diff --git a/Alpha_wrap_2/include/CGAL/Alpha_wrap_2/internal/Point_set_oracle.h b/Alpha_wrap_2/include/CGAL/Alpha_wrap_2/internal/Point_set_oracle.h
index 88f1a700878..b80d45676f7 100644
--- a/Alpha_wrap_2/include/CGAL/Alpha_wrap_2/internal/Point_set_oracle.h
+++ b/Alpha_wrap_2/include/CGAL/Alpha_wrap_2/internal/Point_set_oracle.h
@@ -1,5 +1,4 @@
 // Copyright (c) 2019-2022 Google LLC (USA).
-// Copyright (c) 2025 GeometryFactory (France).
 // All rights reserved.
 //
 // This file is part of CGAL (www.cgal.org).
@@ -110,6 +109,10 @@ public:
   void add_points(const PointRange& points,
                   const CGAL_NP_CLASS& /*np*/ = CGAL::parameters::default_values())
   {
+#ifdef CGAL_AW2_DEBUG
+    std::cout << "Insert into AABB tree (points)..." << std::endl;
+#endif
+
     if(points.empty())
     {
 #ifdef CGAL_AW2_DEBUG
@@ -121,10 +124,6 @@ public:
     const std::size_t old_size = m_points_ptr->size();
     m_points_ptr->insert(std::cend(*m_points_ptr), std::cbegin(points), std::cend(points));
 
-#ifdef CGAL_AW2_DEBUG
-    std::cout << "Insert into AABB tree (points)..." << std::endl;
-#endif
-
     this->tree().rebuild(std::cbegin(*m_points_ptr), std::cend(*m_points_ptr));
 
     // Manually constructing it here purely for profiling reasons: if we keep the lazy approach,
diff --git a/Alpha_wrap_3/examples/Alpha_wrap_3/mixed_inputs_wrap.cpp b/Alpha_wrap_3/examples/Alpha_wrap_3/mixed_inputs_wrap.cpp
index 90b488e0407..4198fe9aa2c 100644
--- a/Alpha_wrap_3/examples/Alpha_wrap_3/mixed_inputs_wrap.cpp
+++ b/Alpha_wrap_3/examples/Alpha_wrap_3/mixed_inputs_wrap.cpp
@@ -100,8 +100,8 @@ int main(int argc, char** argv)
   Oracle oracle(ss_oracle);
 
   oracle.add_triangle_soup(points, faces, CGAL::parameters::default_values());
-  oracle.add_segment_soup(segments, CGAL::parameters::default_values());
-  oracle.add_point_set(ps_points, CGAL::parameters::default_values());
+  oracle.add_segments(segments, CGAL::parameters::default_values());
+  oracle.add_points(ps_points, CGAL::parameters::default_values());
 
   CGAL::Alpha_wraps_3::internal::Alpha_wrapper_3<Oracle> aw3(oracle);
 
diff --git a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Alpha_wrap_3.h b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Alpha_wrap_3.h
index 057e015158b..6eae3e75847 100644
--- a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Alpha_wrap_3.h
+++ b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Alpha_wrap_3.h
@@ -207,13 +207,27 @@ public:
     static_assert(std::is_floating_point<FT>::value);
   }
 
+  void clear()
+  {
+    m_oracle.clear();
+    m_bbox = {};
+    m_alpha = m_sq_alpha = FT(-1);
+    m_offset = m_sq_offset = FT(-1);
+    m_seeds.clear();
+    m_tr.clear();
+    m_queue.clear();
+  }
+
 public:
   const Geom_traits& geom_traits() const { return m_tr.geom_traits(); }
   Oracle& oracle() { return m_oracle; }
   const Oracle& oracle() const { return m_oracle; }
   Triangulation& triangulation() { return m_tr; }
   const Triangulation& triangulation() const { return m_tr; }
+  Alpha_PQ& queue() { return m_queue; }
   const Alpha_PQ& queue() const { return m_queue; }
+  const FT& alpha() const { return m_alpha; }
+  const FT& offset() const { return m_offset; }
 
   double default_alpha() const
   {
@@ -365,6 +379,11 @@ public:
 #endif
     }
 
+ #ifdef CGAL_AW3_DEBUG_DUMP_INTERMEDIATE_WRAPS
+    IO::write_polygon_mesh("final_wrap.off", output_mesh, CGAL::parameters::stream_precision(17));
+    dump_triangulation_faces("final_tr.off", false /*only_boundary_faces*/);
+ #endif
+
     extract_surface(output_mesh, ovpm, !do_enforce_manifoldness);
 
 #ifdef CGAL_AW3_TIMER
@@ -375,11 +394,6 @@ public:
 #ifdef CGAL_AW3_DEBUG
     std::cout << "Alpha wrap vertices:  " << vertices(output_mesh).size() << std::endl;
     std::cout << "Alpha wrap faces:     " << faces(output_mesh).size() << std::endl;
-
- #ifdef CGAL_AW3_DEBUG_DUMP_INTERMEDIATE_WRAPS
-    IO::write_polygon_mesh("final_wrap.off", output_mesh, CGAL::parameters::stream_precision(17));
-    dump_triangulation_faces("final_tr.off", false /*only_boundary_faces*/);
- #endif
 #endif
 
     visitor.on_alpha_wrapping_end(*this);
@@ -501,7 +515,7 @@ private:
   // which yields
   //   r = a * sin(2pi / 5) =~ 0.95105651629515353 * a
   // Faces of an icosahedron are equilateral triangles with size a and circumradius a / sqrt(3)
-  // Since we want faces of the icosahedron to be traversable, we want a such that
+  // Since we want faces of the icosahedron to be traversable, we want `a` such that
   //   a / sqrt(3) > alpha
   // Hence r such that
   //   r / (sqrt(3) * sin(2pi/5)) > alpha
@@ -509,7 +523,7 @@ private:
   //
   // Furthermore, the triangles between edges of the icosahedron and the center of the icosahedron
   // are not equilateral triangles since a is slightly bigger than r. They are
-  // slightly flattened isocele triangles with base 'a' and the circumradius is smaller.
+  // slightly flattened isocele triangles with base `a` and the circumradius is smaller.
   // The circumradius is
   //   r_iso = r² / (2 * h) = r² / (2 * sqrt(r² - (a / 2)²))
   // Since r = a * sin(2pi / 5)
@@ -896,6 +910,16 @@ private:
     return less_squared_radius_of_min_empty_sphere(m_sq_alpha, f, m_tr);
   }
 
+  Steiner_status compute_steiner_point(const Gate& gate,
+                                       Point_2& steiner_point) const
+  {
+    const Face_handle fh = gate.face();
+    const Cell_handle ch = f.first;
+    const int s = f.second;
+    const Cell_handle nh = ch->neighbor(s);
+    return compute_steiner_point(ch, nh, steiner_point);
+  }
+
   bool compute_steiner_point(const Cell_handle ch,
                              const Cell_handle neighbor,
                              Point_3& steiner_point) const
@@ -950,7 +974,7 @@ private:
 #ifdef CGAL_AW3_DEBUG_STEINER_COMPUTATION
         std::cout << "Steiner found through first_intersection(): " << steiner_point << std::endl;
 #endif
-        return true;
+        return Steiner_status::RULE_1;
       }
     }
 
@@ -959,6 +983,10 @@ private:
     {
       // steiner point is the closest point on input from cell centroid with offset
       const Point_3 closest_pt = m_oracle.closest_point(neighbor_cc);
+#ifdef CGAL_AW3_DEBUG_STEINER_COMPUTATION
+      std::cout << "Steiner found through neighboring triangle intersecting the input" << std::endl;
+      std::cout << "Closest point: " << closest_pt << std::endl;
+#endif
       CGAL_assertion(closest_pt != neighbor_cc);
 
       Vector_3 unit = vector(closest_pt, neighbor_cc);
@@ -968,19 +996,18 @@ private:
       steiner_point = translate(closest_pt, unit);
 
 #ifdef CGAL_AW3_DEBUG_STEINER_COMPUTATION
-      std::cout << "Steiner found through neighboring tet intersecting the input: " << steiner_point << std::endl;
-      std::cout << "Closest point: " << closest_pt << std::endl;
-      std::cout << "Direction: " << vector(closest_pt, neighbor_cc) << std::endl;
+      std::cout << "Direction: " << unit << std::endl;
+      std::cout << "Steiner: " << steiner_point << std::endl;
 #endif
 
-      return true;
+      return Steiner_status::RULE_2;
     }
 
 #ifdef CGAL_AW3_DEBUG_STEINER_COMPUTATION
     std::cout << "No Steiner point" << std::endl;
 #endif
 
-    return false;
+    return Steiner_status::NO_STEINER_POINT;
   }
 
 private:
@@ -1043,7 +1070,12 @@ public:
     CGAL_precondition(ch->label() == Cell_label::INSIDE || ch->label() == Cell_label::OUTSIDE);
 
     if(m_tr.is_infinite(nh))
+    {
+#ifdef CGAL_AW3_DEBUG_FACET_STATUS
+      std::cout << "Infinite neighboring cell" << std::endl;
+#endif
       return Facet_status::HAS_INFINITE_NEIGHBOR;
+    }
 
     if(nh->is_outside())
     {
@@ -1104,9 +1136,23 @@ private:
     const FT sqr = smallest_squared_radius_3(f, m_tr);
     const bool is_permissive = (status == Facet_status::HAS_INFINITE_NEIGHBOR ||
                                 status == Facet_status::SCAFFOLDING);
-    m_queue.resize_and_push(Gate(f, sqr, is_permissive));
+    Gate new_gate(f, sqr, is_permissive);
 #else
-    m_queue.push(Gate(f, m_tr));
+    Gate new_gate(f, m_tr);
+#endif
+
+#ifdef CGAL_AW2_COMPUTE_AND_STORE_STEINER_INFO_AT_GATE_CREATION
+    Point_3 steiner_point;
+    Steiner_status steiner_status = compute_steiner_point(new_gate, steiner_point);
+    new_gate.m_steiner_status = steiner_status;
+    if(steiner_status == Steiner_status::RULE_1 || steiner_status == Steiner_status::RULE_2)
+      new_gate.m_steiner_point = steiner_point;
+#endif
+
+#ifdef CGAL_AW3_USE_SORTED_PRIORITY_QUEUE
+  m_queue.resize_and_push(new_gate);
+#else
+  m_queue.push(new_gate);
 #endif
 
 #ifdef CGAL_AW3_DEBUG_QUEUE
@@ -1241,26 +1287,25 @@ private:
 #endif
 
       const Facet& f = gate.facet();
-      CGAL_precondition(!m_tr.is_infinite(f));
-
       const Cell_handle ch = f.first;
       const int s = f.second;
-      CGAL_precondition(ch->is_outside());
-
       const Cell_handle nh = ch->neighbor(s);
-      CGAL_precondition(nh->label() == Cell_label::INSIDE || nh->label() == Cell_label::OUTSIDE);
 
 #ifdef CGAL_AW3_DEBUG_QUEUE
       static int fid = 0;
       std::cout << m_tr.number_of_vertices() << " DT vertices" << std::endl;
       std::cout << m_queue.size() << " facets in the queue" << std::endl;
-      std::cout << "Face " << fid++ << "\n"
-                << "c = " << &*ch << " (" << m_tr.is_infinite(ch) << "), n = " << &*nh << " (" << m_tr.is_infinite(nh) << ")" << "\n"
-                << m_tr.point(ch, Triangulation::vertex_triple_index(s, 0)) << "\n"
-                << m_tr.point(ch, Triangulation::vertex_triple_index(s, 1)) << "\n"
-                << m_tr.point(ch, Triangulation::vertex_triple_index(s, 2)) << std::endl;
+      std::cout << "Face " << fid++ << " ["
+                << m_tr.point(ch, Triangulation::vertex_triple_index(s, 0)) << " "
+                << m_tr.point(ch, Triangulation::vertex_triple_index(s, 1)) << " "
+                << m_tr.point(ch, Triangulation::vertex_triple_index(s, 2)) << "]" << std::endl;
+      std::cout << "c = " << &*ch << " (inf: " << m_tr.is_infinite(ch) << ", label: " << ch->label() << "), n = "
+                          << &*nh << " (inf: " << m_tr.is_infinite(nh) << ", label: " << nh->label() << ")" << "\n"
+
+# ifdef CGAL_AW3_USE_SORTED_PRIORITY_QUEUE
       std::cout << "Priority: " << gate.priority() << " (sq alpha: " << m_sq_alpha << ")" << std::endl;
       std::cout << "Permissiveness: " << gate.is_permissive_facet() << std::endl;
+# endif
 #endif
 
 #ifdef CGAL_AW3_DEBUG_DUMP_EVERY_STEP
@@ -1277,6 +1322,10 @@ private:
       face_out.close();
 #endif
 
+      CGAL_precondition(!m_tr.is_infinite(f));
+      CGAL_precondition(ch->is_outside());
+      CGAL_precondition(nh->label() == Cell_label::INSIDE || nh->label() == Cell_label::OUTSIDE);
+
       visitor.before_facet_treatment(*this, gate);
 
       m_queue.pop();
@@ -1291,7 +1340,8 @@ private:
       }
 
       Point_3 steiner_point;
-      if(compute_steiner_point(ch, nh, steiner_point))
+      Steiner_status ss = compute_steiner_point(ch, nh, steiner_point);
+      if(ss != Steiner_status::NO_STEINER_POINT)
       {
 //        std::cout << CGAL::abs(CGAL::approximate_sqrt(m_oracle.squared_distance(steiner_point)) - m_offset)
 //                  << " vs " << 1e-2 * m_offset << std::endl;
diff --git a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Alpha_wrap_triangulation_cell_base_3.h b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Alpha_wrap_triangulation_cell_base_3.h
index bcca1bd0143..f655bd6ba5c 100644
--- a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Alpha_wrap_triangulation_cell_base_3.h
+++ b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Alpha_wrap_triangulation_cell_base_3.h
@@ -23,13 +23,33 @@ namespace internal {
 enum class Cell_label
 {
   // Cells that have been carved
-  OUTSIDE,
+  OUTSIDE = 0,
   // Cells that have not yet been carved
   INSIDE,
   // OUTSIDE cells that have been labeled "inside" again as to make the result manifold
   MANIFOLD
 };
 
+inline std::string label_string(const Cell_label& label) {
+  switch(label) {
+    case Cell_label::OUTSIDE:
+      return "OUTSIDE";
+    case Cell_label::INSIDE:
+      return "INSIDE";
+    case Cell_label::MANIFOLD:
+      return "MANIFOLD";
+    default:
+      CGAL_assertion_msg(false, "Unknown label");
+      return {};
+  }
+}
+
+inline std::ostream& operator <<(std::ostream& os, const Cell_label& label)
+{
+   os << static_cast<std::underlying_type<Cell_label>::type>(label);
+   return os;
+}
+
 template < typename GT,
            typename Cb = CGAL::Delaunay_triangulation_cell_base_with_circumcenter_3<GT> >
 class Alpha_wrap_triangulation_cell_base_3
diff --git a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Oracle_base.h b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Oracle_base.h
index 5e105f83047..abee36b9e88 100644
--- a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Oracle_base.h
+++ b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Oracle_base.h
@@ -145,7 +145,11 @@ public:
   bool empty() const { return m_tree_ptr->empty(); }
   bool do_call() const { return (!empty() || base().do_call()); }
 
-  void clear() { m_tree_ptr->clear() && base().clear(); }
+  void clear()
+  {
+    m_tree_ptr->clear();
+    base().clear();
+  }
 
 public:
   typename AABB_tree::Bounding_box bbox() const
diff --git a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Point_set_oracle.h b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Point_set_oracle.h
index d35a20c93fa..199e1f09b90 100644
--- a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Point_set_oracle.h
+++ b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Point_set_oracle.h
@@ -97,12 +97,22 @@ public:
   { }
 
 public:
+  void clear()
+  {
+    m_points_ptr->clear();
+    Oracle_base::clear();
+  }
+
   // adds a range of points to the oracle
   template <typename PointRange,
             typename CGAL_NP_TEMPLATE_PARAMETERS>
-  void add_point_set(const PointRange& points,
-                     const CGAL_NP_CLASS& /*np*/ = CGAL::parameters::default_values())
+  void add_points(const PointRange& points,
+                  const CGAL_NP_CLASS& /*np*/ = CGAL::parameters::default_values())
   {
+#ifdef CGAL_AW3_DEBUG
+    std::cout << "Insert into AABB tree (points)..." << std::endl;
+#endif
+
     if(points.empty())
     {
 #ifdef CGAL_AW3_DEBUG
@@ -114,20 +124,18 @@ public:
     const std::size_t old_size = m_points_ptr->size();
     m_points_ptr->insert(std::cend(*m_points_ptr), std::cbegin(points), std::cend(points));
 
-#ifdef CGAL_AW3_DEBUG
-    std::cout << "Insert into AABB tree (points)..." << std::endl;
-#endif
-
-    this->tree().insert(std::next(std::cbegin(*m_points_ptr), old_size), std::cend(*m_points_ptr));
+    this->tree().rebuild(std::cbegin(*m_points_ptr), std::cend(*m_points_ptr));
 
     // Manually constructing it here purely for profiling reasons: if we keep the lazy approach,
     // it will be done at the first treatment of a facet that needs a Steiner point.
     // So if one wanted to bench the flood fill runtime, it would be skewed by the time it takes
     // to accelerate the tree.
     this->tree().accelerate_distance_queries();
-
-    CGAL_postcondition(this->tree().size() == m_points_ptr->size());
   }
+
+#ifdef CGAL_AW2_DEBUG
+    std::cout << "PS Tree: " << this->tree().size() << " primitives" << std::endl;
+#endif
 };
 
 } // namespace internal
diff --git a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Segment_soup_oracle.h b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Segment_soup_oracle.h
index 88d565c6b66..ca3a83bd95a 100644
--- a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Segment_soup_oracle.h
+++ b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Segment_soup_oracle.h
@@ -99,11 +99,21 @@ public:
   { }
 
 public:
+  void clear()
+  {
+    m_segments_ptr->clear();
+    Oracle_base::clear();
+  }
+
   template <typename SegmentRange,
             typename CGAL_NP_TEMPLATE_PARAMETERS>
-  void add_segment_soup(const SegmentRange& segments,
-                        const CGAL_NP_CLASS& /*np*/ = CGAL::parameters::default_values())
+  void add_segments(const SegmentRange& segments,
+                    const CGAL_NP_CLASS& /*np*/ = CGAL::parameters::default_values())
   {
+#ifdef CGAL_AW3_DEBUG
+    std::cout << "Insert into AABB Tree (" << segments.size() << " segments)..." << std::endl;
+#endif
+
     if(segments.empty())
     {
 #ifdef CGAL_AW3_DEBUG
@@ -129,10 +139,7 @@ public:
       m_segments_ptr->push_back(s);
     }
 
-#ifdef CGAL_AW3_DEBUG
-    std::cout << "Insert into AABB tree (segments)..." << std::endl;
-#endif
-    this->tree().insert(std::next(std::cbegin(*m_segments_ptr), old_size), std::cend(*m_segments_ptr));
+    this->tree().rebuild(std::cbegin(*m_segments_ptr), std::cend(*m_segments_ptr));
 
     // Manually constructing it here purely for profiling reasons: if we keep the lazy approach,
     // it will be done at the first treatment of a facet that needs a Steiner point.
@@ -140,7 +147,9 @@ public:
     // to accelerate the tree.
     this->tree().accelerate_distance_queries();
 
-    CGAL_postcondition(this->tree().size() == m_segments_ptr->size());
+#ifdef CGAL_AW3_DEBUG
+    std::cout << "SS Tree: " << this->tree().size() << " primitives" << std::endl;
+#endif
   }
 };
 
diff --git a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Triangle_mesh_oracle.h b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Triangle_mesh_oracle.h
index d7f325e1428..a4c30e6cc88 100644
--- a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Triangle_mesh_oracle.h
+++ b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Triangle_mesh_oracle.h
@@ -130,6 +130,10 @@ public:
     using VPM = typename GetVertexPointMap<TriangleMesh>::const_type;
     using Point_ref = typename boost::property_traits<VPM>::reference;
 
+#ifdef CGAL_AW3_DEBUG
+    std::cout << "Insert into AABB tree (" << faces(tmesh).size() << " faces)..." << std::endl;
+#endif
+
     CGAL_precondition(CGAL::is_triangle_mesh(tmesh));
 
     if(is_empty(tmesh))
@@ -140,10 +144,6 @@ public:
       return;
     }
 
-#ifdef CGAL_AW3_DEBUG
-    std::cout << "Insert into AABB tree (faces)..." << std::endl;
-#endif
-
     VPM vpm = choose_parameter(get_parameter(np, internal_np::vertex_point),
                                get_const_property_map(vertex_point, tmesh));
     static_assert(std::is_same<typename boost::property_traits<VPM>::value_type, Point_3>::value);
@@ -176,7 +176,7 @@ public:
     this->tree().accelerate_distance_queries();
 
 #ifdef CGAL_AW3_DEBUG
-    std::cout << "Tree: " << this->tree().size() << " primitives (" << num_faces(tmesh) << " faces in input)" << std::endl;
+    std::cout << "Tree: " << this->tree().size() << " primitives" << std::endl;
 #endif
   }
 };
diff --git a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Triangle_soup_oracle.h b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Triangle_soup_oracle.h
index d066107f620..947dd6ff8df 100644
--- a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Triangle_soup_oracle.h
+++ b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/Triangle_soup_oracle.h
@@ -130,6 +130,10 @@ public:
 
     using Face = typename boost::range_value<FaceRange>::type;
 
+#ifdef CGAL_AW3_DEBUG
+    std::cout << "Insert into AABB Tree (" << faces.size() << " faces)..." << std::endl;
+#endif
+
     if(points.empty() || faces.empty())
     {
 #ifdef CGAL_AW3_DEBUG
@@ -138,10 +142,6 @@ public:
       return;
     }
 
-#ifdef CGAL_AW3_DEBUG
-    std::cout << "Insert into AABB Tree (triangles)..." << std::endl;
-#endif
-
     PPM pm = choose_parameter<PPM>(get_parameter(np, internal_np::point_map));
     static_assert(std::is_same<typename boost::property_traits<PPM>::value_type, Point_3>::value);
 
@@ -181,15 +181,19 @@ public:
     this->tree().accelerate_distance_queries();
 
 #ifdef CGAL_AW3_DEBUG
-    std::cout << "Tree: " << this->tree().size() << " primitives (" << faces.size() << " faces in input)" << std::endl;
+    std::cout << "TS Tree: " << this->tree().size() << " primitives" << std::endl;
 #endif
   }
 
   template <typename TriangleRange,
             typename CGAL_NP_TEMPLATE_PARAMETERS>
-  void add_triangle_soup(const TriangleRange& triangles,
-                         const CGAL_NP_CLASS& /*np*/ = CGAL::parameters::default_values())
+  void add_triangles(const TriangleRange& triangles,
+                     const CGAL_NP_CLASS& /*np*/ = CGAL::parameters::default_values())
   {
+#ifdef CGAL_AW3_DEBUG
+    std::cout << "Insert into AABB Tree (" << triangles.size() << " triangles)..." << std::endl;
+#endif
+
     if(triangles.empty())
     {
 #ifdef CGAL_AW3_DEBUG
@@ -198,10 +202,6 @@ public:
       return;
     }
 
-#ifdef CGAL_AW3_DEBUG
-    std::cout << "Insert into AABB Tree (triangles)..." << std::endl;
-#endif
-
     typename Geom_traits::Is_degenerate_3 is_degenerate = this->geom_traits().is_degenerate_3_object();
 
     Splitter_base::reserve(triangles.size());
@@ -218,6 +218,9 @@ public:
 
       Splitter_base::split_and_insert_datum(tr, this->tree(), this->geom_traits());
     }
+#ifdef CGAL_AW3_DEBUG
+    std::cout << "TS Tree: " << this->tree().size() << " primitives" << std::endl;
+#endif
   }
 };
 
diff --git a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/gate_priority_queue.h b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/gate_priority_queue.h
index ec48b4f458b..1df539b4093 100644
--- a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/gate_priority_queue.h
+++ b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/gate_priority_queue.h
@@ -22,16 +22,46 @@
 
 #include <cassert>
 #include <iostream>
+#include <optional>
 
 namespace CGAL {
 namespace Alpha_wraps_3 {
 namespace internal {
 
+enum class Steiner_status
+{
+  UNKONWN = 0,
+  NO_STEINER_POINT,
+  RULE_1,
+  RULE_2
+};
+
+#ifdef CGAL_AW2_COMPUTE_AND_STORE_STEINER_INFO_AT_GATE_CREATION
+template <typename Tr>
+struct Gate_steiner_info
+{
+  using Point_3 = typename Tr::Geom_traits::Point_3;
+
+  Steiner_status m_steiner_status = Steiner_status::UNKNOWN;
+  std::optional<Point_3> m_steiner_point = std::nullopt;
+
+  bool has_steiner_point() const { return m_steiner_point.has_value(); }
+  bool has_steiner_from_projection() const { return (m_steiner_status == Steiner_status::RULE_2); }
+  const Point_3& steiner_point() const {
+    CGAL_precondition(has_steiner_point());
+    return *m_steiner_point;
+  }
+};
+#endif
+
 #ifdef CGAL_AW3_USE_SORTED_PRIORITY_QUEUE
 
 // Represents an alpha-traversable facet in the mutable priority queue
 template <typename Tr>
 class Gate
+#ifdef CGAL_AW2_COMPUTE_AND_STORE_STEINER_INFO_AT_GATE_CREATION
+  : public Gate_steiner_info<Tr>
+#endif
 {
   using Facet = typename Tr::Facet;
   using FT = typename Tr::Geom_traits::FT;
@@ -97,6 +127,9 @@ struct Less_gate
 // Represents an alpha-traversable facet in the mutable priority queue
 template <typename Tr>
 class Gate
+#ifdef CGAL_AW2_COMPUTE_AND_STORE_STEINER_INFO_AT_GATE_CREATION
+  : public Gate_steiner_info<Tr>
+#endif
 {
   using Facet = typename Tr::Facet;
   using FT = typename Tr::Geom_traits::FT;
diff --git a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/validation.h b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/validation.h
index 6d9a9191d5f..d5702304de6 100644
--- a/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/validation.h
+++ b/Alpha_wrap_3/include/CGAL/Alpha_wrap_3/internal/validation.h
@@ -65,20 +65,30 @@ bool has_degenerated_faces(const TriangleMesh& mesh,
 // Edge length is bounded by twice the circumradius
 template <typename TriangleMesh,
           typename NamedParameters = parameters::Default_named_parameters>
-bool check_edge_length(const TriangleMesh& output_mesh,
-                       const double alpha,
-                       const NamedParameters& np = CGAL::parameters::default_values())
+bool has_bounded_edge_length(const TriangleMesh& wrap,
+                             const double alpha,
+                             const NamedParameters& np = CGAL::parameters::default_values())
 {
-  const auto sq_alpha_bound = 4 * square(alpha);
-  for(auto e : edges(output_mesh))
+  using Geom_traits = typename GetGeomTraits<TriangleMesh, NamedParameters>::type;
+  using FT = typename Geom_traits::FT;
+  using VPM = typename CGAL::GetVertexPointMap<PolygonMesh, NamedParameters>::type;
+
+  Geom_traits gt = choose_parameter<Geom_traits>(get_parameter(in_np, internal_np::geom_traits));
+  VPM vpm = choose_parameter(get_parameter(np, internal_np::vertex_point),
+                             get_const_property_map(vertex_point, tmesh));
+
+  const FT sq_alpha_bound = 4 * square(alpha);
+  for(auto e : edges(wrap))
   {
-    const auto sqd = Polygon_mesh_processing::squared_edge_length(e, output_mesh, np);
-    if(sqd > sq_alpha_bound) // alpha is the circumradius
+    if(gt.compare_squared_distance(get(vpm, source(e, wrap)),
+                                   get(vpm, target(e, wrap)),
+                                   sq_alpha_bound) == LARGER)
     {
 #ifdef CGAL_AW3_DEBUG
+      const FT sqd = Polygon_mesh_processing::squared_edge_length(e, wrap, np);
       std::cerr << "Error: " << sqd << " greater than " << sq_alpha_bound << std::endl;
-      std::cerr << get(CGAL::vertex_point, output_mesh, source(e, output_mesh)) << std::endl;
-      std::cerr << get(CGAL::vertex_point, output_mesh, target(e, output_mesh)) << std::endl;
+      std::cerr << "source pt: " << get(vpm, source(e, wrap)) << std::endl;
+      std::cerr << "target pt: " << get(vpm, target(e, wrap)) << std::endl;
 #endif
       return false;
     }
@@ -346,13 +356,12 @@ bool is_outer_wrap_of_point_set(const TriangleMesh& wrap,
 
   CGAL::Side_of_triangle_mesh<TriangleMesh, K, OVPM> side_of_wrap(wrap, out_vpm);
 
-  // @speed a single vertex per CC would be sufficient
   for(const auto& p : points)
   {
     if(side_of_wrap(get(in_pm, p)) != CGAL::ON_BOUNDED_SIDE)
     {
 #ifdef CGAL_AW3_DEBUG
-      std::cerr << "Part(s) of the input mesh are outside the wrap: " << get(in_pm, p) << std::endl;
+      std::cerr << "An input point is outside the wrap: " << get(in_pm, p) << std::endl;
 #endif
       return false;
     }
diff --git a/Alpha_wrap_3/include/CGAL/alpha_wrap_3.h b/Alpha_wrap_3/include/CGAL/alpha_wrap_3.h
index 8be78675422..50b540f4eed 100644
--- a/Alpha_wrap_3/include/CGAL/alpha_wrap_3.h
+++ b/Alpha_wrap_3/include/CGAL/alpha_wrap_3.h
@@ -355,7 +355,7 @@ void alpha_wrap_3(const PointRange& points,
   Geom_traits gt = choose_parameter<Geom_traits>(get_parameter(in_np, internal_np::geom_traits));
 
   Oracle oracle(gt);
-  oracle.add_point_set(points, in_np);
+  oracle.add_points(points, in_np);
   AW3 alpha_wrap_builder(oracle);
   alpha_wrap_builder(alpha, offset, alpha_wrap, in_np, out_np);
 }
diff --git a/Alpha_wrap_3/test/Alpha_wrap_3/test_AW3_cavity_initializations.cpp b/Alpha_wrap_3/test/Alpha_wrap_3/test_AW3_cavity_initializations.cpp
index af94eecd25a..eef71c321a1 100644
--- a/Alpha_wrap_3/test/Alpha_wrap_3/test_AW3_cavity_initializations.cpp
+++ b/Alpha_wrap_3/test/Alpha_wrap_3/test_AW3_cavity_initializations.cpp
@@ -105,7 +105,7 @@ void alpha_wrap_triangle_mesh(Mesh& input_mesh,
   }
 
   if(!enforce_manifoldness)
-    assert(AW3::internal::check_edge_length(wrap, alpha));
+    assert(AW3::internal::has_bounded_edge_length(wrap, alpha));
 }
 
 void alpha_wrap_triangle_mesh(Mesh& input_mesh,
diff --git a/Alpha_wrap_3/test/Alpha_wrap_3/test_AW3_manifoldness.cpp b/Alpha_wrap_3/test/Alpha_wrap_3/test_AW3_manifoldness.cpp
index 954fe6163df..32893100730 100644
--- a/Alpha_wrap_3/test/Alpha_wrap_3/test_AW3_manifoldness.cpp
+++ b/Alpha_wrap_3/test/Alpha_wrap_3/test_AW3_manifoldness.cpp
@@ -57,7 +57,7 @@ void alpha_wrap_triangle_manifoldness(Mesh& input_mesh,
     assert(AW3::internal::has_expected_Hausdorff_distance(nm_wrap, input_mesh, alpha, offset));
   }
 
-  assert(AW3::internal::check_edge_length(nm_wrap, alpha));
+  assert(AW3::internal::has_bounded_edge_length(nm_wrap, alpha));
 
   FT base_vol = 0;
   if(!is_closed(nm_wrap))
diff --git a/Alpha_wrap_3/test/Alpha_wrap_3/test_AW3_multiple_calls.cpp b/Alpha_wrap_3/test/Alpha_wrap_3/test_AW3_multiple_calls.cpp
index 35a954f17b2..c9352c2caa9 100644
--- a/Alpha_wrap_3/test/Alpha_wrap_3/test_AW3_multiple_calls.cpp
+++ b/Alpha_wrap_3/test/Alpha_wrap_3/test_AW3_multiple_calls.cpp
@@ -52,7 +52,7 @@ void alpha_wrap_triangle_soup(Points& pr,
 
   // AW3
   Oracle oracle;
-  oracle.add_triangle_soup(pr, fr);
+  oracle.add_triangles(pr, fr);
   AW3::internal::Alpha_wrapper_3<Oracle> aw3(oracle);
 
   Mesh wrap;
@@ -65,7 +65,7 @@ void alpha_wrap_triangle_soup(Points& pr,
   assert(AW3::internal::is_valid_wrap(wrap, false /*manifoldness*/));
   assert(AW3::internal::is_outer_wrap_of_triangle_soup(wrap, pr, fr));
   assert(AW3::internal::has_expected_Hausdorff_distance(wrap, input_mesh, alpha, offset));
-  assert(AW3::internal::check_edge_length(wrap, alpha));
+  assert(AW3::internal::has_bounded_edge_length(wrap, alpha));
 
   alpha *= 2;
   offset *= 2;
@@ -83,7 +83,7 @@ void alpha_wrap_triangle_soup(Points& pr,
   assert(AW3::internal::is_valid_wrap(wrap_2, false /*manifoldness*/));
   assert(AW3::internal::is_outer_wrap_of_triangle_soup(wrap_2, pr, fr));
   assert(AW3::internal::has_expected_Hausdorff_distance(wrap_2, input_mesh, alpha, offset));
-  assert(AW3::internal::check_edge_length(wrap_2, alpha));
+  assert(AW3::internal::has_bounded_edge_length(wrap_2, alpha));
 }
 
 void alpha_wrap_triangle_soup(const std::string& filename)
diff --git a/Alpha_wrap_3/test/Alpha_wrap_3/test_alpha_wrap_3_mesh.cpp b/Alpha_wrap_3/test/Alpha_wrap_3/test_alpha_wrap_3_mesh.cpp
index 6209019a7a4..06b64164430 100644
--- a/Alpha_wrap_3/test/Alpha_wrap_3/test_alpha_wrap_3_mesh.cpp
+++ b/Alpha_wrap_3/test/Alpha_wrap_3/test_alpha_wrap_3_mesh.cpp
@@ -63,7 +63,7 @@ void alpha_wrap_triangle_mesh(Mesh& input_mesh,
   }
 
   if(!enforce_manifoldness)
-    assert(AW3::internal::check_edge_length(wrap, alpha));
+    assert(AW3::internal::has_bounded_edge_length(wrap, alpha));
 }
 
 void alpha_wrap_triangle_mesh(const std::string& filename,
diff --git a/Lab/demo/Lab/Plugins/Alpha_wrap_3/Alpha_wrap_3_plugin.cpp b/Lab/demo/Lab/Plugins/Alpha_wrap_3/Alpha_wrap_3_plugin.cpp
index 3374d7fe619..1896ac74fa7 100644
--- a/Lab/demo/Lab/Plugins/Alpha_wrap_3/Alpha_wrap_3_plugin.cpp
+++ b/Lab/demo/Lab/Plugins/Alpha_wrap_3/Alpha_wrap_3_plugin.cpp
@@ -732,11 +732,11 @@ public Q_SLOTS:
     Oracle oracle(ss_oracle);
 
     if(wrap_triangles)
-      oracle.add_triangle_soup(triangles);
+      oracle.add_triangles(triangles);
     if(wrap_segments)
-      oracle.add_segment_soup(segments);
+      oracle.add_segments(segments);
     if(wrap_points)
-      oracle.add_point_set(points);
+      oracle.add_points(points);
 
     if(!oracle.do_call())
     {