\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - 2D and 3D Linear Geometry Kernel

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CGAL::Point_2< Kernel > Class Template Reference
2D Kernel Objects

`#include <CGAL/Point_2.h>`

## Definition

An object `p` of the class `Point_2` is a point in the two-dimensional Euclidean plane \( \E^2\).

Remember that `Kernel::RT` and `Kernel::FT` denote a `RingNumberType` and a `FieldNumberType`, respectively. For the kernel model `Cartesian<T>`, the two types are the same. For the kernel model `Homogeneous<T>`, `Kernel::RT` is equal to `T`, and `Kernel::FT` is equal to `Quotient<T>`.

Operators

The following operations can be applied on points:

Example

The following declaration creates two points with Cartesian double coordinates.

Point_2< Cartesian<double> > p, q(1.0, 2.0);

The variable `p` is uninitialized and should first be used on the left hand side of an assignment.

p = q;
std::cout << p.x() << " " << p.y() << std::endl;

See Also
  `Kernel::Point_2`

## Related Functions
(Note that these are not member functions.)
bool | operator< (const Point_2< Kernel > &p, const Point_2< Kernel > &q)
| returns true iff `p` is lexicographically smaller than `q`, i.e. either if `p.x() < q.x()` or if `p.x() == q.x()` and `p.y() < q.y()`.
bool | operator> (const Point_2< Kernel > &p, const Point_2< Kernel > &q)
| returns true iff `p` is lexicographically greater than `q`.
bool | operator<= (const Point_2< Kernel > &p, const Point_2< Kernel > &q)
| returns true iff `p` is lexicographically smaller or equal to `q`.
bool | operator>= (const Point_2< Kernel > &p, const Point_2< Kernel > &q)
| returns true iff `p` is lexicographically greater or equal to `q`.
Vector_2< Kernel > | operator- (const Point_2< Kernel > &p, const Point_2< Kernel > &q)
| returns the difference vector between `q` and `p`. More...
Point_2< Kernel > | operator+ (const Point_2< Kernel > &p, const Vector_2< Kernel > &v)
| returns the point obtained by translating `p` by the vector `v`.
Point_2< Kernel > | operator- (const Point_2< Kernel > &p, const Vector_2< Kernel > &v)
| returns the point obtained by translating `p` by the vector -`v`.

## Types
typedef unspecified_type | Cartesian_const_iterator
| An iterator for enumerating the Cartesian coordinates of a point.

## Creation
| Point_2 (const Origin &ORIGIN)
| introduces a variable `p` with Cartesian coordinates \( (0,0)\).
| Point_2 (int x, int y)
| introduces a point `p` initialized to `(x,y)`.
| Point_2 (double x, double y)
| introduces a point `p` initialized to `(x,y)` provided `RT` supports construction from `double`.
| Point_2 (const Kernel::RT &hx, const Kernel::RT &hy, const Kernel::RT &hw=RT(1))
| introduces a point `p` initialized to `(hx/hw,hy/hw)`. More...
| Point_2 (const Kernel::FT &x, const Kernel::FT &y)
| introduces a point `p` initialized to `(x,y)`.

## Operations
bool | operator== (const Point_2< Kernel > &q) const
| Test for equality. More...
bool | operator!= (const Point_2< Kernel > &q) const
| Test for inequality. More...

## Coordinate Access
There are two sets of coordinate access functions, namely to the homogeneous and to the Cartesian coordinates. They can be used independently from the chosen kernel model. Note that you do not loose information with the homogeneous representation, because the `FieldNumberType` is a quotient.
Kernel::RT | hx () const
| returns the homogeneous \( x\) coordinate.
Kernel::RT | hy () const
| returns the homogeneous \( y\) coordinate.
Kernel::RT | hw () const
| returns the homogenizing coordinate.
Kernel::FT | x () const
| returns the Cartesian \( x\) coordinate, that is `hx()`/`hw()`.
Kernel::FT | y () const
| returns the Cartesian \( y\) coordinate, that is `hy()`/`hw()`.

## Convenience Operations
The following operations are for convenience and for compatibility with higher dimensional points. Again they come in a Cartesian and in a homogeneous flavor.
Kernel::RT | homogeneous (int i) const
| returns the i'th homogeneous coordinate of `p`, starting with 0. More...
Kernel::FT | cartesian (int i) const
| returns the i'th Cartesian coordinate of `p`, starting with 0. More...
Kernel::FT | operator[] (int i) const
| returns `cartesian(i)`. More...
Cartesian_const_iterator | cartesian_begin () const
| returns an iterator to the Cartesian coordinates of `p`, starting with the 0th coordinate.
Cartesian_const_iterator | cartesian_end () const
| returns an off the end iterator to the Cartesian coordinates of `p`.
int | dimension () const
| returns the dimension (the constant 2).
Bbox_2 | bbox () const
| returns a bounding box containing `p`. More...
Point_2< Kernel > | transform (const Aff_transformation_2< Kernel > &t) const
| returns the point obtained by applying `t` on `p`.

## Constructor & Destructor Documentation

template<typename Kernel >

CGAL::Point_2< Kernel >::Point_2 | ( | const Kernel::RT & | hx,
|  | const Kernel::RT & | hy,
|  | const Kernel::RT & | hw = `RT(1)`
| ) |  |

introduces a point `p` initialized to `(hx/hw,hy/hw)`.

Precondition
  `hw` \( \neq\) `Kernel::RT(0)`.

## Member Function Documentation

template<typename Kernel >

Bbox_2 CGAL::Point_2< Kernel >::bbox | ( | ) |  | const

returns a bounding box containing `p`.

Note that bounding boxes are not parameterized with whatsoever.

template<typename Kernel >

Kernel::FT CGAL::Point_2< Kernel >::cartesian | ( | int | i) |  | const

returns the i'th Cartesian coordinate of `p`, starting with 0.

Precondition
  \( 0\leq i \leq1\).

template<typename Kernel >

Kernel::RT CGAL::Point_2< Kernel >::homogeneous | ( | int | i) |  | const

returns the i'th homogeneous coordinate of `p`, starting with 0.

Precondition
  \( 0\leq i \leq2\).

template<typename Kernel >

bool CGAL::Point_2< Kernel >::operator!= | ( | const Point_2< Kernel > & | q) |  | const

Test for inequality.

The point can be compared with `ORIGIN`.

template<typename Kernel >

bool CGAL::Point_2< Kernel >::operator== | ( | const Point_2< Kernel > & | q) |  | const

Test for equality.

Two points are equal, iff their \( x\) and \( y\) coordinates are equal. The point can be compared with `ORIGIN`.

template<typename Kernel >

Kernel::FT CGAL::Point_2< Kernel >::operator[] | ( | int | i) |  | const

returns `cartesian(i)`.

Precondition
  \( 0\leq i \leq1\).

## Friends And Related Function Documentation

template<typename Kernel >

Vector_2< Kernel > operator- | ( | const Point_2< Kernel > & | p,
|  | const Point_2< Kernel > & | q
| ) |  |
| related

returns the difference vector between `q` and `p`.

You can substitute `ORIGIN` for either `p` or `q`, but not for both.