CGAL 6.1.1 - 2D Polygons

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##

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

## Function Documentation

## ◆ area_2()

template<class ForwardIterator , class PolygonTraits >

void CGAL::area_2 | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | typename PolygonTraits::FT & | result,
|  | const PolygonTraits & | traits
| ) |  |

`#include <CGAL/Polygon_2_algorithms.h>`

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.

Template Parameters

  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 | must have `Traits::Point_2` as value type.

See also
  `CGAL::polygon_area_2()`
  `PolygonTraits_2`
  `CGAL::orientation_2()`
  `CGAL::Polygon_2`

## ◆ bottom_vertex_2()

template<class ForwardIterator , class PolygonTraits >

ForwardIterator CGAL::bottom_vertex_2 | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | const PolygonTraits & | traits
| ) |  |

`#include <CGAL/Polygon_2_algorithms.h>`

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.

Template Parameters

  Traits | is a model of the concept `PolygonTraits_2`. Only the members `Less_yx_2` and `less_yx_2_object()` are used.
  ForwardIterator | must have `Traits::Point_2` as value type.

See also
  `CGAL::left_vertex_2()`
  `CGAL::right_vertex_2()`
  `CGAL::top_vertex_2()`
  `CGAL::Polygon_2`
  `PolygonTraits_2`

## ◆ bounded_side_2()

template<class ForwardIterator , class Point , class PolygonTraits >

Bounded_side CGAL::bounded_side_2 | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | const Point & | point,
|  | const PolygonTraits & | traits
| ) |  |

`#include <CGAL/Polygon_2_algorithms.h>`

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.

Template Parameters

  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()`
  ForwardIterator | must have `Traits::Point_2` as value type.

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).

See also
  `PolygonTraits_2`
  `CGAL::oriented_side_2()`
  `CGAL::Polygon_2`
  `CGAL::Bounded_side`

Examples
  Polygon/polygon_algorithms.cpp.

## ◆ is_convex_2()

template<class ForwardIterator , class PolygonTraits >

bool CGAL::is_convex_2 | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | const PolygonTraits & | traits
| ) |  |

`#include <CGAL/Polygon_2_algorithms.h>`

Checks if the polygon is convex.

Template Parameters

  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 | must have `PolygonTraits::Point_2` as value type.

See also
  `PolygonTraits_2`
  `CGAL::Polygon_2`

## ◆ is_simple_2()

template<class ForwardIterator , class PolygonTraits >

bool CGAL::is_simple_2 | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | const PolygonTraits & | traits
| ) |  |

`#include <CGAL/Polygon_2_algorithms.h>`

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.

Template Parameters

  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()`
  ForwardIterator | must have `PolygonTraits::Point_2` as value type.

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.

See also
  `PolygonTraits_2`
  `CGAL::Polygon_2`

Examples
  Polygon/polygon_algorithms.cpp, and Polygon/projected_polygon.cpp.

## ◆ left_vertex_2()

template<class ForwardIterator , class PolygonTraits >

ForwardIterator CGAL::left_vertex_2 | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | const PolygonTraits & | traits
| ) |  |

`#include <CGAL/Polygon_2_algorithms.h>`

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.

Template Parameters

  Traits | is a model of the concept `PolygonTraits_2`. Only the members `Less_xy_2` and `less_xy_2_object()` are used.
  ForwardIterator | must have `Traits::Point_2` as value type.

See also
  `CGAL::right_vertex_2()`
  `CGAL::top_vertex_2()`
  `CGAL::bottom_vertex_2()`
  `CGAL::Polygon_2`

## ◆ orientation_2()

template<class ForwardIterator , class Traits >

Orientation CGAL::orientation_2 | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | const Traits & | traits
| ) |  |

`#include <CGAL/Polygon_2_algorithms.h>`

Computes if a polygon is clockwise or counterclockwise oriented.

Precondition
  `is_simple_2(first, last, traits);`

Template Parameters

  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()`
  ForwardIterator | must have`Traits::Point_2` as value type.

See also
  `PolygonTraits_2`
  `CGAL::is_simple_2()`
  `CGAL::Polygon_2`
  `CGAL::Orientation`

## ◆ oriented_side_2()

template<class ForwardIterator , class Point , class Traits >

Oriented_side CGAL::oriented_side_2 | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | const Point & | point,
|  | const Traits & | traits
| ) |  |

`#include <CGAL/Polygon_2_algorithms.h>`

Computes on which side of a polygon a point lies.

Template Parameters

  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()`
  ForwardIterator | must have `PolygonTraits::Point_2` as value type.

See also
  `PolygonTraits_2`
  `CGAL::bounded_side_2()`
  `CGAL::is_simple_2()`
  `CGAL::Polygon_2`
  `Oriented_side`

## ◆ polygon_area_2()

template<class ForwardIterator , class PolygonTraits >

PolygonTraits::FT CGAL::polygon_area_2 | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | const PolygonTraits & | traits
| ) |  |

`#include <CGAL/Polygon_2_algorithms.h>`

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.

Template Parameters

  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 | must have `Traits::Point_2` as value type.

See also
  `PolygonTraits_2`
  `CGAL::orientation_2()`
  `CGAL::Polygon_2`

## ◆ right_vertex_2()

template<class ForwardIterator , class PolygonTraits >

ForwardIterator CGAL::right_vertex_2 | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | const PolygonTraits & | traits
| ) |  |

`#include <CGAL/Polygon_2_algorithms.h>`

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.

Template Parameters

  Traits | is a model of the concept `PolygonTraits_2`. In fact, only the members `Less_xy_2` and `less_xy_2_object()` are used.
  ForwardIterator | must have`Traits::Point_2` as value type.

See also
  `CGAL::left_vertex_2()`
  `CGAL::top_vertex_2()`
  `CGAL::bottom_vertex_2()`
  `CGAL::Polygon_2`

## ◆ top_vertex_2()

template<class ForwardIterator , class PolygonTraits >

ForwardIterator CGAL::top_vertex_2 | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | const PolygonTraits & | traits
| ) |  |

`#include <CGAL/Polygon_2_algorithms.h>`

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.

Template Parameters

  Traits | is a model of the concept `PolygonTraits_2`. Only the members `Less_yx_2` and `less_yx_2_object()` are used.
  ForwardIterator | must have `Traits::Point_2` as value type.

See also
  `CGAL::left_vertex_2()`
  `CGAL::right_vertex_2()`
  `CGAL::bottom_vertex_2()`
  `CGAL::Polygon_2`