CGAL  3D Polyhedral Surface

#include <CGAL/Polyhedron_3.h>
A polyhedral surface Polyhedron_3
consists of vertices V
, edges E
, facets F
and an incidence relation on them.
Each edge is represented by two halfedges with opposite orientations.
Vertices represent points in 3dspace. 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 border halfedges and have no incident facet. An edge is a border edge if one of its halfedges is a border halfedge. A surface is 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 [3]. One implication of this definition is that the polyhedral surface is always an orientable and oriented 2manifold 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 2manifolds 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. Polyhedron_3
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 Polyhedron_3
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 nonborder 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.
Parameters
The full template declaration of Polyhedron_3
states four template parameters:
The first parameter requires a model of the PolyhedronTraits_3
concept as argument, for example CGAL::Polyhedron_traits_3
. The second parameter expects a model of the PolyhedronItems_3
concept. By default, the class CGAL::Polyhedron_items_3
is preselected. The third parameter is a class template. A model of the HalfedgeDS
concept is expected. By default, the class CGAL::HalfedgeDS_default
is preselected, which is a list based implementation of the halfedge data structure. The fourth parameter Alloc
requires a standard allocator for STL container classes. The rebind
mechanism from Alloc
will be used to create appropriate allocators internally. A default is provided with the macro CGAL_ALLOCATOR(int)
from the <CGAL/memory.h>
header file.
CGAL::Polyhedron_3::Vertex
CGAL::Polyhedron_3::Halfedge
CGAL::Polyhedron_3::Facet
PolyhedronTraits_3
CGAL::Polyhedron_traits_3<Kernel>
PolyhedronItems_3
CGAL::Polyhedron_items_3
HalfedgeDS
CGAL::HalfedgeDS_default
CGAL::Polyhedron_incremental_builder_3<HDS>
CGAL::Modifier_base
.Example
This example program instantiates a polyhedron using the default traits class and creates a tetrahedron.
File Polyhedron/polyhedron_prog_simple.cpp
Classes  
class  Facet 
A facet optionally stores a plane equation, and a reference to an incident halfedge that points to the facet. More...  
class  Halfedge 
A halfedge is an oriented edge between two vertices. More...  
class  Vertex 
A vertex optionally stores a point and a reference to an incident halfedge that points to the vertex. More...  
Related Functions  
(Note that these are not member functions.)  
template<class PolyhedronTraits_3 >  
bool  read_off (std::istream &in, Polyhedron_3< PolyhedronTraits_3 > &P) 
template<class PolyhedronTraits_3 >  
std::istream &  operator>> (std::istream &in, Polyhedron_3< PolyhedronTraits_3 > &P) 
template<class PolyhedronTraits_3 >  
bool  write_off (std::ostream &out, Polyhedron_3< PolyhedronTraits_3 > &P) 
template<class PolyhedronTraits_3 >  
std::ostream &  operator<< (std::ostream &out, Polyhedron_3< PolyhedronTraits_3 > &P) 
Types  
typedef unspecified_type  Items 
items class selected for PolyhedronItems_3 .  
typedef unspecified_type  HalfedgeDS 
instantiated halfedge data structure.  
typedef unspecified_type  size_type 
size type of HalfedgeDS .  
typedef unspecified_type  difference_type 
difference type of HalfedgeDS .  
typedef unspecified_type  iterator_category 
iterator category of HalfedgeDS for all iterators.  
typedef unspecified_type  circulator_category 
circulator category of all circulators; bidirectional category if the Items::Halfedge provides a Halfedge::prev() member function, otherwise forward category.  
typedef unspecified_type  allocator_type 
allocator type Alloc .  
Handles, Iterators, and Circulators  
The following handles, iterators, and circulators have appropriate nonmutable counterparts, i.e., The mutable types are assignable to their nonmutable counterparts. Both circulators are assignable to the All handles are model of  
typedef unspecified_type  Point_3 
point stored in vertices.  
typedef unspecified_type  Plane_3 
plane equation stored in facets (if supported).  
typedef unspecified_type  Vertex_handle 
handle to vertex.  
typedef unspecified_type  Halfedge_handle 
handle to halfedge.  
typedef unspecified_type  Facet_handle 
handle to facet.  
typedef unspecified_type  Vertex_iterator 
iterator over all vertices.  
typedef unspecified_type  Halfedge_iterator 
iterator over all halfedges.  
typedef unspecified_type  Facet_iterator 
iterator over all facets.  
typedef unspecified_type  Halfedge_around_vertex_circulator 
circulator of halfedges around a vertex (cw).  
typedef unspecified_type  Halfedge_around_facet_circulator 
circulator of halfedges around a facet (ccw).  
typedef unspecified_type  Edge_iterator 
iterator over all edges (every other halfedge).  
typedef unspecified_type  Point_iterator 
iterator over all points.  
typedef unspecified_type  Plane_iterator 
iterator over all plane equations.  
Types for Tagging Optional Features  
The following types are equal to either  
typedef unspecified_type  Supports_vertex_halfedge 
Vertex::halfedge() .  
typedef unspecified_type  Supports_vertex_point 
Vertex::point() .  
typedef unspecified_type  Supports_halfedge_prev 
Halfedge::prev() .  
typedef unspecified_type  Supports_halfedge_vertex 
Halfedge::vertex() .  
typedef unspecified_type  Supports_halfedge_facet 
Halfedge::facet() .  
typedef unspecified_type  Supports_facet_halfedge 
Facet::halfedge() .  
typedef unspecified_type  Supports_facet_plane 
Facet::plane() .  
typedef unspecified_type  Supports_removal 
supports removal of individual elements.  
Creation  
Polyhedron_3 (const Traits &traits=Traits())  
Default constructor.  
Polyhedron_3 (size_type v, size_type h, size_type f, const Traits &traits=Traits())  
Constructor of a polyhedron with storage reserved for v vertices, h halfedges, and f facets. More...  
void  reserve (size_type v, size_type h, size_type f) 
reserve storage for v vertices, h halfedges, and f facets. More...  
Halfedge_handle  make_tetrahedron () 
a tetrahedron is added to the polyhedral surface. More...  
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 p1 , p2 , p3 , and p4 . More...  
Halfedge_handle  make_triangle () 
a triangle with border edges is added to the polyhedral surface. More...  
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 p1 , p2 , and p3 . More...  
Access Member Functions  
bool  empty () const 
returns true if the polyhedron is empty.  
size_type  size_of_vertices () const 
number of vertices.  
size_type  size_of_halfedges () const 
number of halfedges (incl. border halfedges).  
size_type  size_of_facets () const 
number of facets.  
size_type  capacity_of_vertices () const 
space reserved for vertices.  
size_type  capacity_of_halfedges () const 
space reserved for halfedges.  
size_type  capacity_of_facets () const 
space reserved for facets.  
size_t  bytes () const 
bytes used for the polyhedron.  
size_t  bytes_reserved () const 
bytes reserved for the polyhedron.  
allocator_type  get_allocator () const 
allocator object.  
Vertex_iterator  vertices_begin () 
iterator over all vertices.  
Vertex_iterator  vertices_end () 
pasttheend iterator.  
Halfedge_iterator  halfedges_begin () 
iterator over all halfedges.  
Halfedge_iterator  halfedges_end () 
pasttheend iterator.  
Facet_iterator  facets_begin () 
iterator over all facets (excluding holes).  
Facet_iterator  facets_end () 
pasttheend iterator.  
Edge_iterator  edges_begin () 
iterator over all edges.  
Edge_iterator  edges_end () 
pasttheend iterator.  
Point_iterator  points_begin () 
iterator over all points.  
Point_iterator  points_end () 
pasttheend iterator.  
Plane_iterator  planes_begin () 
iterator over all plane equations.  
Plane_iterator  planes_end () 
pasttheend iterator.  
const Traits &  traits () const 
returns the traits class.  
Combinatorial Predicates  
bool  is_closed () const 
returns true if there are no border edges.  
bool  is_pure_bivalent () const 
returns true if all vertices have exactly two incident edges.  
bool  is_pure_trivalent () const 
returns true if all vertices have exactly three incident edges.  
bool  is_pure_triangle () const 
returns true if all facets are triangles.  
bool  is_pure_quad () const 
returns true if all facets are quadrilaterals.  
bool  is_triangle (Halfedge_const_handle h) const 
true iff the connected component denoted by h is a triangle.  
bool  is_tetrahedron (Halfedge_const_handle h) const 
true iff the connected component denoted by h is a tetrahedron.  
Euler Operators (Combinatorial Modifications)  
The following Euler operations modify consistently the combinatorial structure of the polyhedral surface. The geometry remains unchanged.  
Halfedge_handle  split_facet (Halfedge_handle h, Halfedge_handle g) 
splits the facet incident to h and g into two facets with a new diagonal between the two vertices denoted by h and g respectively. More...  
Halfedge_handle  join_facet (Halfedge_handle h) 
joins the two facets incident to h . More...  
Halfedge_handle  split_vertex (Halfedge_handle h, Halfedge_handle g) 
splits the vertex incident to h and g into two vertices, the old vertex remains and a new copy is created, and connects them with a new edge. More...  
Halfedge_handle  join_vertex (Halfedge_handle h) 
joins the two vertices incident to h . More...  
Halfedge_handle  split_edge (Halfedge_handle h) 
splits the halfedge h into two halfedges inserting a new vertex that is a copy of h>opposite()>vertex() . More...  
Halfedge_handle  flip_edge (Halfedge_handle h) 
performs an edge flip. More...  
Halfedge_handle  create_center_vertex (Halfedge_handle h) 
barycentric triangulation of h>facet() . More...  
Halfedge_handle  erase_center_vertex (Halfedge_handle g) 
reverses create_center_vertex() . More...  
Euler Operators Modifying Genus  
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 j runs on the backside of the three dimensional figure above). More...  
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 . More...  
Modifying Facets and Holes  
Halfedge_handle  make_hole (Halfedge_handle h) 
removes the incident facet of h and changes all halfedges incident to the facet into border edges. More...  
Halfedge_handle  fill_hole (Halfedge_handle h) 
fills a hole with a newly created facet. More...  
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 . More...  
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 . More...  
Erasing  
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. More...  
void  erase_connected_component (Halfedge_handle h) 
removes the vertices, halfedges, and facets that belong to the connected component of h . More...  
unsigned int  keep_largest_connected_components (unsigned int nb_components_to_keep) 
Erases the small connected components and the isolated vertices. More...  
void  clear () 
removes all vertices, halfedges, and facets.  
Operations with Border Halfedges  
Advanced Halfedges incident to a hole are called border halfedges. A halfedge is a border edge if itself or its opposite halfedge are border halfedges. The only requirement to work with border halfedges is that the  
void  normalize_border () 
sorts halfedges such that the nonborder edges precede the border edges. More...  
size_type  size_of_border_halfedges () const 
number of border halfedges. More...  
size_type  size_of_border_edges () const 
number of border edges. More...  
Halfedge_iterator  border_halfedges_begin () 
halfedge iterator starting with the border edges. More...  
Edge_iterator  border_edges_begin () 
edge iterator starting with the border edges. More...  
Miscellaneous  
void  inside_out () 
reverses facet orientations (incl. plane equations if supported).  
bool  is_valid (bool verbose=false, int level=0) const 
returns true if the polyhedral surface is combinatorially consistent. More...  
bool  normalized_border_is_valid (bool verbose=false) const 
returns true if the border halfedges are in normalized representation, which is when enumerating all halfedges with the iterator: The nonborder edges precede the border edges and for border edges, the second halfedge is the border halfedge. More...  
void  delegate (CGAL::Modifier_base< HDS > &m) 
This is an advanced function. More...  
CGAL::Polyhedron_3< Traits >::Polyhedron_3  (  size_type  v, 
size_type  h,  
size_type  f,  
const Traits &  traits = Traits() 

) 
Constructor of a polyhedron with storage reserved for v
vertices, h
halfedges, and f
facets.
The reservation sizes are a hint for optimizing storage allocation.
Halfedge_handle CGAL::Polyhedron_3< Traits >::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.
h>is_border()
, g>is_border()
, h != g
, h>next() != g
, and g
can be reached along the same hole starting with h
.Halfedge_handle CGAL::Polyhedron_3< Traits >::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.
h>is_border()
, g>is_border()
, h != g
, and g
can be reached along the same hole starting with h
.Edge_iterator CGAL::Polyhedron_3< Traits >::border_edges_begin  (  ) 
edge iterator starting with the border edges.
The range [edges_begin(), border_edges_begin()
) denotes all nonborder edges. The range [border_edges_begin(), edges_end()
) denotes all border edges.
normalize_border()
call still valid, see above. Halfedge_iterator CGAL::Polyhedron_3< Traits >::border_halfedges_begin  (  ) 
halfedge iterator starting with the border edges.
The range [halfedges_begin(), border_halfedges_begin()
) denotes all nonborder halfedges. The range [border_halfedges_begin(), halfedges_end()
) denotes all border edges.
normalize_border()
call still valid, see above. Halfedge_handle CGAL::Polyhedron_3< Traits >::create_center_vertex  (  Halfedge_handle  h) 
barycentric triangulation of h>facet()
.
Creates a new vertex, a copy of h>vertex()
, and connects it to each vertex incident to h>facet()
splitting h>facet()
into triangles. h
remains incident to the original facet, all other triangles are copies of this facet. Returns the halfedge h>next()
after the operation, i.e., a halfedge pointing to the new vertex. The time is proportional to the size of the facet.
h
is not a border halfedge.void CGAL::Polyhedron_3< Traits >::delegate  (  CGAL::Modifier_base< HDS > &  m) 
This is an advanced function.
calls the Modifier_base::operator()()
of the modifier m
.
Halfedge_handle CGAL::Polyhedron_3< Traits >::erase_center_vertex  (  Halfedge_handle  g) 
reverses create_center_vertex()
.
Erases the vertex pointed to by g
and all incident halfedges thereby merging all incident facets. Only g>facet()
remains. The neighborhood of g>vertex()
may not be triangulated, it can have larger facets. Returns the halfedge g>prev()
. Thus, the invariant h == erase_center_vertex( create_center_vertex(h))
holds if h
is not a border halfedge. The time is proportional to the sum of the size of all incident facets.
g>vertex()
is a hole. There are at least two distinct facets incident to the facets that are incident to 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.)Supports_removal
must be CGAL::Tag_true
.void CGAL::Polyhedron_3< Traits >::erase_connected_component  (  Halfedge_handle  h) 
removes the vertices, halfedges, and facets that belong to the connected component of h
.
Supports_removal
must be CGAL::Tag_true
. void CGAL::Polyhedron_3< Traits >::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 make_hole(h)
for a more specialized variant.
h>is_border() == false
. Supports_removal
must be CGAL::Tag_true
Halfedge_handle CGAL::Polyhedron_3< Traits >::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
.
h.is_border()
. Halfedge_handle CGAL::Polyhedron_3< Traits >::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.
h != Halfedge_handle()
and both facets incident to h
are triangles. bool CGAL::Polyhedron_3< Traits >::is_valid  (  bool  verbose = false , 
int  level = 0 

)  const 
returns true
if the polyhedral surface is combinatorially consistent.
If verbose
is true
, statistics are printed to cerr
. For level == 1
the normalization of the border edges is checked too. This method checks in particular level 3 of CGAL::Halfedge_data_structure_decorator::is_valid()
from CGAL::HalfedgeDS_const_decorator
and that each facet is at least a triangle and that the two incident facets of a nonborder edge are distinct.
Halfedge_handle CGAL::Polyhedron_3< Traits >::join_facet  (  Halfedge_handle  h) 
joins the two facets incident to h
.
The facet incident to h>opposite()
gets removed. Both facets might be holes. Returns the predecessor of h
around the facet. The invariant 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 h>prev()
.
h
is at least three (no antennas).Supports_removal
must be CGAL::Tag_true
.Halfedge_handle CGAL::Polyhedron_3< Traits >::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 join_loop( h, split_loop( h, i, j))
returns h
and keeps the polyhedron unchanged.
h
and g
are different and have equal degree (i.e., number of edges). Supports_removal
must be CGAL::Tag_true
.Halfedge_handle CGAL::Polyhedron_3< Traits >::join_vertex  (  Halfedge_handle  h) 
joins the two vertices incident to h
.
The vertex denoted by h>opposite()
gets removed. Returns the predecessor of h
around the vertex, i.e., h>opposite()>prev()
. The invariant 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 h>prev()
and h>opposite()>prev()
.
h
is at least four (no multiedges).Supports_removal
must be CGAL::Tag_true
.unsigned int CGAL::Polyhedron_3< Traits >::keep_largest_connected_components  (  unsigned int  nb_components_to_keep) 
Erases the small connected components and the isolated vertices.
Keep nb_components_to_keep
largest connected components. Returns the number of connected components erased (ignoring isolated vertices).
The polyhedron type must support vertices, halfedges, and removal operations.
Halfedge_handle CGAL::Polyhedron_3< Traits >::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 erase_facet(h)
for a more generalized variant.
Supports_removal
must be CGAL::Tag_true
. Halfedge_handle CGAL::Polyhedron_3< Traits >::make_tetrahedron  (  ) 
a tetrahedron is added to the polyhedral surface.
Returns a halfedge of the tetrahedron.
Halfedge_handle CGAL::Polyhedron_3< Traits >::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 p1
, p2
, p3
, and p4
.
Returns that halfedge of the tetrahedron which incident vertex is initialized to p1
. The incident vertex of the next halfedge is p2
, and the vertex thereafter is p3
. The remaining fourth vertex is initialized to p4
.
Halfedge_handle CGAL::Polyhedron_3< Traits >::make_triangle  (  ) 
a triangle with border edges is added to the polyhedral surface.
Returns a nonborder halfedge of the triangle.
Halfedge_handle CGAL::Polyhedron_3< Traits >::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 p1
, p2
, and p3
.
Returns that nonborder halfedge of the triangle which incident vertex is initialized to p1
. The incident vertex of the next halfedge is p2
, and the vertex thereafter is p3
.
void CGAL::Polyhedron_3< Traits >::normalize_border  (  ) 
sorts halfedges such that the nonborder 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.
bool CGAL::Polyhedron_3< Traits >::normalized_border_is_valid  (  bool  verbose = false )  const 
returns true
if the border halfedges are in normalized representation, which is when enumerating all halfedges with the iterator: The nonborder edges precede the border edges and for border edges, the second halfedge is the border halfedge.
The halfedge iterator border_halfedges_begin()
denotes the first border edge. If verbose
is true
, statistics are printed to cerr
.
void CGAL::Polyhedron_3< Traits >::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 capacity
is already greater than the requested size nothing happens. If the capacity
changes all iterators and circulators might invalidate.
size_type CGAL::Polyhedron_3< Traits >::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 size_of_border_halfedges()
.
normalize_border()
call still valid, see above. size_type CGAL::Polyhedron_3< Traits >::size_of_border_halfedges  (  )  const 
number of border halfedges.
normalize_border()
call still valid, see above. Halfedge_handle CGAL::Polyhedron_3< Traits >::split_edge  (  Halfedge_handle  h) 
splits the halfedge h
into two halfedges inserting a new vertex that is a copy of h>opposite()>vertex()
.
Is equivalent to split_vertex( h>prev(), h>opposite())>opposite()
. The call of prev()
can make this method slower than a direct call of split_vertex()
if the previous halfedge is already known and computing it would be costly when the halfedge data structure does not support the prev()
member function. Returns the new halfedge hnew
pointing to the inserted vertex. The new halfedge is followed by the old halfedge, i.e., hnew>next() == h
.
Halfedge_handle CGAL::Polyhedron_3< Traits >::split_facet  (  Halfedge_handle  h, 
Halfedge_handle  g  
) 
splits the facet incident to h
and g
into two facets with a new diagonal between the two vertices denoted by h
and g
respectively.
The second (new) facet is a copy of the first facet. Returns 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 h
to g
around the facet.
h
and g
are incident to the same facet. h != g
(no loops). h>next() != g
and g>next() != h
(no multiedges).Halfedge_handle CGAL::Polyhedron_3< Traits >::split_loop  (  Halfedge_handle  h, 
Halfedge_handle  i,  
Halfedge_handle  j  
) 
cuts the polyhedron into two parts along the cycle (h,i,j)
(edge 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 hopposite()
.
h
, i
, j
denote distinct, consecutive vertices of the polyhedron and form a cycle: i.e., h>vertex() == i>opposite()>vertex()
, \( \ldots\) , j>vertex() == h>opposite()>vertex()
. The six facets incident to (h,i,j)
are all distinct.Halfedge_handle CGAL::Polyhedron_3< Traits >::split_vertex  (  Halfedge_handle  h, 
Halfedge_handle  g  
) 
splits the vertex incident to h
and g
into two vertices, the old vertex remains and a new copy is created, and connects them with a new edge.
Let hnew
be 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 hnew
, g>next()>opposite()
, \( \ldots\) , h
remains around the old vertex, while the halfedge sequence hnew>opposite()
, h>next()>opposite()
(before the split), \( \ldots\) , g
is regrouped around the new vertex. The split returns hnew
, i.e., the new halfedge incident to the old vertex. The time is proportional to the distance from h
to g
around the vertex.
h
and g
are incident to the same vertex. h != g
(antennas are not allowed).split_vertex(h,h>next()>opposite())
which is equivalent to an edge split of the halfedge h>next()
that creates a new vertex on the halfedge h>next()
. See also split_edge(h)
below.