\subsubsection{Connectivity Operators}

The polyhedron provides a series of atomic operators to modify the
connectivity of the polyhedral surface:
\begin{itemize}
\item split or join of two facets,
\item split or join of two vertices,
\item split or join of two loops,
\item split of an edge.
\end{itemize}

Furthermore, more operators are provided to work with surfaces with
boundaries, to create or delete holes, add a facet to the border,
etc. We refere to the references manual for precise definitions and
illustratives figures\footnote{See
\href{http://www.cgal.org/Manual/doc_html/basic_lib/Polyhedron/Chapter_main.html}{Euler 
operators}}.

\subsubsection{Incremental Builder}
\label{sec:builder}

The utility class \verb+Polyhedron_incremental_builder_3+ helps in
creating polyhedral surfaces from a list of points followed by a list
of facets that are represented as indices into the point list. This is
particularly useful for implementing file reader for common file
formats. In Section~\ref{sec:subdivision_builder}, we use the
incremental builder to implement the quad-triangle subdivision
scheme.\\

In the following example, the incremental builder is used to create a
simple triangle. \verb+Build_triangle+ is such a function object
derived from \verb+Modifier_base<HalfedgeDS>+. The
\verb+delegate()+ member function of the polyhedron accepts this function
object and calls its \verb+operator()+ with a reference to its
internally used halfedge data structure. Thus, this member function in
\verb+Build_triangle+ can create the triangle in the 
halfedge data structure.

{ \scriptsize
\begin{verbatim}
// examples/Polyhedron/polyhedron_prog_incr_builder.C

#include <CGAL/Cartesian.h>
#include <CGAL/Polyhedron_incremental_builder_3.h>
#include <CGAL/Polyhedron_3.h>

// A modifier creating a triangle with 
// the incremental builder.

template <class HDS>
class Build_triangle
  : public CGAL::Modifier_base<HDS> 
{
public:
  Build_triangle() {}

  void operator()(HDS& hds) 
  {
    // Postcondition: `hds' is a valid polyhedral surface.
    CGAL::Polyhedron_incremental_builder_3<HDS> B(hds, true);
    B.begin_surface(3, 1, 6);
    typedef typename HDS::Vertex   Vertex;
    typedef typename Vertex::Point Point;
    B.add_vertex(Point(0, 0, 0));
    B.add_vertex(Point(1, 0, 0));
    B.add_vertex(Point(0, 1, 0));
    B.begin_facet();
    B.add_vertex_to_facet(0);
    B.add_vertex_to_facet(1);
    B.add_vertex_to_facet(2);
    B.end_facet();
    B.end_surface();
  }
};

typedef CGAL::Cartesian<double>     Kernel;
typedef CGAL::Polyhedron_3<Kernel>  Polyhedron;
typedef Polyhedron::HalfedgeDS      HalfedgeDS;

Polyhedron P;
Build_triangle<HalfedgeDS> triangle;
P.delegate(triangle);
CGAL_assertion(P.is_triangle(P.halfedges_begin()));

\end{verbatim}}