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CGAL 4.8 - 2D and 3D Linear Geometry Kernel

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Functions
CGAL::intersection() (2D/3D Linear Kernel)
CGAL::intersection()

##

## Functions
template<typename Kernel >
cpp11::result_of < Kernel::Intersect_23(Type1, Type2)>::type | CGAL::intersection (Type1< Kernel > obj1, Type2< Kernel > obj2)
| Two objects `obj1` and `obj2` intersect if there is a point `p` that is part of both `obj1` and `obj2`. More...
template<typename Kernel >
boost::optional < boost::variant< Point_3, Line_3, Plane_3 > > | CGAL::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 a point, a line, a plane, or empty.

## Function Documentation

template<typename Kernel >

cpp11::result_of<Kernel::Intersect_23(Type1, Type2)>::type CGAL::intersection | ( | Type1< Kernel > | obj1,
|  | Type2< Kernel > | obj2
| ) |  |

Two objects `obj1` and `obj2` intersect if there is a point `p` that is part of both `obj1` and `obj2`.

The intersection region of those two objects is defined as the set of all points `p` that are part of both `obj1` and `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 intersection region is the complete segment.

Here, `Intersect_23` means either `Intersect_2` or `Intersect_3`, depending on the arguments.

The following tables give the possible values for `Type1` and `Type2`.

2D Intersections

The return type can be obtained through `CGAL::cpp11::result_of<Kernel::Intersect_2(A, B)>::type`. It is equivalent to `boost::optional< boost::variant< T... > >`, the last column in the table providing the template parameter pack.

Type1 | Type2 | Return Type: `T...`
--- | --- | ---
Iso_rectangle_2 | Iso_rectangle_2 | Iso_rectangle_2
Iso_rectangle_2 | Line_2 | Point_2, or Segment_2
Iso_rectangle_2 | Ray_2 | Point_2, or Segment_2
Iso_rectangle_2 | Segment_2 | Point_2, or Segment_2
Iso_rectangle_2 | Triangle_2 | Point_2, or Segment_2, or Triangle_2, or std::vector<Point_2>
Line_2 | Line_2 | Point_2, or Line_2
Line_2 | Ray_2 | Point_2, or Ray_2
Line_2 | Segment_2 | Point_2, or Segment_2
Line_2 | Triangle_2 | Point_2, or Segment_2
Ray_2 | Ray_2 | Point_2, or Segment_2, or Ray_2
Ray_2 | Segment_2 | Point_2, or Segment_2
Ray_2 | Triangle_2 | Point_2, or Segment_2
Segment_2 | Segment_2 | Point_2, or Segment_2
Segment_2 | Triangle_2 | Point_2, or Segment_2
Triangle_2 | Triangle_2 | Point_2, or Segment_2, or Triangle_2, or std::vector<Point_2>

3D Intersections

The return type can be obtained through `CGAL::cpp11::result_of<Kernel::Intersect_3(A, B)>::type`. It is equivalent to `boost::optional< boost::variant< T... > >`, the last column in the table providing the template parameter pack.

Type1 | Type2 | Return Type: `T...`
--- | --- | ---
Line_3 | Line_3 | Point_3, or Line_3
Line_3 | Plane_3 | Point_3, or Line_3
Line_3 | Ray_3 | Point_3, or Ray_3
Line_3 | Segment_3 | Point_3, or Segment_3
Line_3 | Triangle_3 | Point_3, or Segment_3
Plane_3 | Plane_3 | Line_3, or Plane_3
Plane_3 | Ray_3 | Point_3, or Ray_3
Plane_3 | Segment_3 | Point_3, or Segment_3
Plane_3 | Sphere_3 | Point_3, or Circle_3
Plane_3 | Triangle_3 | Point_3, or Segment_3, or Triangle_3
Ray_3 | Ray_3 | Point_3, or Ray_3, or Segment_3
Ray_3 | Segment_3 | Point_3, or Segment_3
Ray_3 | Triangle_3 p | Point_3, or Segment_3
Segment_3 | Segment_3 | Point_3, or Segment_3
Segment_3 | Triangle_3 | Point_3, or Segment_3
Sphere_3 | Sphere_3 | Point_3, or Circle_3, or Sphere_3
Triangle_3 | Triangle_3 | Point_3, or Segment_3, or Triangle_3, or std::vector < Point_3 >

Examples

The following examples demonstrate the most common use of `intersection()` functions with the 2D and 3D Linear Kernel.

In the first two examples we intersect a segment and a line. The result type can be obtained with `CGAL::cpp11::result_of`. It looks simpler if you use a C++ compiler which supports `auto`, but you must anyways know that the result type is a `boost::optional<boost::variant<..> >`, in order to unpack the point or segment.

`boost::optional` comes in as there might be no intersection. `boost::variant` comes in as, if there is an intersection, it is either a point or a segment.

As explained in the boost manual pages for `boost::variant`, there are two ways to access the variants. The first examples uses `boost::get`.

File Kernel_23/intersection_get.cpp

#include <CGAL/Exact_predicates_exact_constructions_kernel.h>
#include <CGAL/intersections.h>
typedef CGAL::Exact_predicates_exact_constructions_kernel K;
typedef K::Point_2 Point_2;
typedef K::Segment_2 Segment_2;
typedef K::Line_2 Line_2;
typedef K::Intersect_2 Intersect_2;
int main()
{
Segment_2 seg(Point_2(0,0), Point_2(2,2));
Line_2 lin(1,-1,0);
CGAL::cpp11::result_of<Intersect_2(Segment_2, Line_2)>::type
result = intersection(seg, lin);
if (result) {
if (const Segment_2* s = boost::get<Segment_2>(&*result)) {
std::cout << *s << std::endl;
} else {
const Point_2* p = boost::get<Point_2 >(&*result);
std::cout << *p << std::endl;
}
}
return 0;
}

The second example uses `boost::apply_visitor`.

File Kernel_23/intersection_visitor.cpp

#include <CGAL/Exact_predicates_exact_constructions_kernel.h>
#include <CGAL/intersections.h>
typedef CGAL::Exact_predicates_exact_constructions_kernel K;
typedef K::Point_2 Point_2;
typedef K::Segment_2 Segment_2;
typedef K::Line_2 Line_2;
typedef K::Intersect_2 Intersect_2;
struct Intersection_visitor {
typedef void result_type;
void operator()(const Point_2& p) const
{
std::cout << p << std::endl;
}
void operator()(const Segment_2& s) const
{
std::cout << s << std::endl;
}
};
int main()
{
Segment_2 seg(Point_2(0,0), Point_2(1,1));
Line_2 lin(1,-1,0);
// with C++11 support
// auto result = intersection(seg, lin);
// without C++11
CGAL::cpp11::result_of<Intersect_2(Segment_2, Line_2)>::type
result = intersection(seg, lin);
if (result) {
boost::apply_visitor(Intersection_visitor(), *result);
} else {
// no intersection
}
return 0;
}

A third example shows the use of the intersection function as a plain function call and with `Dispatch_output_iterator`, combined with a standard library algorithm.

File Kernel_23/intersections.cpp

#include <CGAL/Exact_predicates_exact_constructions_kernel.h>
#include <CGAL/iterator.h>
#include <CGAL/point_generators_2.h>
#include <boost/bind.hpp>
typedef CGAL::Exact_predicates_exact_constructions_kernel K;
typedef K::Point_2 Point;
typedef K::Segment_2 Segment;
typedef CGAL::Creator_uniform_2<double,Point> Pt_creator;
typedef CGAL::Random_points_on_segment_2<Point,Pt_creator> P1;
typedef CGAL::Random_points_on_circle_2<Point,Pt_creator> P2;
typedef CGAL::Creator_uniform_2< Point, Segment> Seg_creator;
typedef CGAL::Join_input_iterator_2< P1, P2, Seg_creator> Seg_iterator;
struct Intersector{
typedef CGAL::cpp11::result_of<K::Intersect_2(Segment,Segment)>::type result_type;
const Segment& s;
K::Intersect_2 intersect;
Intersector(const Segment& seg): s(seg) {}
result_type
operator() ( const Segment& other) const
{
return intersect(s, other);
}
};
int main()
{
std::vector<Segment> input;
// Prepare point generator for the horizontal segment, length 200.
P1 p1( Point(-100,0), Point(100,0));
// Prepare point generator for random points on circle, radius 250.
P2 p2( 250);
// Create segments.
Seg_iterator g( p1, p2);
CGAL::cpp11::copy_n( g, 200, std::back_inserter(input));
// splitting results with Dispatch_output_iterator
std::vector<Point> points;
std::vector<Segment> segments;
typedef CGAL::Dispatch_output_iterator<
CGAL::cpp11::tuple<Point,Segment>, CGAL::cpp11::tuple< std::back_insert_iterator<std::vector<Point> >,
std::back_insert_iterator<std::vector<Segment> > > >
Dispatcher;
Dispatcher disp = CGAL::dispatch_output<Point,Segment>( std::back_inserter(points),
std::back_inserter(segments) );
// intersects the first segment of input with all other segments
// The resulting points or segments are written in the vectors with the same names
std::transform( input.begin(), input.end(), disp,
Intersector(input.front()) );
std::cout << "Point intersections: " << points.size() << std::endl;
std::cout << "Segment intersections: " << segments.size() << std::endl;
return 0;
}

`#include <CGAL/intersections.h>`