diff --git a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/GarlandHeckbert_plane_policies.h b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/GarlandHeckbert_plane_policies.h
index 8fbc2cba3fc..2d77132ec59 100644
--- a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/GarlandHeckbert_plane_policies.h
+++ b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/GarlandHeckbert_plane_policies.h
@@ -32,6 +32,15 @@ class Plane_quadric_calculator
 public:
   Plane_quadric_calculator() { }
 
+  template <typename VertexPointMap>
+  Mat_4 construct_quadric_from_vertex(typename boost::graph_traits<TriangleMesh>::vertex_descriptor v,
+                                      const TriangleMesh& tmesh,
+                                      const VertexPointMap point_map,
+                                      const GeomTraits& gt) const
+  {
+    return Mat_4::Zero();
+  }
+
   template <typename VertexPointMap>
   Mat_4 construct_quadric_from_edge(typename boost::graph_traits<TriangleMesh>::halfedge_descriptor he,
                                     const TriangleMesh& tmesh,
diff --git a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/GarlandHeckbert_probabilistic_plane_policies.h b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/GarlandHeckbert_probabilistic_plane_policies.h
index e533b381ded..f1f2b102c90 100644
--- a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/GarlandHeckbert_probabilistic_plane_policies.h
+++ b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/GarlandHeckbert_probabilistic_plane_policies.h
@@ -74,6 +74,15 @@ public:
   }
 
 public:
+  template <typename VertexPointMap>
+  Mat_4 construct_quadric_from_vertex(typename boost::graph_traits<TriangleMesh>::vertex_descriptor v,
+                                      const TriangleMesh& tmesh,
+                                      const VertexPointMap point_map,
+                                      const GeomTraits& gt) const
+  {
+    return Mat_4::Zero();
+  }
+
   template <typename VertexPointMap>
   Mat_4 construct_quadric_from_edge(const halfedge_descriptor he,
                                     const TriangleMesh& tmesh,
diff --git a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/GarlandHeckbert_probabilistic_triangle_policies.h b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/GarlandHeckbert_probabilistic_triangle_policies.h
index d8ef45f13f0..c08d66c5642 100644
--- a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/GarlandHeckbert_probabilistic_triangle_policies.h
+++ b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/GarlandHeckbert_probabilistic_triangle_policies.h
@@ -48,7 +48,7 @@ private:
 
   // this is only used when we fall back to probabilistic planes for the discontinuous edges,
   // the actual triangle quadric calculation only uses the normal variance
-  static constexpr FT position_variance_factor = 1;
+  static constexpr FT position_variance_factor = 0.1;
 
   Face_variance_map m_face_variance_map;
 
@@ -74,6 +74,15 @@ public:
   }
 
 public:
+  template <typename VertexPointMap>
+  Mat_4 construct_quadric_from_vertex(typename boost::graph_traits<TriangleMesh>::vertex_descriptor v,
+                                      const TriangleMesh& tmesh,
+                                      const VertexPointMap point_map,
+                                      const GeomTraits& gt) const
+  {
+    return Mat_4::Zero();
+  }
+
   // we don't have a sensible way to construct a triangle quadric
   // from an edge, so we fall back to probabilistic plane quadrics here
   template<typename VertexPointMap>
diff --git a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/GarlandHeckbert_triangle_policies.h b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/GarlandHeckbert_triangle_policies.h
index 1e8b2eaee10..1f340e233ea 100644
--- a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/GarlandHeckbert_triangle_policies.h
+++ b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/GarlandHeckbert_triangle_policies.h
@@ -32,6 +32,15 @@ class Triangle_quadric_calculator
 public:
   Triangle_quadric_calculator() { }
 
+  template <typename VertexPointMap>
+  Mat_4 construct_quadric_from_vertex(typename boost::graph_traits<TriangleMesh>::vertex_descriptor v,
+                                      const TriangleMesh& tmesh,
+                                      const VertexPointMap point_map,
+                                      const GeomTraits& gt) const
+  {
+    return Mat_4::Zero();
+  }
+
   template <typename VertexPointMap>
   Mat_4 construct_quadric_from_edge(typename boost::graph_traits<TriangleMesh>::halfedge_descriptor he,
                                     const TriangleMesh& tmesh,
diff --git a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/internal/GarlandHeckbert_functions.h b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/internal/GarlandHeckbert_functions.h
index 6d954dd8b65..135d2c2e531 100644
--- a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/internal/GarlandHeckbert_functions.h
+++ b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/internal/GarlandHeckbert_functions.h
@@ -14,6 +14,8 @@
 
 #include <CGAL/license/Surface_mesh_simplification.h>
 
+#include <CGAL/Polygon_mesh_processing/compute_normal.h>
+
 #include <CGAL/Surface_mesh_simplification/internal/Common.h>
 #include <CGAL/boost/graph/helpers.h>
 
@@ -508,6 +510,77 @@ construct_prob_triangle_quadric_from_face(typename boost::graph_traits<TriangleM
   return ret;
 }
 
+////////////////////////////////////////////////////////////////////////////////////////////////////
+/// LINE QUADRICS (Liu et al. 2025)
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+template <typename GeomTraits>
+typename GarlandHeckbert_matrix_types<GeomTraits>::Mat_4
+construct_line_quadric_from_normal(const typename GeomTraits::Vector_3& normal,
+                                           const typename GeomTraits::Point_3& point,
+                                           const GeomTraits& gt)
+{
+  typedef typename GeomTraits::FT                                              FT;
+  typedef typename GeomTraits::Vector_3                                        Vector_3;
+
+  typedef typename GarlandHeckbert_matrix_types<GeomTraits>::Row_4             Row_4;
+
+  auto cp = gt.construct_cross_product_vector_3_object();
+  auto plane = gt.construct_plane_3_object();
+  auto c_point = gt.construct_point_3_object();
+  auto base= gt.construct_base_vector_3_object();
+
+  Vector_3 x=base(plane(c_point(0,0,0),normal), 1);
+  CGAL::Polygon_mesh_processing::internal::normalize(x, gt);
+  Vector_3 y=cp(x,normal);
+
+  return construct_classic_plane_quadric_from_normal(x, point, gt)+construct_classic_plane_quadric_from_normal(y, point, gt);
+}
+
+template <typename TriangleMesh, typename VertexPointMap, typename GeomTraits>
+typename GarlandHeckbert_matrix_types<GeomTraits>::Mat_4
+construct_line_quadric_from_vertex(const typename boost::graph_traits<TriangleMesh>::vertex_descriptor v,
+                                           const TriangleMesh& mesh,
+                                           const VertexPointMap vpm,
+                                           const GeomTraits& gt)
+{
+  typedef typename GeomTraits::Vector_3                                        Vector_3;
+
+  const Vector_3 normal = Polygon_mesh_processing::compute_vertex_normal(v, mesh, parameters::geom_traits(gt).vertex_point_map(vpm));
+  return construct_line_quadric_from_normal(normal, get(vpm, v), gt);
+}
+
+// template <typename TriangleMesh, typename VertexPointMap, typename GeomTraits>
+// typename GarlandHeckbert_matrix_types<GeomTraits>::Mat_4
+// construct_classic_line_quadric_from_edge(typename boost::graph_traits<TriangleMesh>::halfedge_descriptor he,
+//                                           const TriangleMesh& mesh,
+//                                           const VertexPointMap vpm,
+//                                           const GeomTraits& gt)
+// {
+//   typedef typename GeomTraits::Vector_3                                        Vector_3;
+
+//   const Vector_3 normal = construct_edge_normal(he, mesh, vpm, gt);
+
+//   // use this normal to construct the quadric analogously to constructing quadric
+//   // from the normal of the face
+//   return construct_classic_line_quadric_from_normal(normal, get(vpm, target(he, mesh)), gt);
+// }
+
+// template <typename TriangleMesh, typename VertexPointMap, typename GeomTraits>
+// typename GarlandHeckbert_matrix_types<GeomTraits>::Mat_4
+// construct_classic_line_quadric_from_face(typename boost::graph_traits<TriangleMesh>::face_descriptor f,
+//                                           const TriangleMesh& mesh,
+//                                           const VertexPointMap vpm,
+//                                           const GeomTraits& gt)
+// {
+//   auto normal = construct_unit_normal_from_face(f, mesh, vpm, gt);
+
+//   // get any point of the face
+//   const auto p = get(vpm, target(halfedge(f, mesh), mesh));
+
+//   return construct_classic_line_quadric_from_normal(normal, p, gt);
+// }
+
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 /// PROB VARIANCE
 ////////////////////////////////////////////////////////////////////////////////////////////////////
diff --git a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/internal/GarlandHeckbert_policy_base.h b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/internal/GarlandHeckbert_policy_base.h
index 590747a252b..2f8d6cfbc78 100644
--- a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/internal/GarlandHeckbert_policy_base.h
+++ b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/internal/GarlandHeckbert_policy_base.h
@@ -129,6 +129,15 @@ public:
   }
 
 public:
+  template <typename VertexPointMap>
+  Mat_4 construct_quadric(const vertex_descriptor v,
+                          const TriangleMesh& tmesh,
+                          const VertexPointMap vpm,
+                          const GeomTraits& gt) const
+  {
+    return quadric_calculator().construct_quadric_from_vertex(v, tmesh, vpm, gt);
+  }
+
   template <typename VertexPointMap>
   Mat_4 construct_quadric(const halfedge_descriptor he,
                           const TriangleMesh& tmesh,
@@ -157,7 +166,7 @@ public:
     Mat_4 zero_mat = Mat_4::Zero();
 
     for(vertex_descriptor v : vertices(tmesh))
-      put(vcm(), v, zero_mat);
+      put(vcm(), v, construct_quadric(v, tmesh, vpm, gt));
 
     for(face_descriptor f : faces(tmesh))
     {