diff --git a/AABB_tree/include/CGAL/AABB_indexed_triangle_primitive_2.h b/AABB_tree/include/CGAL/AABB_indexed_triangle_primitive_2.h
index 61ad6ae95eb..d5e0359fdc8 100644
--- a/AABB_tree/include/CGAL/AABB_indexed_triangle_primitive_2.h
+++ b/AABB_tree/include/CGAL/AABB_indexed_triangle_primitive_2.h
@@ -85,11 +85,11 @@ namespace internal {
  * \tparam IndexIterator is a model of `ForwardIterator` with its value type being a `RandomAccessRange` of size 3 with an index type as `value_type`, e.g., `uint8_t`, `uint16_t` or int.
  * \tparam PointRange is a model of `RandomAccessRange`. Its value type needs to be compatible to PointMap or `Point_2` in the default case.
  * \tparam CacheDatum is either `CGAL::Tag_true` or `CGAL::Tag_false`. In the former case,
- *           the datum is stored in the primitive, while in the latter it is
- *           constructed on the fly to reduce the memory footprint.
- *           The default is `CGAL::Tag_false` (datum is not stored).
+ *         the datum is stored in the primitive, while in the latter it is
+ *         constructed on the fly to reduce the memory footprint.
+ *         The default is `CGAL::Tag_false` (datum is not stored).
  * \tparam PointMap is a model of `ReadablePropertyMap` with its key type being the value type of `PointRange` and the value type being a `Point_2`.
-             The default is `Identity_property_map<typename PointRange::value_type>`.
+ *         The default is `Identity_property_map<typename PointRange::value_type>`.
  *
  * \sa `AABBPrimitive`
  * \sa `AABB_primitive<Id,ObjectPropertyMap,PointPropertyMapPolyhedron,ExternalPropertyMaps,CacheDatum>`
diff --git a/AABB_tree/include/CGAL/AABB_polyline_segment_primitive_2.h b/AABB_tree/include/CGAL/AABB_polyline_segment_primitive_2.h
index 660da7d5533..7022478a1ff 100644
--- a/AABB_tree/include/CGAL/AABB_polyline_segment_primitive_2.h
+++ b/AABB_tree/include/CGAL/AABB_polyline_segment_primitive_2.h
@@ -24,16 +24,16 @@
 namespace CGAL {
 
 namespace internal {
-  template <class GeomTraits, class Iterator>
+  template <class GeomTraits, class Iterator, class PointMap>
   struct Segment_2_from_point_iterator_property_map {
     //classical typedefs
     typedef Iterator key_type;
     typedef typename GeomTraits::Segment_2 value_type;
     typedef typename GeomTraits::Segment_2 reference; // The segments are created on the fly, so working with references is not possible.
     typedef boost::readable_property_map_tag category;
-    typedef Segment_2_from_point_iterator_property_map<GeomTraits, Iterator> Self;
+    typedef Segment_2_from_point_iterator_property_map<GeomTraits, Iterator, PointMap> Self;
 
-    Segment_2_from_point_iterator_property_map(Iterator b, Iterator e) : begin(b), end(e) {}
+    Segment_2_from_point_iterator_property_map(Iterator b, Iterator e, PointMap& pmap) : begin(b), end(e), pmap(pmap) {}
     Segment_2_from_point_iterator_property_map() {}
 
     inline friend reference // Cannot return reference as the Segment does not exist, only the points exist.
@@ -41,12 +41,35 @@ namespace internal {
     {
       Iterator it2 = std::next(it);
       if (it2 == s.end)
-        return typename GeomTraits::Construct_segment_2()(*it, *s.begin);
+        return typename GeomTraits::Construct_segment_2()(get(s.pmap, *it), get(s.pmap, *s.begin));
       else
-        return typename GeomTraits::Construct_segment_2()(*it, *it2 );
+        return typename GeomTraits::Construct_segment_2()(get(s.pmap, *it), get(s.pmap, *it2));
     }
 
     Iterator begin, end;
+    PointMap pmap;
+  };
+
+  template <class GeomTraits, class Iterator, class PointMap>
+  struct Point_from_iterator_property_map {
+    //classical typedefs
+    typedef Iterator key_type;
+    typedef typename PointMap::value_type value_type;
+    typedef const value_type reference;
+
+    typedef boost::readable_property_map_tag category;
+    typedef Point_from_iterator_property_map<GeomTraits, Iterator, PointMap> Self;
+
+    Point_from_iterator_property_map() {}
+    Point_from_iterator_property_map(PointMap& pmap) :  pmap(pmap) {}
+
+    inline friend reference
+      get(Self s, key_type it)
+    {
+      return get(s.pmap, *it);
+    }
+
+    PointMap pmap;
   };
 }//namespace internal
 
@@ -64,11 +87,13 @@ namespace internal {
  *         It also provides the functor `Construct_segment_2` that has an operator taking two `Point_2`
  *         and returning a `Segment_2`.
  * \tparam Iterator is a model of `ForwardIterator` with its value type convertible to `GeomTraits::Point_2`
- * \tparam PointRange is a model of `ForwardRange` with its value type convertible to `GeomTraits::Point_2`
+ * \tparam PointRange is a model of `ConstRange`. Its value type needs to be compatible to PointMap or `Point_2` in the default case.
  * \tparam CacheDatum is either `CGAL::Tag_true` or `CGAL::Tag_false`. In the former case,
- *           the datum is stored in the primitive, while in the latter it is
- *           constructed on the fly to reduce the memory footprint.
- *           The default is `CGAL::Tag_false` (datum is not stored).
+ *         the datum is stored in the primitive, while in the latter it is
+ *         constructed on the fly to reduce the memory footprint.
+ *         The default is `CGAL::Tag_false` (datum is not stored).
+ * \tparam PointMap is a model of `ReadablePropertyMap` with its key type being the value type of `PointRange` and the value type being a `Point_2`.
+ *         The default is `Identity_property_map<typename PointRange::value_type>`.
  *
  * \sa `AABBPrimitive`
  * \sa `AABB_primitive<Id,ObjectPropertyMap,PointPropertyMapPolyhedron,ExternalPropertyMaps,CacheDatum>`
@@ -82,27 +107,28 @@ namespace internal {
 template < class GeomTraits,
            class Iterator,
            class PointRange,
-           class CacheDatum=Tag_false>
+           class CacheDatum = Tag_false,
+           class PointMap = Identity_property_map<typename PointRange::value_type>>
 class AABB_polyline_segment_primitive_2
 #ifndef DOXYGEN_RUNNING
   : public AABB_primitive<  Iterator,
-                            internal::Segment_2_from_point_iterator_property_map<GeomTraits, Iterator>,
-                            Input_iterator_property_map<Iterator>,
+                            internal::Segment_2_from_point_iterator_property_map<GeomTraits, Iterator, PointMap>,
+                            internal::Point_from_iterator_property_map<GeomTraits, Iterator, PointMap>,
                             Tag_true,
                             CacheDatum >
 #endif
 {
   typedef AABB_primitive< Iterator,
-                          internal::Segment_2_from_point_iterator_property_map<GeomTraits, Iterator>,
-                          Input_iterator_property_map<Iterator>,
+                          internal::Segment_2_from_point_iterator_property_map<GeomTraits, Iterator, PointMap>,
+                          internal::Point_from_iterator_property_map<GeomTraits, Iterator, PointMap>,
                           Tag_true,
                           CacheDatum > Base;
 public:
   AABB_polyline_segment_primitive_2(Iterator it, PointRange& poly) : Base(it) {}
 
   /// \internal
-  static typename Base::Shared_data construct_shared_data(PointRange& range) {
-    return std::make_pair(internal::Segment_2_from_point_iterator_property_map<GeomTraits, Iterator>(range.begin(), range.end()), Input_iterator_property_map<Iterator>());
+  static typename Base::Shared_data construct_shared_data(PointRange& range, PointMap pmap = PointMap()) {
+    return std::make_pair(internal::Segment_2_from_point_iterator_property_map<GeomTraits, Iterator, PointMap>(range.begin(), range.end(), pmap), internal::Point_from_iterator_property_map<GeomTraits, Iterator, PointMap>(pmap));
   }
 };