\begin{ccRefFunction}{intersection}

Depending on which \cgal\ \ccHtmlNoLinksFrom{kernel} is used,
different versions of this global function are available. This is
described below.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{With the basic 2D and 3D Kernel} (see Chapter~\ref{chapter-kernel-23})

\ccInclude{CGAL/intersections.h}

\ccUnchecked{
\ccRefLabel{Kernel::intersection}
\ccFunction{Object intersection(Type1<Kernel> obj1, Type2<Kernel> obj2);}
{Two objects \ccStyle{obj1} and \ccStyle{obj2} intersect if there is a point
\ccStyle{p} that is part of both \ccStyle{obj1} and \ccStyle{obj2}.
The \ccHtmlNoLinksFrom{intersection} region of those two objects is defined as the set of all
points \ccStyle{p} that are part of both \ccStyle{obj1} and \ccStyle{obj2}.
Note that for objects like triangles and polygons that enclose a
bounded region, this region is considered part of the object.
If a segment lies completely inside a triangle, then those two objects
intersect and the \ccHtmlNoLinksFrom{intersection} region is the complete segment.
}}

The possible value for types \ccStyle{Type1} and \ccStyle{Type2} and the
possible return values wrapped in \ccc{Object} are the
following:


\begin{ccTexOnly}
\begin{longtable}[c]{|l|l|l|}
%\caption{All available intersection computations}\\
\multicolumn{3}{l}{\sl \ \ }
\endfirsthead
\multicolumn{3}{l}{\sl continued}
\endhead
\hline
type A & type B & \parbox{4 cm}{\vspace{1 mm}{return type}} \\
\hline
\ccStyle{Line_2} & \ccStyle{Line_2} & \parbox{4 cm}{\vspace{1 mm}
    \ccStyle{Point_2}
    \\ \ccStyle{Line_2}
  \vspace{1 mm}} \\
\hline
\ccStyle{Segment_2} & \ccStyle{Line_2} & \parbox{4 cm}{\vspace{1 mm}
    \ccStyle{Point_2}
    \\ \ccStyle{Segment_2}
  \vspace{1 mm}} \\
\hline
\ccStyle{Segment_2} & \ccStyle{Segment_2} & \parbox{4 cm}{\vspace{1 mm}
    \ccStyle{Point_2}
    \\ \ccStyle{Segment_2}
  \vspace{1 mm}} \\
\hline
\ccStyle{Ray_2} & \ccStyle{Line_2} & \parbox{4 cm}{\vspace{1 mm}
    \ccStyle{Point_2}
    \\ \ccStyle{Ray_2}
  \vspace{1 mm}} \\
\hline
\ccStyle{Ray_2} & \ccStyle{Segment_2} & \parbox{4 cm}{\vspace{1 mm}
    \ccStyle{Point_2}
    \\ \ccStyle{Segment_2}
  \vspace{1 mm}} \\
\hline
\ccStyle{Ray_2} & \ccStyle{Ray_2} & \parbox{4 cm}{\vspace{1 mm}
    \ccStyle{Point_2}
    \\ \ccStyle{Segment_2}
    \\ \ccStyle{Ray_2}
  \vspace{1 mm}} \\
\hline
\ccStyle{Triangle_2} & \ccStyle{Line_2} & \parbox{4 cm}{\vspace{1 mm}
    \ccStyle{Point_2}
    \\ \ccStyle{Segment_2}
  \vspace{1 mm}} \\
\hline
\ccStyle{Triangle_2} & \ccStyle{Segment_2} & \parbox{4 cm}{\vspace{1 mm}
    \ccStyle{Point_2}
    \\ \ccStyle{Segment_2}
  \vspace{1 mm}} \\
\hline
\ccStyle{Triangle_2} & \ccStyle{Ray_2} & \parbox{4 cm}{\vspace{1 mm}
    \ccStyle{Point_2}
    \\ \ccStyle{Segment_2}
  \vspace{1 mm}} \\
\hline
\ccStyle{Triangle_2} & \ccStyle{Triangle_2} & \parbox{4 cm}{\vspace{1 mm}
    \ccStyle{Point_2}
    \\ \ccStyle{Segment_2}
    \\ \ccStyle{Triangle_2}
    \\ \ccStyle{std::vector<Point_2>}
  \vspace{1 mm}} \\
\hline
\ccStyle{Iso_rectangle_2} & \ccStyle{Line_2} & \parbox{4 cm}{\vspace{1 mm}
    \ccStyle{Point_2}
    \\ \ccStyle{Segment_2}
  \vspace{1 mm}} \\
\hline
\ccStyle{Iso_rectangle_2} & \ccStyle{Segment_2} & \parbox{4 cm}{\vspace{1 mm}
    \ccStyle{Point_2}
    \\ \ccStyle{Segment_2}
  \vspace{1 mm}} \\
\hline
\ccStyle{Iso_rectangle_2} & \ccStyle{Ray_2} & \parbox{4 cm}{\vspace{1 mm}
    \ccStyle{Point_2}
    \\ \ccStyle{Segment_2}
  \vspace{1 mm}} \\
\hline
\ccStyle{Iso_rectangle_2} & \ccStyle{Iso_rectangle_2} & \parbox{4 cm}{\vspace{1 mm}
    \ccStyle{Iso_rectangle_2}
  \vspace{1 mm}} \\
\hline
{\ccStyle{Line_3}} & {\ccStyle{Line_3}} & \parbox{4 cm}{\vspace{1 mm}
\ccStyle{Point_3} \\
 \ccStyle{Line_3} 
\vspace{1 mm}} \\
\hline
{\ccStyle{Plane_3}} & {\ccStyle{Line_3}} & \parbox{4 cm}{\vspace{1 mm}
\ccStyle{Point_3} \\
 \ccStyle{Line_3} 
\vspace{1 mm}} \\
\hline
{\ccStyle{Plane_3}} & {\ccStyle{Ray_3}} & \parbox{4 cm}{\vspace{1 mm}
\ccStyle{Point_3} \\
 \ccStyle{Ray_3}
\vspace{1 mm}}  \\
\hline
{\ccStyle{Plane_3}} & {\ccStyle{Segment_3}} & \parbox{4 cm}{\vspace{1 mm}
\ccStyle{Point_3} \\
 \ccStyle{Segment_3}
\vspace{1 mm}}  \\
\hline
{\ccStyle{Plane_3}} & {\ccStyle{Plane_3}} & \parbox{4 cm}{\vspace{1 mm}
\ccStyle{Line_3} \\
 \ccStyle{Plane_3}
\vspace{1 mm}}  \\
\hline
{\ccStyle{Plane_3}} & {\ccStyle{Sphere_3}} & \parbox{4 cm}{\vspace{1 mm}
\ccStyle{Point_3} \\
 \ccStyle{Circle_3}
\vspace{1 mm}}  \\
\hline
{\ccStyle{Sphere_3}} & {\ccStyle{Sphere_3}} & \parbox{4 cm}{\vspace{1 mm}
\ccStyle{Point_3} \\
 \ccStyle{Circle_3} \\
 \ccStyle{Sphere_3}
\vspace{1 mm}}  \\
\hline
\end{longtable}
\end{ccTexOnly}

\begin{ccHtmlOnly}
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR> <TH> type A </TH>
 <TH> type B </TH>
 <TH> return type </TH>
</TR>
<TR>
    <TD VALIGN="CENTER" > Line_2 </TD>
    <TD VALIGN="CENTER" > Line_2 </TD>
    <TD><TABLE>
	<TR><TD>Point_2</TD></TR>
	<TR><TD>Line_2</TD></TR>
        </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Segment_2 </TD>
    <TD VALIGN="CENTER" > Line_2 </TD>
    <TD><TABLE>
	<TR><TD>Point_2</TD></TR>
	<TR><TD>Segment_2</TD></TR>
      </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Segment_2 </TD>
    <TD VALIGN="CENTER" > Segment_2 </TD>
    <TD><TABLE>
	<TR><TD>Point_2</TD></TR>
	<TR><TD>Segment_2</TD></TR>
      </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Ray_2 </TD>
    <TD VALIGN="CENTER" > Line_2 </TD>
    <TD><TABLE>
	<TR><TD>Point_2</TD></TR>
	<TR><TD>Ray_2</TD></TR>
      </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Ray_2 </TD>
    <TD VALIGN="CENTER" > Segment_2 </TD>
    <TD><TABLE>
	<TR><TD>Point_2</TD></TR>
	<TR><TD>Segment_2</TD></TR>
      </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Ray_2 </TD>
    <TD VALIGN="CENTER" > Ray_2 </TD>
    <TD><TABLE>
	<TR><TD>Point_2</TD></TR>
	<TR><TD>Segment_2</TD></TR>
	<TR><TD>Ray_2</TD></TR>
      </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Triangle_2 </TD>
    <TD VALIGN="CENTER" > Line_2 </TD>
    <TD><TABLE>
	<TR><TD>Point_2</TD></TR>
	<TR><TD>Segment_2</TD></TR>
      </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Triangle_2 </TD>
    <TD VALIGN="CENTER" > Segment_2 </TD>
    <TD><TABLE>
	<TR><TD>Point_2</TD></TR>
	<TR><TD>Segment_2</TD></TR>
      </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Triangle_2 </TD>
    <TD VALIGN="CENTER" > Ray_2 </TD>
    <TD><TABLE>
	<TR><TD>Point_2</TD></TR>
	<TR><TD>Segment_2</TD></TR>
      </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Triangle_2 </TD>
    <TD VALIGN="CENTER" > Triangle_2 </TD>
    <TD><TABLE>
	<TR><TD>Point_2</TD></TR>
	<TR><TD>Segment_2</TD></TR>
	<TR><TD>Triangle_2</TD></TR>
	<TR><TD>std::vector&lt;Point_2&gt;</TD></TR>
      </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Iso_rectangle_2 </TD>
    <TD VALIGN="CENTER" > Line_2 </TD>
    <TD><TABLE>
	<TR><TD>Point_2</TD></TR>
	<TR><TD>Segment_2</TD></TR>
      </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Iso_rectangle_2 </TD>
    <TD VALIGN="CENTER" > Segment_2 </TD>
    <TD><TABLE>
	<TR><TD>Point_2</TD></TR>
	<TR><TD>Segment_2</TD></TR>
      </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Iso_rectangle_2 </TD>
    <TD VALIGN="CENTER" > Ray_2 </TD>
    <TD><TABLE>
	<TR><TD>Point_2</TD></TR>
	<TR><TD>Segment_2</TD></TR>
      </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Iso_rectangle_2 </TD>
    <TD VALIGN="CENTER" > Iso_rectangle_2 </TD>
    <TD><TABLE>
	<TR><TD>Iso_rectangle_2</TD></TR>
      </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Line_3 </TD>
    <TD VALIGN="CENTER" > Line_3 </TD>
    <TD><TABLE>
	<TR><TD>Point_3</TD></TR>
	<TR><TD>Line_3</TD></TR>
        </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Plane_3 </TD>
    <TD VALIGN="CENTER" > Line_3 </TD>
    <TD><TABLE>
	<TR><TD>Point_3</TD></TR>
	<TR><TD>Line_3</TD></TR>
        </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Plane_3 </TD>
    <TD VALIGN="CENTER" > Ray_3 </TD>
    <TD><TABLE>
	<TR><TD>Point_3</TD></TR>
	<TR><TD>Ray_3</TD></TR>
        </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Plane_3 </TD>
    <TD VALIGN="CENTER" > Segment_3 </TD>
    <TD><TABLE>
	<TR><TD>Point_3</TD></TR>
	<TR><TD>Segment_3</TD></TR>
        </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Plane_3 </TD>
    <TD VALIGN="CENTER" > Plane_3 </TD>
    <TD><TABLE>
	<TR><TD>Line_3</TD></TR>
	<TR><TD>Plane_3</TD></TR>
        </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Plane_3 </TD>
    <TD VALIGN="CENTER" > Sphere_3 </TD>
    <TD><TABLE>
	<TR><TD>Point_3</TD></TR>
	<TR><TD>Circle_3</TD></TR>
        </TABLE></TD>
</TR>
<TR>
    <TD VALIGN="CENTER" > Sphere_3 </TD>
    <TD VALIGN="CENTER" > Sphere_3 </TD>
    <TD><TABLE>
	<TR><TD>Point_3</TD></TR>
	<TR><TD>Circle_3</TD></TR>
	<TR><TD>Sphere_3</TD></TR>
        </TABLE></TD>
</TR>
</TABLE>
</DIV>
\end{ccHtmlOnly}

There is also an intersection function between 3 planes.

\ccFunction{Object intersection(const Plane_3<Kernel>& pl1,
                                const Plane_3<Kernel>& pl2,
                                const Plane_3<Kernel>& pl3);}
{returns the intersection of 3 planes, which can be either a point, a line,
a plane, or empty.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{With the 2D Circular Kernel} (see Chapter~\ref{chapter-circular-kernel}) 

\ccInclude{CGAL/Circular_kernel_intersections.h}

If this kernel is used, in addition to the function and the
combination of 2D types described above, another version of the function
is provided.

Since both the number of intersections, if any, and their type,
depend on the arguments, the function returns an output
iterator on \ccc{Object}'s, as presented below. 

\ccFunction{template < class OutputIterator >
OutputIterator
intersection(const  Type1 &obj1, const Type2 &obj2,
		OutputIterator intersections);}
{Copies in the output iterator the intersection elements between the
two objects. \ccc{intersections} iterates on
elements of type \ccc{CGAL::Object}, in lexicographic order,}

where \ccStyle{Type1} and \ccStyle{Type2} can both be either
\begin{itemize}
\item {} \ccStyle{Line_2<CircularKernel>} or
\item {} \ccStyle{Line_arc_2<CircularKernel>} or
\item {} \ccStyle{Circle_2<CircularKernel>} or
\item {} \ccStyle{Circular_arc_2<CircularKernel>}.
\end{itemize} 

Depending on the types \ccStyle{Type1} and \ccStyle{Type2}, these
elements can be assigned to
\begin{itemize}
\item {} \ccStyle{std::pair<Circular_arc_point_2<CircularKernel>, unsigned>},
where the unsigned integer is the multiplicity of the corresponding
intersection point between \ccc{obj1} and \ccc{obj2},
\item {} \ccStyle{Circular_arc_2<CircularKernel>} in case of an overlap of 
two circular arcs,
\item {} \ccStyle{Line_arc_2<CircularKernel>} in case of an overlap of two 
line segments or
\item {} \ccStyle{Line_2<CircularKernel>} or 
\ccStyle{Circle_2<CircularKernel>} in case of two equal input lines or circles.
\end{itemize} 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{With the 3D Spherical Kernel} (see Chapter~\ref{chapter-spherical-kernel}) 

\ccInclude{CGAL/Spherical_kernel_intersections.h}

If this kernel is used, in addition to the function and the
combination of 3D types described above, two other versions of the function
are provided.

Since both the number of intersections, if any, and their type,
depend on the arguments, the functions return an output
iterator on \ccStyle{Object}'s, as presented below. 

The \textbf{first function} is:

\ccFunction{template < class OutputIterator >
OutputIterator
intersection(const  Type1 &obj1, const Type2 &obj2,
		OutputIterator intersections);}
{Copies in the output iterator the intersection elements between the
two objects. \ccc{intersections} iterates on
elements of type \ccStyle{CGAL::Object}, in lexicographic order,
when this ordering is defined on the computed objects,}

where \ccStyle{Type1} and \ccStyle{Type2} can both be either:
\begin{itemize}
\item {} \ccStyle{Sphere_3<SphericalKernel>},
\item {} \ccStyle{Plane_3<SphericalKernel>},
\item {} \ccStyle{Line_3<SphericalKernel>},
\item {} \ccStyle{Circle_3<SphericalKernel>},
\item {} \ccStyle{Line_arc_3<SphericalKernel>} or
\item {} \ccStyle{Circular_arc_3<SphericalKernel>},
\end{itemize} 

and depending on the types \ccStyle{Type1} and \ccStyle{Type2}, the computed 
\ccStyle{CGAL::Object}s can be assigned to 
\begin{itemize}
\item {} \ccStyle{std::pair<Circular_arc_point_3<SphericalKernel>, unsigned>},
where the unsigned integer is the multiplicity of the corresponding
intersection point between \ccc{obj1} and \ccc{obj2},
\item {} \ccStyle{Type1}, when \ccStyle{Type1} and \ccStyle{Type2} are equal, 
and if the two objets \ccc{obj1} and \ccc{obj2} are equal,
\item {} \ccStyle{Line_3<SphericalKernel>} or 
\ccStyle{Circle_3<SphericalKernel>} when \ccStyle{Type1} and \ccStyle{Type2} 
are two-dimensional objets intersecting along a curve (2 planes, or 2
spheres, or one plane and one sphere),
\item {} \ccStyle{Circular_arc_3<SphericalKernel>} in case of an overlap of 
two circular arcs or
\item {} \ccStyle{Line_arc_3<SphericalKernel>} in case of an overlap of two 
line segments. 
\end{itemize} 

The \textbf{second function} is:

\ccFunction{template < class OutputIterator >
OutputIterator
intersection(const Type1 &obj1, const Type2 &obj2, const Type3 &obj3,
	OutputIterator intersections);}
{Copies in the output iterator the intersection elements between the
three objects. \ccc{intersections} iterates on
elements of type \ccStyle{CGAL::Object}, in lexicographic order 
when this ordering is defined on the computed objects}

where \ccStyle{Type1}, \ccStyle{Type2} and \ccStyle{Type3}
can be either
\begin{itemize}
\item {} \ccStyle{Sphere_3<SphericalKernel>} or
\item {} \ccStyle{Plane_3<SphericalKernel>}
\end{itemize}

and depending of these types, the computed \ccStyle{CGAL::Object}s can be 
assigned to 
\begin{itemize}
\item {} \ccStyle{std::pair<Circular_arc_point_3<SphericalKernel>, unsigned>},
where the unsigned integer is the multiplicity of the corresponding
intersection point,
\item {} \ccStyle{Circle_3<SphericalKernel>} or
\item {} \ccStyle{Type1}, when \ccStyle{Type1}, \ccStyle{Type2} and 
\ccc{Type3} are equal, and if the three objets \ccc{obj1} and \ccc{obj2} 
and \ccc{obj3} are equal.
\end{itemize} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ccExample

The following example demonstrates the most common use of 
\ccc{intersection} routines with the basic 2D and 3D Kernels.
\begin{verbatim}
#include <CGAL/intersections.h>

void foo(CGAL::Segment_2<Kernel> seg, CGAL::Line_2<Kernel> line)
{
    CGAL::Object result = CGAL::intersection(seg, line);
    if (const CGAL::Point_2<Kernel> *ipoint = CGAL::object_cast<CGAL::Point_2<Kernel> >(&result)) {
\end{verbatim}%
\ccHtmlLinksOff%
\begin{verbatim}
        // handle the point intersection case with *ipoint.
\end{verbatim}%
\ccHtmlLinksOn%
\begin{verbatim}
    } else
        if (const CGAL::Segment_2<Kernel> *iseg = CGAL::object_cast<CGAL::Segment_2<Kernel> >(&result)) {
\end{verbatim}%
\ccHtmlLinksOff%
\begin{verbatim}
            // handle the segment intersection case with *iseg.
\end{verbatim}%
\ccHtmlLinksOn%
\begin{verbatim}
        } else {
\end{verbatim}%
\ccHtmlLinksOff%
\begin{verbatim}
            // handle the no intersection case.
        }
}
\end{verbatim}%
\ccHtmlLinksOn%

Examples illustrating the use of this function in the case of the 2D
Circular Kernel and the 3D Spherical Kernel are presented respectively
in Chapters~\ref{chapter-circular-kernel}
and~\ref{chapter-spherical-kernel}.

\ccSeeAlso
% \ccRefIdfierPage{CGAL::assign} \\
\ccRefIdfierPage{CGAL::do_intersect} \\ 
\ccRefIdfierPage{CGAL::Object} \\

\end{ccRefFunction}
