% +------------------------------------------------------------------------+ % | Reference manual page: Polyhedron_3.tex % +------------------------------------------------------------------------+ % | 17.03.1999 Lutz Kettner % | Package: Polyhedron % | \RCSdef{\RCSPolyhedronRev}{$Id$} \RCSdefDate{\RCSPolyhedronDate}{$Date$} % +------------------------------------------------------------------------+ \ccRefPageBegin %%RefPage: end of header, begin of main body % +------------------------------------------------------------------------+ \begin{ccRefClass}{Polyhedron_3} \ccDefinition A polyhedral surface \ccClassTemplateName\ consists of vertices $V$, edges $E$, facets $F$ and an incidence relation on them. Each edge is represented by two halfedges with opposite orientations. \begin{ccTexOnly} \vspace{-7mm} \begin{center} \parbox{0.4\textwidth}{% \includegraphics[width=0.4\textwidth]{Polyhedron_ref/fig/halfedge}% } \end{center} \vspace{-5mm} \end{ccTexOnly} \begin{ccHtmlOnly}
Halfedge Diagram

\end{ccHtmlOnly} Vertices represent points in 3d-space. Edges are straight line segments between two endpoints. Facets are planar polygons without holes defined by the circular sequence of halfedges along their boundary. The polyhedral surface itself can have holes. The halfedges along the boundary of a hole are called {\em border halfedges\/} and have no incident facet. An edge is a {\em border edge\/} if one of its halfedges is a border halfedge. A surface is {\em closed\/} if it contains no border halfedges. A closed surface is a boundary representation for polyhedra in three dimensions. The convention is that the halfedges are oriented counterclockwise around facets as seen from the outside of the polyhedron. An implication is that the halfedges are oriented clockwise around the vertices. The notion of the solid side of a facet as defined by the halfedge orientation extends to polyhedral surfaces with border edges although they do not define a closed object. If normal vectors are considered for the facets, normals point outwards (following the right hand rule). The strict definition can be found in~\cite{k-ugpdd-99}. One implication of this definition is that the polyhedral surface is always an orientable and oriented 2-manifold with border edges, i.e., the neighborhood of each point on the polyhedral surface is either homeomorphic to a disc or to a half disc, except for vertices where many holes and surfaces with boundary can join. Another implication is that the smallest representable surface is a triangle (for polyhedral surfaces with border edges) or a tetrahedron (for polyhedra). Boundary representations of orientable 2-manifolds are closed under Euler operations. They are extended with operations that create or close holes in the surface. Other intersections besides the incidence relation are not allowed, although they are not automatically handled, since self intersections are not easy to check efficiently. \ccClassTemplateName\ does only maintain the combinatorial integrity of the polyhedral surface (using Euler operations) and does not consider the coordinates of the points or any geometric information. The class \ccClassTemplateName\ can represent polyhedral surfaces as well as polyhedra. The interface is designed in such a way that it is easy to ignore border edges and work only with polyhedra. The sequence of edges can be ordered in the data structure on request such that the sequence starts with the non-border edges and ends with the border edges. Border edges are then itself ordered such that the halfedge which is incident to the facet comes first and the halfedge incident to the hole comes thereafter. This normalization step counts simultaneously the number of border edges. This number is zero if and only if the surface is a closed polyhedron. Note that this class does not maintain this counter nor the halfedge order during further modifications. There is no automatic caching done for auxiliary information. \ccInclude{CGAL/Polyhedron_3.h} \ccParameters The full template declaration of \ccClassTemplateName\ states four template parameters: \begin{tabbing} \ccc{template <} \=\ccc{class PolyhedronTraits_3,}\\ \>\ccc{class PolyhedronItems_3 = CGAL::Polyhedron_items_3,}\\ \>\ccc{template < class T, class I>} \ccc{class HalfedgeDS = CGAL::HalfedgeDS_default,}\\ \>\ccc{class Alloc = CGAL_ALLOCATOR(int)>}\\ \ccc{class Polyhedron_3;} \end{tabbing} The first parameter requires a model of the \ccc{PolyhedronTraits_3} concept as argument, for example \ccc{CGAL::Polyhedron_traits_3}. The second parameter expects a model of the \ccc{PolyhedronItems_3} concept. By default, the class \ccc{CGAL::Polyhedron_items_3} is preselected. The third parameter is a class template. A model of the \ccc{HalfedgeDS} concept is expected. By default, the class \ccc{CGAL::HalfedgeDS_default} is preselected, which is a list based implementation of the halfedge data structure. The fourth parameter \ccc{Alloc} requires a standard allocator for \stl\ container classes. The \ccc{rebind} mechanism from \ccc{Alloc} will be used to create appropriate allocators internally. A default is provided with the macro \ccc{CGAL_ALLOCATOR(int)} from the \ccc{} header file. \ccTypes \ccTwo{Polyhedron_3:: circulator_category}{} \ccNestedType{Traits}{traits class selected for \ccc{PolyhedronTraits_3}.} \ccGlue \ccNestedType{Items}{items class selected for \ccc{PolyhedronItems_3}.} \ccGlue \ccNestedType{HalfedgeDS}{instantiated halfedge data structure.} \ccNestedType{size_type}{size type of \ccc{HalfedgeDS}.} \ccGlue \ccNestedType{difference_type}{difference type of \ccc{HalfedgeDS}.} \ccGlue \ccNestedType{iterator_category}{iterator category of \ccc{HalfedgeDS} for all iterators.} \ccGlue \ccNestedType{circulator_category}{circulator category of all circulators; bidirectional category if the \ccc{Items::Halfedge} provides a \ccc{prev()} member function, otherwise forward category.} \ccGlue \ccNestedType{allocator_type}{allocator type \ccc{Alloc}.} \ccNestedType{Vertex}{vertex type.} \ccGlue \ccNestedType{Halfedge}{halfedge type.} \ccGlue \ccNestedType{Facet}{facet type.} \ccNestedType{Point_3}{point stored in vertices.} \ccGlue \ccNestedType{Plane_3}{plane equation stored in facets (if supported).} %\ccGlue %\ccNestedType{Normal}{normal vector stored in facets (if supported).} The following handles, iterators, and circulators have appropriate non-mutable counterparts, i.e., \ccc{const_handle}, \ccc{const_iterator}, and \ccc{const_circulator}. The mutable types are assignable to their non-mutable counterparts. Both circulators are assignable to the \ccc{Halfedge_iterator}. The iterators are assignable to the respective handle types. Wherever the handles appear in function parameter lists, the corresponding iterators can be used as well. For convenience, the \ccc{Edge_iterator} enumerates every other halfedge. It is based on the \ccc{CGAL::N_step_adaptor} class. For convenience, the \ccc{Point_iterator} enumerates all points in the polyhedral surface in the same order as the \ccc{Vertex_iterator}, but with the value type \ccc{Point}. It is based on the \ccc{CGAL::Iterator_project} adaptor. Similarly, a \ccc{Plane_iterator} is provided. \ccNestedType{Vertex_handle}{handle to vertex.} \ccGlue \ccNestedType{Halfedge_handle}{handle to halfedge.} \ccGlue \ccNestedType{Facet_handle}{handle to facet.} \ccNestedType{Vertex_iterator}{iterator over all vertices.} \ccGlue \ccNestedType{Halfedge_iterator}{iterator over all halfedges.} \ccGlue \ccNestedType{Facet_iterator}{iterator over all facets.} \ccNestedType{Halfedge_around_vertex_circulator}{circulator of halfedges around a vertex (cw).} \ccGlue \ccNestedType{Halfedge_around_facet_circulator}{circulator of halfedges around a facet (ccw).} \ccNestedType{Edge_iterator}{iterator over all edges (every other halfedge).} \ccGlue \ccNestedType{Point_iterator}{iterator over all points.} \ccGlue \ccNestedType{Plane_iterator}{iterator over all plane equations.} % +-----------------------------------+ \begin{ccAdvanced} \ccHeading{Types for Tagging Optional Features} \ccTwo{Polyhedron_3:: Supports_vertex_halfedge}{} The following types are equal to either \ccStyle{CGAL::Tag_true} or \ccStyle{CGAL::Tag_false}, depending on whether the named feature is supported or not. \ccNestedType{Supports_vertex_halfedge}{\ccc{Vertex::halfedge()}.} \ccGlue \ccNestedType{Supports_vertex_point}{\ccc{Vertex::point()}.} \ccGlue \ccNestedType{Supports_halfedge_prev}{\ccc{Halfedge::prev()}.} \ccGlue \ccNestedType{Supports_halfedge_vertex}{\ccc{Halfedge::vertex()}.} \ccGlue \ccNestedType{Supports_halfedge_facet}{\ccc{Halfedge::facet()}.} \ccGlue \ccNestedType{Supports_facet_halfedge}{\ccc{Facet::halfedge()}.} \ccGlue \ccNestedType{Supports_facet_plane}{\ccc{Facet::plane()}.} \ccGlue \ccNestedType{Supports_removal}{supports removal of individual elements.} %\ccGlue %\ccNestedType{Supports_facet_normal}{\ccc{Facet::normal()}.} \end{ccAdvanced} \ccCreation \ccCreationVariable{P} \ccThree{Halfedge_iterator}{AP.m}{} \ccThreeToTwo \ccConstructor{Polyhedron_3(const Traits& traits = Traits());}{} \ccConstructor{Polyhedron_3( size_type v, size_type h, size_type f, const Traits& traits = Traits());} {a polyhedron \ccVar\ with storage reserved for $v$ vertices, $h$ halfedges, and $f$ facets. The reservation sizes are a hint for optimizing storage allocation.} \ccMethod{void reserve( size_type v, size_type h, size_type f);} {reserve storage for $v$ vertices, $h$ halfedges, and $f$ facets. The reservation sizes are a hint for optimizing storage allocation. If the \ccc{capacity} is already greater than the requested size nothing happens. If the \ccc{capacity} changes all iterators and circulators might invalidate.} \ccMethod{Halfedge_handle make_tetrahedron();}{a tetrahedron is added to the polyhedral surface. Returns a halfedge of the tetrahedron.} \ccMethod{Halfedge_handle make_tetrahedron(const Point& p1, const Point& p2, const Point& p3, const Point& p4);}{ a tetrahedron is added to the polyhedral surface with its vertices initialized to $p_1, p_2, p_3$, and $p_4$. Returns that halfedge of the tetrahedron which incident vertex is initialized to $p_1$. The incident vertex of the next halfedge is $p_2$, and the vertex thereafter is $p_3$. The remaining fourth vertex is initialized to $p_4$.} \vspace{-3mm} \ccMethod{Halfedge_handle make_triangle();}{a triangle with border edges is added to the polyhedral surface. Returns a non-border halfedge of the triangle.} \vspace{-3mm} \ccMethod{Halfedge_handle make_triangle(const Point& p1, const Point& p2, const Point& p3);}{ a triangle with border edges is added to the polyhedral surface with its vertices initialized to $p_1, p_2$, and $p_3$. Returns that non-border halfedge of the triangle which incident vertex is initialized to $p_1$. The incident vertex of the next halfedge is $p_2$, and the vertex thereafter is $p_3$.} \vspace{-3mm} % +-----------------------------------+ \ccHeading{Access Member Functions} \ccThree{Halfedge_iterator}{P.size_of_border_halfedges();}{} \ccMethod{bool empty() const;}{returns true if \ccVar\ is empty.} \ccMethod{size_type size_of_vertices() const;} {number of vertices.} \ccGlue \ccMethod{size_type size_of_halfedges() const;} {number of halfedges (incl.\ border halfedges).} \ccGlue \ccMethod{size_type size_of_facets() const;} {number of facets.} \ccMethod{size_type capacity_of_vertices() const;} {space reserved for vertices.} \ccGlue \ccMethod{size_type capacity_of_halfedges() const;} {space reserved for halfedges.} \ccGlue \ccMethod{size_type capacity_of_facets() const;} {space reserved for facets.} \ccMethod{size_t bytes() const;} {bytes used for the polyhedron.} \ccGlue \ccMethod{size_t bytes_reserved() const;} {bytes reserved for the polyhedron.} \ccMethod{allocator_type get_allocator() const;}{allocator object.} \ccMethod{Vertex_iterator vertices_begin();}{iterator over all vertices.} \ccGlue \ccMethod{Vertex_iterator vertices_end();}{past-the-end iterator.} \ccGlue \ccMethod{Halfedge_iterator halfedges_begin();}{iterator over all halfedges.} \ccGlue \ccMethod{Halfedge_iterator halfedges_end();}{past-the-end iterator.} \ccGlue \ccMethod{Facet_iterator facets_begin();}{iterator over all facets (excluding holes).} \ccGlue \ccMethod{Facet_iterator facets_end();}{past-the-end iterator.} \ccMethod{Edge_iterator edges_begin();}{iterator over all edges.} \ccGlue \ccMethod{Edge_iterator edges_end();}{past-the-end iterator.} \ccMethod{Point_iterator points_begin();}{iterator over all points.} \ccGlue \ccMethod{Point_iterator points_end();}{past-the-end iterator.} \ccMethod{Plane_iterator planes_begin();}{iterator over all plane equations.} \ccGlue \ccMethod{Plane_iterator planes_end();}{past-the-end iterator.} \ccMethod{const Traits& traits() const;}{returns the traits class.} % +-----------------------------------+ \ccHeading{Combinatorial Predicates} \ccThree{Halfedge_handle}{P.split_f}{} \ccMethod{bool is_closed() const;}{returns \ccc{true} if there are no border edges.} \ccMethod{bool is_pure_bivalent() const;}{returns \ccc{true} if all vertices have exactly two incident edges.} \ccMethod{bool is_pure_trivalent() const;}{returns \ccc{true} if all vertices have exactly three incident edges.} \ccMethod{bool is_pure_triangle() const;}{returns \ccc{true} if all facets are triangles.} \ccMethod{bool is_pure_quad() const;}{returns \ccc{true} if all facets are quadrilaterals.} % +-----------------------------------+ %\ccHeading{Geometric Predicates} %\ccThree{Halfedge_handle}{P.split_f}{} \ccMethod{bool is_triangle( Halfedge_const_handle h) const;}{\ccc{true} iff the connected component denoted by $h$ is a triangle.} \ccMethod{bool is_tetrahedron( Halfedge_const_handle h) const;}{\ccc{true} iff the connected component denoted by $h$ is a tetrahedron.} % +-----------------------------------+ \ccHeading{Euler Operators (Combinatorial Modifications)} \label{sectionPolyhedronEuler} \ccThree{Halfedge_handle}{P.split_f}{} The following Euler operations modify consistently the combinatorial structure of the polyhedral surface. The geometry remains unchanged. \begin{ccTexOnly} \begin{center} \parbox{\textwidth}{% \includegraphics[width=\textwidth]{Polyhedron_ref/fig/euler}% } \end{center} \end{ccTexOnly} \begin{ccHtmlOnly}
Euler Operator: Facet

\end{ccHtmlOnly} \ccMethod{Halfedge_handle split_facet( Halfedge_handle h, Halfedge_handle g);} {splits the facet incident to \ccc{h} and \ccc{g} into two facets with a new diagonal between the two vertices denoted by \ccc{h} and \ccc{g} respectively. The second (new) facet is a copy of the first facet. Returns \ccc{h->next()} after the operation, i.e., the new diagonal. The new face is to the right of the new diagonal, the old face is to the left. The time is proportional to the distance from \ccc{h} to \ccc{g} around the facet. \ccPrecond \ccc{h} and \ccc{g} are incident to the same facet. \ccc{h != g} (no loops). \ccc{h->next() != g} and \ccc{g->next() != h} (no multi-edges).} \ccMethod{Halfedge_handle join_facet( Halfedge_handle h);} {joins the two facets incident to $h$. The facet incident to \ccc{h->opposite()} gets removed. Both facets might be holes. Returns the predecessor of $h$ around the facet. The invariant \ccc{join_facet( split_facet( h, g))} returns $h$ and keeps the polyhedron unchanged. The time is proportional to the size of the facet removed and the time to compute \ccc{h->prev()}. \ccPrecond The degree of both vertices incident to $h$ is at least three (no antennas). \ccCommentHeading{Requirement} \ccc{Supports_removal} $\equiv$ \ccc{CGAL::Tag_true}.} \begin{ccHtmlOnly}
Euler Operator: Vertex

\end{ccHtmlOnly} \ccMethod{Halfedge_handle split_vertex( Halfedge_handle h, Halfedge_handle g);} {splits the vertex incident to \ccc{h} and \ccc{g} into two vertices, the old vertex remains and a new copy is created, and connects them with a new edge. Let \ccc{hnew} be \ccc{h->next()->opposite()} after the split, i.e., a halfedge of the new edge. The split regroups the halfedges around the two vertices. The halfedge sequence \ccc{hnew}, \ccc{g->next()->opposite()}, \ldots, \ccc{h} remains around the old vertex, while the halfedge sequence \ccc{hnew->opposite()}, \ccc{h->next()->opposite()} (before the split), \ldots, \ccc{g} is regrouped around the new vertex. The split returns \ccc{hnew}, i.e., the new halfedge incident to the old vertex. The time is proportional to the distance from \ccc{h} to \ccc{g} around the vertex. \ccPrecond \ccc{h} and \ccc{g} are incident to the same vertex. \ccc{h != g} (antennas are not allowed). \ccCommentHeading{Note} A special application of the split is \ccc{split_vertex(h,h->next()->opposite())} which is equivalent to an edge split of the halfedge \ccc{h->next()} that creates a new vertex on the halfedge \ccc{h->next()}. See also \ccc{split_edge(h)} below.} \ccMethod{Halfedge_handle join_vertex( Halfedge_handle h);} {joins the two vertices incident to $h$. The vertex denoted by \ccc{h->opposite()} gets removed. Returns the predecessor of $h$ around the vertex, i.e., \ccc{h->opposite()->prev()}. The invariant \ccc{join_vertex( split_vertex( h, g))} returns $h$ and keeps the polyhedron unchanged. The time is proportional to the degree of the vertex removed and the time to compute \ccc{h->prev()} and \ccc{h->opposite()->prev()}. \ccPrecond The size of both facets incident to $h$ is at least four (no multi-edges). \ccCommentHeading{Requirement} \ccc{Supports_removal} $\equiv$ \ccc{CGAL::Tag_true}.} \ccMethod{Halfedge_handle split_edge( Halfedge_handle h);}{ splits the halfedge \ccc{h} into two halfedges inserting a new vertex that is a copy of \ccc{h->opposite()->vertex()}. Is equivalent to \ccc{split_vertex( h->prev(), h->opposite())}. The call of \ccc{prev()} can make this method slower than a direct call of \ccc{split_vertex()} if the previous halfedge is already known and computing it would be costly when the halfedge data structure does not support the \ccc{prev()} member function. Returns the new halfedge \ccc{hnew} pointing to the inserted vertex. The new halfedge is followed by the old halfedge, i.e., \ccc{hnew->next() == h}.} \ccMethod{Halfedge_handle flip_edge( Halfedge_handle h);} {performs an edge flip. It returns $h$ after rotating the edge $h$ one vertex in the direction of the face orientation. \ccPrecond \ccc{h != Halfedge_handle()} and both facets incident to $h$ are triangles.} \begin{ccTexOnly} \begin{center} \parbox{0.52\textwidth}{% \includegraphics[width=0.52\textwidth]{Polyhedron_ref/fig/euler_center}% } \end{center} \end{ccTexOnly} \begin{ccHtmlOnly}
Euler Operator: Center Vertex

\end{ccHtmlOnly} \ccMethod{Halfedge_handle create_center_vertex( Halfedge_handle h);} {barycentric triangulation of \ccc{h->facet()}. Creates a new vertex, a copy of \ccc{h->vertex()}, and connects it to each vertex incident to \ccc{h->facet()} splitting \ccc{h->facet()} into triangles. \ccc{h} remains incident to the original facet, all other triangles are copies of this facet. Returns the halfedge \ccc{h->next()} after the operation, i.e., a halfedge pointing to the new vertex. The time is proportional to the size of the facet. \ccPrecond \ccc{h} is not a border halfedge.} \ccMethod{Halfedge_handle erase_center_vertex( Halfedge_handle g);} {reverses \ccc{create_center_vertex}. Erases the vertex pointed to by \ccc{g} and all incident halfedges thereby merging all incident facets. Only \ccc{g->facet()} remains. The neighborhood of \ccc{g->vertex()} may not be triangulated, it can have larger facets. Returns the halfedge \ccc{g->prev()}. Thus, the invariant \ccc{h == erase_center_vertex( create_center_vertex(h))} holds if \ccc{h} is not a border halfedge. The time is proportional to the sum of the size of all incident facets. \ccPrecond None of the incident facets of \ccc{g->vertex()} is a hole. There are at least two distinct facets incident to the facets that are incident to \ccc{g->vertex()}. (This prevents the operation from collapsing a volume into two facets glued together with opposite orientations, such as would happen with any vertex of a tetrahedron.) \ccCommentHeading{Requirement} \ccc{Supports_removal} $\equiv$ \ccc{CGAL::Tag_true}.} % +-----------------------------------+ \ccHeading{Euler Operators Modifying Genus} \begin{ccTexOnly} \begin{center} \parbox{0.636\textwidth}{% \includegraphics[width=0.636\textwidth]% {Polyhedron_ref/fig/euler_loop}% } \end{center} \end{ccTexOnly} \begin{ccHtmlOnly}
Euler Operator: Loop

\end{ccHtmlOnly} \ccMethod{Halfedge_handle split_loop( Halfedge_handle h, Halfedge_handle i, Halfedge_handle j);} {cuts the polyhedron into two parts along the cycle $(h,i,j)$ (edge \ccc{j} runs on the backside of the three dimensional figure above). Three new vertices (one copy for each vertex in the cycle) and three new halfedges (one copy for each halfedge in the cycle), and two new triangles are created. $h,i,j$ will be incident to the first new triangle. The return value will be the halfedge incident to the second new triangle which is the copy of \ccc{h-opposite()}. \ccPrecond $h,i,j$ denote distinct, consecutive vertices of the polyhedron and form a cycle: i.e., \ccc{h->vertex() == i->opposite()->vertex()}, \ldots, \ccc{j->vertex() == h->opposite()->vertex()}. The six facets incident to $h,i,j$ are all distinct. } \ccMethod{Halfedge_handle join_loop( Halfedge_handle h, Halfedge_handle g);} {glues the boundary of the two facets denoted by $h$ and $g$ together and returns $h$. Both facets and the vertices along the facet denoted by $g$ gets removed. Both facets may be holes. The invariant \ccc{join_loop( h, split_loop( h, i, j))} returns $h$ and keeps the polyhedron unchanged. \ccPrecond The facets denoted by $h$ and $g$ are different and have equal degree (i.e., number of edges). \ccCommentHeading{Requirement} \ccc{Supports_removal} $\equiv$ \ccc{CGAL::Tag_true}.} % +-----------------------------------+ \ccHeading{Modifying Facets and Holes} \ccMethod{Halfedge_handle make_hole( Halfedge_handle h);} {removes the incident facet of $h$ and changes all halfedges incident to the facet into border edges. Returns $h$. See \ccc{erase_facet(h)} for a more generalized variant. \ccPrecond None of the incident halfedges of the facet is a border edge. \ccCommentHeading{Requirement} \ccc{Supports_removal} $\equiv$ \ccc{CGAL::Tag_true}.} \ccMethod{Halfedge_handle fill_hole( Halfedge_handle h);}{ fills a hole with a newly created facet. Makes all border halfedges of the hole denoted by $h$ incident to the new facet. Returns $h$. \ccPrecond \ccc{h.is_border()}.} \begin{ccTexOnly} \begin{center} \parbox{\textwidth}{% \includegraphics[width=\textwidth]{Polyhedron_ref/fig/add_facet}% } \end{center} \end{ccTexOnly} \begin{ccHtmlOnly}
Modifying Facets and Holes: add_vertex_and_facet_to_border()

\end{ccHtmlOnly} \ccMethod{Halfedge_handle add_vertex_and_facet_to_border( Halfedge_handle h, Halfedge_handle g);} {creates a new facet within the hole incident to $h$ and $g$ by connecting the tip of $g$ with the tip of $h$ with two new halfedges and a new vertex and filling this separated part of the hole with a new facet, such that the new facet is incident to $g$. Returns the halfedge of the new edge that is incident to the new facet and the new vertex. \ccPrecond \ccc{h->is_border()}, \ccc{g->is_border()}, \ccc{h != g}, and $g$ can be reached along the same hole starting with $h$.} \begin{ccHtmlOnly}
Modifying Facets and Holes: add_facet_to_border()

\end{ccHtmlOnly} \ccMethod{Halfedge_handle add_facet_to_border( Halfedge_handle h, Halfedge_handle g);} {creates a new facet within the hole incident to $h$ and $g$ by connecting the vertex denoted by $g$ with the vertex denoted by $h$ with a new halfedge and filling this separated part of the hole with a new facet, such that the new facet is incident to $g$. Returns the halfedge of the new edge that is incident to the new facet. \ccPrecond \ccc{h->is_border()}, \ccc{g->is_border()}, \ccc{h != g}, \ccc{h->next() != g}, and $g$ can be reached along the same hole starting with $h$.} % +-----------------------------------+ \ccHeading{Erasing} \ccMethod{void erase_facet( Halfedge_handle h);} {removes the incident facet of $h$ and changes all halfedges incident to the facet into border edges or removes them from the polyhedral surface if they were already border edges. If this creates isolated vertices they get removed as well. See \ccc{make_hole(h)} for a more specialized variant. \ccPrecond \ccc{h->is_border() == false}. \ccCommentHeading{Requirement} \ccc{Supports_removal} $\equiv$ \ccc{CGAL::Tag_true}.} \ccMethod{void erase_connected_component( Halfedge_handle h);} {removes the vertices, halfedges, and facets that belong to the connected component of $h$. \ccCommentHeading{Requirement} \ccc{Supports_removal} $\equiv$ \ccc{CGAL::Tag_true}.} \ccMethod{void clear();} {removes all vertices, halfedges, and facets.} % +-----------------------------------+ \ccHeading{Operations with Border Halfedges} \begin{ccAdvanced} Halfedges incident to a hole are called {\em border halfedges}. An halfedge is a {\em border edge\/} if itself or its opposite halfedge are border halfedges. The only requirement to work with border halfedges is that the \ccc{Halfedge} class provides a member function \ccc{is_border()} returning a \ccc{bool}. Usually, the halfedge data structure supports facets and a \ccc{NULL} facet pointer will indicate a border halfedge, but this is not the only possibility. The \ccc{is_border()} predicate divides the edges into two classes, the border edges and the non-border edges. The following normalization reorganizes the sequential storage of the edges such that the non-border edges precede the border edges, and that for each border edge the latter one of the two halfedges is a border halfedge (the first one is a non-border halfedge in conformance with the polyhedral surface definition). The normalization stores the number of border halfedges and the halfedge iterator the border edges start at within the data structure. Halfedge insertion or removal and changing the border status of a halfedge invalidate these values. They are not automatically updated. \ccThree{Halfedge_iterator}{P.split_f}{} \ccMethod{void normalize_border();} {sorts halfedges such that the non-border edges precede the border edges. For each border edge the halfedge iterator will reference the halfedge incident to the facet right before the halfedge incident to the hole.} \ccMethod{size_type size_of_border_halfedges() const;} {number of border halfedges. \ccPrecond last \ccc{normalize_border()} call still valid, see above.} \ccMethod{size_type size_of_border_edges() const;} {number of border edges. Since each border edge of a polyhedral surface has exactly one border halfedge, this number is equal to \ccc{size_of_border_halfedges()}. \ccPrecond last \ccc{normalize_border()} call still valid, see above.} \ccMethod{Halfedge_iterator border_halfedges_begin();} {halfedge iterator starting with the border edges. The range [\ccStyle{halfedges_begin(), border_halfedges_begin()}) denotes all non-border halfedges. The range [\ccStyle{border_halfedges_begin(), halfedges_end()}) denotes all border edges. \ccPrecond last \ccc{normalize_border()} call still valid, see above.} \ccMethod{Edge_iterator border_edges_begin();} {edge iterator starting with the border edges. The range [\ccStyle{edges_begin(), border_edges_begin()}) denotes all non-border edges. The range [\ccStyle{border_edges_begin(), edges_end()}) denotes all border edges. \ccPrecond last \ccc{normalize_border()} call still valid, see above.} \end{ccAdvanced} % +-----------------------------------+ \ccHeading{Miscellaneous} \ccMethod{void inside_out();} {reverses facet orientations (incl.\ plane equations if supported).} \ccMethod{bool is_valid( bool verbose = false, int level = 0) const;} {returns \ccc{true} if the polyhedral surface is combinatorially consistent. If \ccc{verbose} is \ccc{true}, statistics are printed to \ccc{cerr}. For \ccc{level == 1} the normalization of the border edges is checked too. This method checks in particular level 3 of \ccc{CGAL::Halfedge_data_structure_decorator::is_valid} from page~\ref{pageHalfedgeDSconstDecoratorRef} and that each facet is at least a triangle and that the two incident facets of a non-border edge are distinct.} \ccMethod{bool normalized_border_is_valid( bool verbose = false) const;}{% returns \ccc{true} if the border halfedges are in normalized representation, which is when enumerating all halfedges with the iterator: The non-border edges precede the border edges and for border edges, the second halfedge is the border halfedge. The halfedge iterator \ccc{border_halfedges_begin()} denotes the first border edge. If \ccc{verbose} is \ccc{true}, statistics are printed to \ccc{cerr}. } \newpage \begin{ccAdvanced} \ccMethod{void delegate( CGAL::Modifier_base& m);} {calls the \ccc{operator()} of the modifier $m$. See \ccc{CGAL::Modifier_base} in the Support Library Manual for a description of modifier design and its usage. \ccPrecond The polyhedral surface must be valid when the modifier returns from execution.} \end{ccAdvanced} \ccSeeAlso \ccRefIdfierPage{CGAL::Polyhedron_3::Vertex}\\ \ccRefIdfierPage{CGAL::Polyhedron_3::Halfedge}\\ \ccRefIdfierPage{CGAL::Polyhedron_3::Facet}\\ \ccRefConceptPage{PolyhedronTraits_3}\\ \ccRefIdfierPage{CGAL::Polyhedron_traits_3}\\ \ccRefConceptPage{PolyhedronItems_3}\\ \ccRefIdfierPage{CGAL::Polyhedron_items_3}\\ \ccRefConceptPage{HalfedgeDS}\\ \ccRefIdfierPage{CGAL::HalfedgeDS_default}\\ \ccRefIdfierPage{CGAL::Polyhedron_incremental_builder_3}\\ \ccc{CGAL::Modifier_base} in the Support Library Reference Manual. \ccExample This example program instantiates a polyhedron using the default traits class and creates a tetrahedron. \ccIncludeExampleCode{Polyhedron/polyhedron_prog_simple.C} \end{ccRefClass} % +------------------------------------------------------------------------+ %%RefPage: end of main body, begin of footer \ccRefPageEnd % EOF % +------------------------------------------------------------------------+