diff --git a/Polyline_simplification_2/doc/Polyline_simplification_2/Concepts/PolylineSimplificationCostFunction.h b/Polyline_simplification_2/doc/Polyline_simplification_2/Concepts/PolylineSimplificationCostFunction.h
index 2533372d079..6df2dd06d0f 100644
--- a/Polyline_simplification_2/doc/Polyline_simplification_2/Concepts/PolylineSimplificationCostFunction.h
+++ b/Polyline_simplification_2/doc/Polyline_simplification_2/Concepts/PolylineSimplificationCostFunction.h
@@ -26,11 +26,11 @@ public:
 /*! 
 Given a vertex in constraint iterator `viq` computes `vip=std::prev(viq)` and `vir=std::next(vir)`, and the cost of removing vertex `*viq`, replacing edges `(*vip,*viq)` and `(*viq,*vir)` with edge `(*vip,*vir)`. 
 
-\param ct The underlying constrained Delaunay triangulation with constraint hierarchy which embeds the polyline constraints
+\param ct The underlying constrained Delaunay triangulation which embeds the polyline constraints
 \param viq The vertex in constraint iterator of the vertex to remove
 \returns The cost for removing `*viq`. The value `boost::none` can be returned to indicate an infinite or uncomputable cost.
 
-\tparam CDT must be `CGAL::Constrained_Delaunay_triangulation_2` with a vertex type that
+\tparam CDT must be `CGAL::Constrained_triangulation_plus_2` with a vertex type that
 is model of `PolylineSimplificationVertexBase_2`. `CDT::Geom_traits` must be model of
 the concept `ConstrainedDelaunayTriangulationTraits_2`.
 
diff --git a/Polyline_simplification_2/doc/Polyline_simplification_2/Concepts/PolylineSimplificationStopPredicate.h b/Polyline_simplification_2/doc/Polyline_simplification_2/Concepts/PolylineSimplificationStopPredicate.h
index d2b84ec28ce..62e84c66f48 100644
--- a/Polyline_simplification_2/doc/Polyline_simplification_2/Concepts/PolylineSimplificationStopPredicate.h
+++ b/Polyline_simplification_2/doc/Polyline_simplification_2/Concepts/PolylineSimplificationStopPredicate.h
@@ -16,13 +16,13 @@ public :
   /*!
   Indicates if the simplification must be stopped.
   This is called right before each vertex is about to be removed.
-  \param ct The underlying constrained Delaunay triangulation with constraint hierarchy which embeds the polyline constraints
+  \param ct The underlying constrained Delaunay triangulation which embeds the polyline constraints
   \param q The current vertex about to be removed
   \param cost The associated cost for removing the current vertex (as given by `PolylineSimplificationCostFunction`)
   \param initial_count The initial number of vertices in the entire polyline set (including intersection vertices not in any source polyline)
   \param current_count The current number of vertices
   \return `true` if the algorithm should stop, `false` if it should continue.
-\tparam CDT must be `CGAL::Constrained_Delaunay_triangulation_2` with a vertex type that
+\tparam CDT must be `CGAL::Constrained_triangulation_plus_2` with a vertex type that
 is model of `PolylineSimplificationVertexBase_2`.
   */
   template<class CDT>
diff --git a/Polyline_simplification_2/doc/Polyline_simplification_2/Polyline_simplification_2.txt b/Polyline_simplification_2/doc/Polyline_simplification_2/Polyline_simplification_2.txt
index ca09e72818b..d27c7fd1bc6 100644
--- a/Polyline_simplification_2/doc/Polyline_simplification_2/Polyline_simplification_2.txt
+++ b/Polyline_simplification_2/doc/Polyline_simplification_2/Polyline_simplification_2.txt
@@ -86,7 +86,7 @@ The maximum squared distance is the maximum of the squared Euclidean
 distances between each point on the original polyline between `p` and
 `r` and the line segment `(p,r)`. Let \f$s_0,...,s_n\f$ be the points
 strictly between `p` and `r` on the original polyline.  The cost of
-removing vertex `q` is: \f$ v_1 = \max \{ squared\_distance((p,r), s_i) |
+removing vertex `q` is: \f$ v_1 = \max \{ \mathrm{squared\_distance}((p,r), s_i) |
 i=0,..,n\} \f$
 
 \cgalFigureBegin{figure_maxDist,maxDist.png}
@@ -104,9 +104,11 @@ the candidate vertex `q` and all its adjacent vertices (except `p` and
 regions of the same polyline.
 
 Let \f$t_0,...,t_m\f$ be the points of the vertices adjacent to vertex
-`q`, different from `p` and `r` and let \f$ v_2 = \min \{ squared\_distance((p,r),
+`q`, different from `p` and `r` and let \f$ v_2 = \min \{ \mathrm{squared\_distance}((p,r),
 t_i) | i=0,..,n\}\f$. The cost of removing vertex `q` is
-\f$v_1/v_2\f$.  See also Figure \cgalFigureRef{figure_scaledAndHybrid}.
+\f$v_1/v_2\f$, and it is +infinity when the set of points \f${t_1..t_m}\f$ 
+is empty (possible if the point `q` is on the convex hull).
+See also Figure \cgalFigureRef{figure_scaledAndHybrid}.
 
 This distance measure gives lower priority to vertices with close
 neighboring polylines.
diff --git a/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h b/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h
index 9e33a2c5a80..3462b8404e5 100644
--- a/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h
+++ b/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h
@@ -307,7 +307,7 @@ operator()()
 
 Simplifies a single polygon.
 
-\tparam Traits must be a model of `PolylineSimplificationTraits_2`
+\tparam Traits must be a model of `ConstrainedDelaunayTriangulationTraits_2`
 \tparam CostFunction must be a model of `PolylineSimplificationCostFunction`.
 \tparam StopFunction must be a model of `PolylineSimplificationStopPredicate`
 
@@ -409,9 +409,9 @@ Simplifies a single polyline in a triangulation with polylines as constraints.
 \param cid The constraint identifier of the polyline constraint to simplify
 \param cost The cost function
 \param stop The stop function
-\param remove_points  If `true` the function `ct.remove_points_without_corresponding_vertex()` is called.
+\param remove_points  If `true` the function \link CGAL::Constrained_triangulation_plus_2::remove_points_without_corresponding_vertex() `ct.remove_points_without_corresponding_vertex()` \endlink is called.
 \returns the number of removed vertices
-\tparam CDT  must be `CGAL::Constrained_Delaunay_triangulation_2` with a vertex type that
+\tparam CDT  must be `CGAL::Constrained_triangulation_plus_2` with a vertex type that
 is model of  `PolylineSimplificationVertexBase_2`.
 \tparam CostFunction must be a model of `PolylineSimplificationCostFunction`
 \tparam StopFunction must be a model of `PolylineSimplificationStopPredicate`
@@ -440,9 +440,9 @@ Simplifies all polylines in a triangulation with polylines as constraints.
 \param ct The underlying constrained Delaunay triangulation which embeds the polyline constraints
 \param cost The cost function
 \param stop The stop function
-\param remove_points If `true` the function `ct.remove_points_without_corresponding_vertex()` is called.
+\param remove_points If `true` the function \link CGAL::Constrained_triangulation_plus_2::remove_points_without_corresponding_vertex() `ct.remove_points_without_corresponding_vertex()`\endlink is called.
 \returns the number of removed vertices
-\tparam CDT  must be `CGAL::Constrained_Delaunay_triangulation_2` with a vertex type that
+\tparam CDT  must be `CGAL::Constrained_triangulation_plus_2` with a vertex type that
 is model of  `PolylineSimplificationVertexBase_2`.
 \tparam CostFunction must be a model of `PolylineSimplificationCostFunction`
 \tparam StopFunction must be a model of `PolylineSimplificationStopPredicate`