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			std::list<Vertex> vertex_db;

			std::list<HalfEdge> halfedge_db;

			std::vector<VertexIter> unused_vertices;

			std::vector<FaceIter> unused_faces;

			std::vector<HalfEdgeIter> unused_halfedges;

			/** Remove a face if it contains only two edges.

					This is an auxiliary function called from collapse_halfedge. */

			void remove_face_if_degenerate(HalfEdgeIter);

			/** Empty copy constructor.

					Copying a manifold will not work, since faces, vertices, and

					halfedges contain iterators pointing to other faces, vertices,

					and halfedges. These pointers would need to be changed if the 

					mesh were copied. In other words, we would need to walk the

					entire mesh. This may be required but it should not be an 

					operation that is easily invoked by calling the copy constructor.

					Hence, this operator is made private.		*/

			Manifold(const Manifold&) {}

			/** Empty assignment operator.

					The assignment operator is private for the same reason that the

					copy constructor is private. */

			const Manifold& operator=(const Manifold&) {return *this;}

			/** Return the bounding box of the manifold. The arguments pmin and pmax

					will contain the extreme corners of the box when the function 

					returns.	*/

			void get_bbox(CGLA::Vec3f& pmin, CGLA::Vec3f& pmax);

			/** Get a bounding sphere. When the function returns c contains the

					centre and r the radius. */ 

			void get_bsphere(CGLA::Vec3f& c, float& r);

			/// Return the number of faces.

			size_t no_faces() const {return face_db.size();}

			/// Return the number of halfedges.

			size_t no_halfedges() const {return halfedge_db.size();}

			/// Return the number of vertices

			size_t no_vertices() const {return vertex_db.size();}

			/** Create a new vertex.

					The argument is the position of the vertex, and the 

					function returns an iterator pointing to the vertex.

					When a vertex v is initially created, is_used(v) will

					return false.

			*/

			VertexIter create_vertex(const CGLA::Vec3f& pos)

				{

					vertex_db.push_back(Vertex(pos));

					return --vertex_db.end();

				}

			/** Create a new face.

					An iterator to the face is returned. When a face f is initially

					created, is_used(f) will return false. */

			FaceIter create_face()

				{

					face_db.push_back(Face());

					return --face_db.end();

				}

			/** Create a new halfedge. An iterator to the halfedge is returned.

					When h is initially created, is_used(h) returns false. */

			HalfEdgeIter create_halfedge()

				{

					halfedge_db.push_back(HalfEdge());

					return --halfedge_db.end();

				}

			/// Clear the manifold - removing all data.

			void clear();

			/** Remove unused vertices, edges and faces from the database.

					If delayed_erase mode is enabled, then until this function 

					has been called, erased vertices, edges, and faces are just marked

					as unused but not removed from their respective lists. */

			void remove_unused();

			/** Delay the actual removal of vertices, faces, and edges that

					are erased.  In many cases, it is a problem if one cannot

					test whether a vertex, halfedge, or face indicated by an

					iterator is in use or has been removed from the mesh. One

					solution to this problem is to delay the actual removal of

					the vertex, edge or face. Instead when calling, say,

					erase_face(f), all the iterators of f are assigned the

					appropriate NULL value to indicate that they are not

					pointing at anything, and f is added to a vector of faces

					that need to be physically removed from the face_db.

					Since f is not erased, all iterators pointing at f remain

					valid! And, importantly, it is possible to test whether f is

					in use by calling is_used(f).

					When remove_unused is called, the physical removal takes

					place. Calling immediate_erase switches Manifold back to the

					state where entities are removed as soon as the appropriate

					erase function is called, and at the same time calls

					remove_unused.

			*/

			void delayed_erase()

				{

					erase_immediately = false;

				}

			/** Immediately remove erased entities.

					Calling immediate_erase switches Manifold back to the state

					where entities are removed as soon as the appropriate erase

					function is called.

					See delayed_erase for more details, and note that

					immediate_erase is the default mode.  */

			void immediate_erase()

				{

					erase_immediately = true;

					remove_unused();

				}

			/** Test whether the vertex indicated by the argument v is used.

					This function returns true if the vertex appears to have a 

					valid outgoing halfedge iterator. */

			bool is_used(VertexIter v) const

				{

					if(v->he == NULL_HALFEDGE_ITER)

						return false;

					return true;

				}

			/** Test whether the face indicated by the argument f is used.

					This function returns true if the face appears to have a 

					valid halfedge iterator. */

			bool is_used(FaceIter f) const

				{

					if(f->last == NULL_HALFEDGE_ITER)

						return false;

					return true;

				}

			/** Test whether the halfedge indicated by the argument h is used.

					This function returns true if the halfedge appears to have a 

					valid vertex iterator. */

			bool is_used(HalfEdgeIter h) const

				{

					if(h->vert == NULL_VERTEX_ITER)

						return false;

					return true;

				}

			/** Erase halfedge h. 

					In general, you should never call this function but use the 

					collapse_halfedge function instead since blindly erasing geometry

					is most likely to invalidate the Mesh. Quite possibly this function

					will be removed from the public interface. 

					Note that if delayed_erase has been called, this function does

					not immediately remove anything from the mesh. Instead the halfedge

					is reset to its initial state. Thus, iterators are not invalidated,

					and it is possible to test whether h is used calling:

					is_used(h). when remove_unused is called, the actual removal

					takes place.

			*/

			void erase_halfedge(HalfEdgeIter h);

			/** Erase vertex v.

					In general, you should never call this function but use the 

					collapse_halfedge function to collapse the vertex away.

					Blindly erasing is extremely likely to invalidate the

					Mesh. Quite possibly this function will be removed from the

					public interface. 

					Note that if delayed_erase has been called, this function does

					not immediately remove anything from the mesh. Instead the halfedge

					is reset to its initial state. Thus, iterators are not invalidated,

					and it is possible to test whether v is used calling:

					is_used(v). when remove_unused is called, the actual removal

					takes place.

			*/

			void erase_vertex(VertexIter v);

			/** Erase face f.

					In general, you should never call this function but use 

					collapse_halfedge in conjunction with other functions to

					remove the face through (Euler) operations which preserve

					the mesh in a valid state.

					Blindly erasing is extremely likely to invalidate the

					Mesh. Quite possibly this function will be removed from the

					public interface. 

					Note that if delayed_erase has been called, this function does

					not immediately remove anything from the mesh. Instead the face

					is reset to its initial state. Thus, iterators are not invalidated,

					and it is possible to test whether f is used calling:

					is_used(f). when remove_unused is called, the actual removal

					takes place.

			*/

			void erase_face(FaceIter f);

			/// Return iterator pointing to beyond the last halfedge.

			HalfEdgeIter halfedges_end() { return halfedge_db.end(); }

			/// Return iterator pointing to the first vertex.

			VertexIter vertices_begin() { return vertex_db.begin();}

			/// Return iterator pointing to beyond the last vertex.

			VertexIter vertices_end() { return vertex_db.end(); }

			/// Return iterator pointing to the first face.

			FaceIter faces_begin() { return face_db.begin();	}

			/// Return iterator pointing to beyond the last face.

			FaceIter faces_end() { return face_db.end(); }

			/** \brief HalfEdge collapse precondition.

			The argument h is the halfedge we want to collapse. 

			If this function does not return true, it is illegal to collapse

			h. The reason is that the collapse would violate the manifold property

			of the mesh.

			The test is as follows.

			1. For the two vertices adjacent to the edge, we generate

			a list of all their neighbouring vertices. We then generate a 

			list of the vertices that occur in both these lists. That is, 

			we find all vertices connected by edges to both endpoints

			of the edge and store these in a list.

			2. For both faces incident on the edge, check whether they

			are triangular. If this is the case, the face will be removed,

			and it is ok that the the third vertex is connected to both 

			endpoints. Thus the third vertex in such a face is removed

			from the list generated in 1.

			3. If the list is now empty, all is well. Otherwise, there

			would be a vertex in the new mesh with two edges connecting

			it to the same vertex. Return false.

			4. TETRAHEDRON TEST:

			If the valency of both vertices is 

			three, and the incident faces are triangles, we also disallow

			the operation. Reason: It introduces a vertex of valency two

			and if the final two polygons incident on the vertices 

			that are adjacent to the edge being collapsed are triangles, then

			the construction will collapse

			5. VALENCY 4 TEST:

			If a face adjacent to the edge being collapsed is a triangle,

			it will disappear, and the valency of the final vertex beloning to

			this edge will be reduced by one. Hence this valency should be at

			least 4. A valency three vertex will be reduced to a valency two

			vertex which is considered illegal.

			6. PREVENT MERGING HOLES:

			We do not want to collapse an edge if its end points are boundary

			vertices, but its two adjacent faces are not NULL faces, because

			in that case we create a vertex where two holes meet. A non manifold

			situation.	*/

			bool collapse_precond(HalfEdgeIter h);

			/** Collapse the halfedge h.

			The argument h is the halfedge being removed. The vertex 

			v=h->opp->vert is the one being removed while h->vert survives.

			The final argument is a boolean indicating whether the vertex

			that survives the collapse should have a position which is the

			average of its own position and the vertex that is removed.

			By default false meaning that the surviving vertex retains it 

			position.

			This function is not guaranteed to keep the mesh sane unless,

			collapse_precond has returned true !!

			*/

			void collapse_halfedge(HalfEdgeIter h, bool=false);

			/** Split a face.

					The face, f, is split by connecting two

					vertices v0 and v1 (the next two arguments). The vertices of

					the old face between v0 and v1 (in counter clockwise order) 

					continue to belong to f. The vertices between v1 and 

					v0 belong to the new face. An iterator to the new face is 

					returned. */

			FaceIter split_face(FaceIter f, VertexIter v0, VertexIter v1);

			/** Insert a new vertex on halfedge h.

					The new halfedge is insterted as the previous edge to h.

					An iterator to the inserted vertex is	returned. */

			VertexIter Manifold::split_edge(HalfEdgeIter h);

			/** Triangulate a polygonal face by repeatedly calling split_face.

					Triangulate iteratively splits triangles off a polygon. The first 

					triangle split off is the one connecting f->last->vert and

					f->last->next->next->vert.

			*/

			void triangulate(FaceIter f);

			/** Triangulate a polygon, f, by inserting a vertex at the barycenter.

					All vertices in f are connected to the new vertex.				

					The word "safe" means that it is less likely to create flipped

					triangles than the normal triangulate. On the other hand, it 

					introduces more vertices and probably makes the triangles more

					acute. This function simply calls face_insert_point.

					The inserted vertex is returned.

			*/

			VertexIter safe_triangulate(FaceIter f);

			/** Insert a new vertex, v, in a face, f.

					This operation splits an N-gon into N triangles.

					In the current implementation the original face is erased. 

					This means that the iterator is not valid after the function

					returns. 

			*/

			void face_insert_point(FaceIter f, VertexIter v);

			/** Merges two faces into a single polygon. The first face is f.  The

					second face is adjacent to f along the halfedge h. This function

					returns true if the merging was possible and false

					otherwise. Currently merge only fails if the mesh is already

					illegal. Thus it should, in fact, never fail. */

			bool merge_faces(FaceIter f, HalfEdgeIter h);

			/** Flip an edge h. Returns false if flipping cannot be performed.

					This is either because one of the two adjacent faces is not

					a triangle, or because either end point has valency three or

					because the vertices that will be connected already are. */

			bool flip(HalfEdgeIter h);

			/** Performs a series of tests to check that this is a valid manifold.

					This function is not rigorously constructed but seems to catch

					all problems so far. The function returns true if the mesh is 

					valid and false otherwise.

			*/

			bool is_valid();

			/** Give each vertex a unique id corresponding to its iterator

					position */

			void enumerate_vertices();

			/** Give each halfedge a unique id corresponding to its iterator

					position */

			void enumerate_halfedges();

			/** Give each face a unique id corresponding to its iterator

					position */

			void enumerate_faces();