CGAL 4.7 - 2D Polygons
Global Functions

## Functions

template<class ForwardIterator , class PolygonTraits >
ForwardIterator CGAL::left_vertex_2 (ForwardIterator first, ForwardIterator last, const PolygonTraits &traits)
Returns an iterator to the leftmost point from the range [first,last). More...

template<class ForwardIterator , class PolygonTraits >
ForwardIterator CGAL::right_vertex_2 (ForwardIterator first, ForwardIterator last, const PolygonTraits &traits)
Returns an iterator to the rightmost point from the range [first,last). More...

template<class ForwardIterator , class PolygonTraits >
ForwardIterator CGAL::top_vertex_2 (ForwardIterator first, ForwardIterator last, const PolygonTraits &traits)
Returns an iterator to the topmost point from the range [first,last). More...

template<class ForwardIterator , class PolygonTraits >
ForwardIterator CGAL::bottom_vertex_2 (ForwardIterator first, ForwardIterator last, const PolygonTraits &traits)
Returns an iterator to the bottommost point from the range [first,last). More...

template<class ForwardIterator , class PolygonTraits >
void CGAL::area_2 (ForwardIterator first, ForwardIterator last, typename PolygonTraits::FT &result, const PolygonTraits &traits)
Computes the signed area of the polygon defined by the range of points [first,last). More...

template<class ForwardIterator , class PolygonTraits >
PolygonTraits::FT CGAL::polygon_area_2 (ForwardIterator first, ForwardIterator last, const PolygonTraits &traits)
Computes the signed area of the polygon defined by the range of points [first,last). More...

template<class ForwardIterator , class PolygonTraits >
bool CGAL::is_convex_2 (ForwardIterator first, ForwardIterator last, const PolygonTraits &traits)
Checks if the polygon is convex. More...

template<class ForwardIterator , class PolygonTraits >
bool CGAL::is_simple_2 (ForwardIterator first, ForwardIterator last, const PolygonTraits &traits)
Checks if the polygon defined by the iterator range [first,last) is simple, that is, if the edges do not intersect, except consecutive edges in their common vertex. More...

template<class ForwardIterator , class Point , class Traits >
Oriented_side CGAL::oriented_side_2 (ForwardIterator first, ForwardIterator last, const Point &point, const Traits &traits)
Computes on which side of a polygon a point lies. More...

template<class ForwardIterator , class Point , class PolygonTraits >
Bounded_side CGAL::bounded_side_2 (ForwardIterator first, ForwardIterator last, const Point &point, const PolygonTraits &traits)
Computes if a point lies inside a polygon. More...

template<class ForwardIterator , class Traits >
Orientation CGAL::orientation_2 (ForwardIterator first, ForwardIterator last, const Traits &traits)
Computes if a polygon is clockwise or counterclockwise oriented. More...

## Function Documentation

template<class ForwardIterator , class PolygonTraits >
 void CGAL::area_2 ( ForwardIterator first, ForwardIterator last, typename PolygonTraits::FT & result, const PolygonTraits & traits )

Computes the signed area of the polygon defined by the range of points [first,last).

The area is returned in the parameter result. The sign is positive for counterclockwise polygons, negative for clockwise polygons. If the polygon is not simple, the area is not well defined. The functionality is also available by the polygon_area_2() function, which returns the area instead of taking it as a parameter.

Requires:

Traits is a model of the concept PolygonTraits_2. Only the following members of this traits class are used:

• Compute_area_2 : Computes the signed area of the oriented triangle defined by 3 Point_2 passed as arguments.
• FT
• compute_area_2_object()

The value type of ForwardIterator must be Traits::Point_2.

CGAL::polygon_area_2()
PolygonTraits_2
CGAL::orientation_2()
CGAL::Polygon_2

#include <CGAL/Polygon_2_algorithms.h>

template<class ForwardIterator , class PolygonTraits >
 ForwardIterator CGAL::bottom_vertex_2 ( ForwardIterator first, ForwardIterator last, const PolygonTraits & traits )

Returns an iterator to the bottommost point from the range [first,last).

In case of a tie, the point with the smallest x-coordinate is taken.

Requires:

Traits is a model of the concept PolygonTraits_2. Only the members Less_yx_2 and less_yx_2_object() are used.

The value type of ForwardIterator must be Traits::Point_2.

CGAL::left_vertex_2()
CGAL::right_vertex_2()
CGAL::top_vertex_2()
CGAL::Polygon_2
PolygonTraits_2

#include <CGAL/Polygon_2_algorithms.h>

template<class ForwardIterator , class Point , class PolygonTraits >
 Bounded_side CGAL::bounded_side_2 ( ForwardIterator first, ForwardIterator last, const Point & point, const PolygonTraits & traits )

Computes if a point lies inside a polygon.

The polygon is defined by the sequence of points [first,last). Being inside is defined by the odd-even rule. If we take a ray starting at the point and extending to infinity (in any direction), we count the number of intersections. If this number is odd, the point is inside, otherwise it is outside. If the point is on a polygon edge, a special value is returned. A simple polygon divides the plane in an unbounded and a bounded region. According to the definition points in the bounded region are inside the polygon.

Requires:

Traits is a model of the concept PolygonTraits_2. Only the following members of this traits class are used:

• Compare_x_2
• Compare_y_2
• Orientation_2
• compare_x_2_object()
• compare_y_2_object()
• orientation_2_object()

The value type of ForwardIterator must be Traits::Point_2,

### Implementation

The running time is linear in the number of vertices of the polygon. A horizontal ray is taken to count the number of intersections. Special care is taken that the result is correct even if there are degeneracies (if the ray passes through a vertex).

PolygonTraits_2
CGAL::oriented_side_2()
CGAL::Polygon_2
CGAL::Bounded_side

#include <CGAL/Polygon_2_algorithms.h>

Examples:
Polygon/polygon_algorithms.cpp.
template<class ForwardIterator , class PolygonTraits >
 bool CGAL::is_convex_2 ( ForwardIterator first, ForwardIterator last, const PolygonTraits & traits )

Checks if the polygon is convex.

Requires:

Traits is a model of the concept PolygonTraits_2. Only the following members of this traits class are used:

• Less_xy_2
• Orientation_2
• less_xy_2_object
• orientation_2_object

ForwardIterator::value_type should be PolygonTraits::Point_2,

PolygonTraits_2
CGAL::Polygon_2

#include <CGAL/Polygon_2_algorithms.h>

template<class ForwardIterator , class PolygonTraits >
 bool CGAL::is_simple_2 ( ForwardIterator first, ForwardIterator last, const PolygonTraits & traits )

Checks if the polygon defined by the iterator range [first,last) is simple, that is, if the edges do not intersect, except consecutive edges in their common vertex.

Requires:

Traits is a model of the concept PolygonTraits_2. Only the following members of this traits class are used:

• Point_2
• Less_xy_2
• Orientation_2
• less_xy_2_object()
• orientation_2_object()

The value type of ForwardIterator must be PolygonTraits::Point_2,

### Implementation

The simplicity test is implemented by means of a plane sweep algorithm. The algorithm is quite robust when used with inexact number types. The running time is O(n log n), where n is the number of vertices of the polygon.

PolygonTraits_2
CGAL::Polygon_2

#include <CGAL/Polygon_2_algorithms.h>

Examples:
Polygon/polygon_algorithms.cpp, and Polygon/projected_polygon.cpp.
template<class ForwardIterator , class PolygonTraits >
 ForwardIterator CGAL::left_vertex_2 ( ForwardIterator first, ForwardIterator last, const PolygonTraits & traits )

Returns an iterator to the leftmost point from the range [first,last).

In case of a tie, the point with the smallest y-coordinate is taken.

Requires:

Traits is a model of the concept PolygonTraits_2. Only the members Less_xy_2 and less_xy_2_object() are used.

The value type of ForwardIterator must be Traits::Point_2,

CGAL::right_vertex_2()
CGAL::top_vertex_2()
CGAL::bottom_vertex_2()
CGAL::Polygon_2

#include <CGAL/Polygon_2_algorithms.h>

template<class ForwardIterator , class Traits >
 Orientation CGAL::orientation_2 ( ForwardIterator first, ForwardIterator last, const Traits & traits )

Computes if a polygon is clockwise or counterclockwise oriented.

Precondition
is_simple_2(first, last, traits);
Requires:

Traits is a model of the concept PolygonTraits_2. Only the following members of this traits class are used:

• Less_xy_2
• less_xy_2_object()
• orientation_2_object()

The value type of ForwardIterator must be Traits::Point_2,

PolygonTraits_2
CGAL::is_simple_2()
CGAL::Polygon_2
CGAL::Orientation

#include <CGAL/Polygon_2_algorithms.h>

template<class ForwardIterator , class Point , class Traits >
 Oriented_side CGAL::oriented_side_2 ( ForwardIterator first, ForwardIterator last, const Point & point, const Traits & traits )

Computes on which side of a polygon a point lies.

Requires:

Traits is a model of the concept PolygonTraits_2. Only the following members of this traits class are used:

• Less_xy_2
• Compare_x_2
• Compare_y_2
• Orientation_2
• less_xy_2_object()
• compare_x_2_object()
• compare_y_2_object()
• orientation_2_object()

The value type of ForwardIterator must be PolygonTraits::Point_2,

PolygonTraits_2
CGAL::bounded_side_2()
CGAL::is_simple_2()
CGAL::Polygon_2
Oriented_side

#include <CGAL/Polygon_2_algorithms.h>

template<class ForwardIterator , class PolygonTraits >
 PolygonTraits::FT CGAL::polygon_area_2 ( ForwardIterator first, ForwardIterator last, const PolygonTraits & traits )

Computes the signed area of the polygon defined by the range of points [first,last).

The sign is positive for counterclockwise polygons, negative for clockwise polygons. If the polygon is not simple, the area is not well defined.

Requires:

Traits is a model of the concept PolygonTraits_2. Only the following members of this traits class are used:

• Compute_area_2 : Computes the signed area of the oriented triangle defined by 3 Point_2 passed as arguments.
• FT
• compute_area_2_object

ForwardIterator::value_type should be Traits::Point_2,

PolygonTraits_2
CGAL::orientation_2()
CGAL::Polygon_2

#include <CGAL/Polygon_2_algorithms.h>

template<class ForwardIterator , class PolygonTraits >
 ForwardIterator CGAL::right_vertex_2 ( ForwardIterator first, ForwardIterator last, const PolygonTraits & traits )

Returns an iterator to the rightmost point from the range [first,last).

In case of a tie, the point with the largest y-coordinate is taken.

Requires:

Traits is a model of the concept PolygonTraits_2. In fact, only the members Less_xy_2 and less_xy_2_object() are used.

The value type of ForwardIterator must be Traits::Point_2.

CGAL::left_vertex_2()
CGAL::top_vertex_2()
CGAL::bottom_vertex_2()
CGAL::Polygon_2

#include <CGAL/Polygon_2_algorithms.h>

template<class ForwardIterator , class PolygonTraits >
 ForwardIterator CGAL::top_vertex_2 ( ForwardIterator first, ForwardIterator last, const PolygonTraits & traits )

Returns an iterator to the topmost point from the range [first,last).

In case of a tie, the point with the largest x-coordinate is taken.

Requires:

Traits is a model of the concept PolygonTraits_2. Only the members Less_yx_2 and less_yx_2_object() are used.

The value type of ForwardIterator must be Traits::Point_2.

CGAL::left_vertex_2()
CGAL::right_vertex_2()
CGAL::bottom_vertex_2()
CGAL::Polygon_2
#include <CGAL/Polygon_2_algorithms.h>