\n";
+#endif
+      auto is_inside_segment = [](const Segment& s, const Point& q)
+      {
+        return Vector_3(q,s.source()) * Vector_3(q,s.target()) <=0;
+      };
+
+      bool src2_in_s1 = is_inside_segment(s1, s2.source());
+      bool tgt2_in_s1 = is_inside_segment(s1, s2.target());
+      bool src1_in_s2 = is_inside_segment(s2, s1.source());
+      bool tgt1_in_s2 = is_inside_segment(s2, s1.target());
+
+      if (src1_in_s2 && tgt1_in_s2) return boost::make_optional(variant_type(s1));
+      if (src2_in_s1 && tgt2_in_s1) return boost::make_optional(variant_type(s2));
+
+      if (src1_in_s2)
+      {
+        if (src2_in_s1)
+        {
+          if (cross_product(normal, Vector_3(s1.source(), s2.source())) != NULL_VECTOR)
+            return boost::make_optional(variant_type(Segment(s1.source(), s2.source())));
+          else
+            return boost::make_optional(variant_type((s1.source())));
+        }
+        if (tgt2_in_s1)
+        {
+          if (cross_product(normal, Vector_3(s1.source(), s2.target())) != NULL_VECTOR)
+            return boost::make_optional(variant_type(Segment(s1.source(), s2.target())));
+          else
+            return boost::make_optional(variant_type(s1.source()));
+        }
+        // should never get here with a Kernel with exact constructions
+        return boost::make_optional(variant_type(s1.source()));
+      }
+      if (tgt1_in_s2)
+      {
+        if (src2_in_s1)
+        {
+          if (cross_product(normal, Vector_3(s1.target(), s2.source())) != NULL_VECTOR)
+            return boost::make_optional(variant_type(Segment(s1.target(), s2.source())));
+          else
+            return boost::make_optional(variant_type(s1.target()));
+        }
+        if (tgt2_in_s1)
+        {
+          if (cross_product(normal, Vector_3(s1.target(), s2.target())) != NULL_VECTOR)
+            return boost::make_optional(variant_type(Segment(s1.target(), s2.target())));
+          else
+            return boost::make_optional(variant_type(s1.target()));
+        }
+        // should never get here with a Kernel with exact constructions
+        return boost::make_optional(variant_type(s1.target()));
+      }
       return boost::none;
     }
     return boost::none;
diff --git a/Triangulation_2/test/Triangulation_2/test_cdt_2_projection_traits_special_case.cpp b/Triangulation_2/test/Triangulation_2/test_cdt_2_projection_traits_special_case.cpp
index 26693b95a28..bf163771380 100644
--- a/Triangulation_2/test/Triangulation_2/test_cdt_2_projection_traits_special_case.cpp
+++ b/Triangulation_2/test/Triangulation_2/test_cdt_2_projection_traits_special_case.cpp
@@ -60,6 +60,67 @@ bool test(std::string test_name)
   return cdt.is_valid();
 }
 
+
+template <class K, typename CDT_2_traits>
+bool test_segment_intersections(std::string test_name)
+{
+  std::cerr << "Testing " << test_name << std::endl;
+
+  CDT_2_traits traits(typename K::Vector_3(0,0,1));
+  typename CDT_2_traits::Intersect_2 intersect = traits.intersect_2_object();
+
+  typedef typename CDT_2_traits::Segment_2 Segment_2;
+  typedef typename CDT_2_traits::Point_2 Point_2;
+
+  // test point intersection
+  Segment_2 s1( Point_2(0,0,0), Point_2(1,1,0) ),
+            s2( Point_2(0,1,0), Point_2(1,0,0) );
+
+  CGAL::Object o = intersect(s1,s2);
+  assert( o.is<Point_2>() );
+
+// test segment overlap
+  Point_2 pts[4] = { Point_2(0,0,0), Point_2(1,0,1), Point_2(2,0,2), Point_2(4,0,3) };
+
+  o = intersect( Segment_2(pts[0], pts[1]), Segment_2(pts[2], pts[3]) );
+  assert( o.empty() );
+
+  // pure overlap
+  o = intersect( Segment_2(pts[0], pts[2]), Segment_2(pts[1], pts[3]) );
+  assert( o.is<Segment_2>() );
+  o = intersect( Segment_2(pts[0], pts[2]), Segment_2(pts[3], pts[1]) );
+  assert( o.is<Segment_2>() );
+  o = intersect( Segment_2(pts[2], pts[0]), Segment_2(pts[1], pts[3]) );
+  assert( o.is<Segment_2>() );
+  o = intersect( Segment_2(pts[2], pts[0]), Segment_2(pts[3], pts[1]) );
+  assert( o.is<Segment_2>() );
+  // segment fully included
+  o = intersect( Segment_2(pts[0], pts[3]), Segment_2(pts[1], pts[2]) );
+  assert( o.is<Segment_2>() );
+  assert( CGAL::object_cast<Segment_2>(o) == Segment_2(pts[1], pts[2]) );
+  // segment fully included with shared vertex
+  o = intersect( Segment_2(pts[0], pts[1]), Segment_2(pts[0], pts[2]) );
+  assert( o.is<Segment_2>() );
+  assert( CGAL::object_cast<Segment_2>(o) == Segment_2(pts[0], pts[1]) );
+  // segment sharing a vertex
+  o = intersect( Segment_2(pts[0], pts[1]), Segment_2(pts[1], pts[2]) );
+  assert( o.is<Point_2>() );
+  assert( CGAL::object_cast<Point_2>(o) == pts[1]);
+
+  // degenerate segment
+  Segment_2 sd(Point_2(1,0,6), Point_2(1,0,7));
+  o = intersect( Segment_2(pts[0], pts[2]), sd );
+  assert( o.is<Point_2>() );
+  o = intersect( Segment_2(pts[0], pts[1]), sd );
+  assert( o.is<Point_2>() );
+  o = intersect( Segment_2(pts[1], pts[0]), sd );
+  assert( o.is<Point_2>() );
+  o = intersect( Segment_2(pts[2], pts[3]), sd );
+  assert( o.empty() );
+
+  return true;
+}
+
 int main()
 {
   std::cerr.precision(17);
@@ -75,5 +136,12 @@ int main()
       test<CGAL::Epeck,
            CGAL::Triangulation_2_projection_traits_3<Epeck> >
       ("CDT_2 in a 3D plane, with Epeck");
+  ok = ok && test_segment_intersections<CGAL::Epick,
+           CGAL::Triangulation_2_projection_traits_3<Epick> >
+      ("CDT_2 traits intersection with Epick");
+  ok = ok && test_segment_intersections<CGAL::Epeck,
+           CGAL::Triangulation_2_projection_traits_3<Epeck> >
+      ("CDT_2 traits intersection with Epeck");
+
   return ok ? 0 : 1;
 }