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

CGAL 4.8 - 2D and 3D Linear Geometry Kernel

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CGAL::Vector_2< Kernel > Class Template Reference
2D Kernel Objects

`#include <CGAL/Vector_2.h>`

## Definition

An object `v` of the class `Vector_2` is a vector in the two-dimensional vector space \( \mathbb{R}^2\).

Geometrically spoken, a vector is the difference of two points \( p_2\), \( p_1\) and denotes the direction and the distance from \( p_1\) to \( p_2\).

CGAL defines a symbolic constant NULL_VECTOR. We will explicitly state where you can pass this constant as an argument instead of a vector initialized with zeros.

See Also
  `Kernel::Vector_2`

## Public Member Functions
Vector_2< Kernel > | operator- () const
| returns the opposite vector.
Kernel::FT | operator* (const Vector_2< Kernel > &w) const
| returns the scalar product (= inner product) of the two vectors.
Vector_2< Kernel > | operator/ (const Kernel::RT &s) const
| Division by a scalar.
Kernel::FT | squared_length () const
| returns the squared length of `v`.
Vector_2< Kernel > | operator* (const Vector_2< Kernel > &v, const Kernel::RT &s)
| Multiplication with a scalar from the right.
Vector_2< Kernel > | operator* (const Vector_2< Kernel > &v, const Kernel::FT &s)
| Multiplication with a scalar from the right.
Vector_2< Kernel > | operator* (const Kernel::RT &s, const Vector_2< Kernel > &v)
| Multiplication with a scalar from the left.
Vector_2< Kernel > | operator* (const Kernel::FT &s, const Vector_2< Kernel > &v)
| Multiplication with a scalar from the left.

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

## Creation
| Vector_2 (const Point_2< Kernel > &a, const Point_2< Kernel > &b)
| introduces the vector `b-a`.
| Vector_2 (const Segment_2< Kernel > &s)
| introduces the vector `s.target()-s.source()`.
| Vector_2 (const Ray_2< Kernel > &r)
| introduces the vector having the same direction as `r`.
| Vector_2 (const Line_2< Kernel > &l)
| introduces the vector having the same direction as `l`.
| Vector_2 (const Null_vector &NULL_VECTOR)
| introduces a null vector `v`.
| Vector_2 (int x, int y)
| introduces a vector `v` initialized to `(x,y)`.
| Vector_2 (double x, double y)
| introduces a vector `v` initialized to `(x,y)`.
| Vector_2 (const Kernel::RT &hx, const Kernel::RT &hy, const Kernel::RT &hw=RT(1))
| introduces a vector `v` initialized to `(hx/hw,hy/hw)`. More...
| Vector_2 (const Kernel::FT &x, const Kernel::FT &y)
| introduces a vector `v` initialized to `(x,y)`.

## Operations
bool | operator== (const Vector_2< Kernel > &w) const
| Test for equality: two vectors are equal, iff their \( x\) and \( y\) coordinates are equal. More...
bool | operator!= (const Vector_2< Kernel > &w) 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 `x`-coordinate of `v`, that is `hx()`/`hw()`.
Kernel::FT | y () const
| returns the `y`-coordinate of `v`, that is `hy()`/`hw()`.

## Convenience Operators
The following operations are for convenience and for compatibility with higher dimensional vectors. Again they come in a Cartesian and homogeneous flavor.
Kernel::RT | homogeneous (int i) const
| returns the i'th homogeneous coordinate of `v`, starting with 0. More...
Kernel::FT | cartesian (int i) const
| returns the i'th Cartesian coordinate of `v`, starting at 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 `v`, starting with the 0th coordinate.
Cartesian_const_iterator | cartesian_end () const
| returns an off the end iterator to the Cartesian coordinates of `v`.
int | dimension () const
| returns the dimension (the constant 2).
Direction_2< Kernel > | direction () const
| returns the direction which passes through `v`.
Vector_2< Kernel > | transform (const Aff_transformation_2< Kernel > &t) const
| returns the vector obtained by applying `t` on `v`.
Vector_2< Kernel > | perpendicular (const Orientation &o) const
| returns the vector perpendicular to `v` in clockwise or counterclockwise orientation.

## Operators
Vector_2< Kernel > | operator+ (const Vector_2< Kernel > &w) const
| Addition.
Vector_2< Kernel > | operator- (const Vector_2< Kernel > &w) const
| Subtraction.

## Constructor & Destructor Documentation

template<typename Kernel >

CGAL::Vector_2< Kernel >::Vector_2 | ( | const Kernel::RT & | hx,
|  | const Kernel::RT & | hy,
|  | const Kernel::RT & | hw = `RT(1)`
| ) |  |

introduces a vector `v` initialized to `(hx/hw,hy/hw)`.

Precondition
  \( hw\neq0\).

## Member Function Documentation

template<typename Kernel >

Kernel::FT CGAL::Vector_2< Kernel >::cartesian | ( | int | i) |  | const

returns the i'th Cartesian coordinate of `v`, starting at 0.

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

template<typename Kernel >

Kernel::RT CGAL::Vector_2< Kernel >::homogeneous | ( | int | i) |  | const

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

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

template<typename Kernel >

bool CGAL::Vector_2< Kernel >::operator!= | ( | const Vector_2< Kernel > & | w) |  | const

Test for inequality.

You can compare a vector with the `NULL_VECTOR`.

template<typename Kernel >

bool CGAL::Vector_2< Kernel >::operator== | ( | const Vector_2< Kernel > & | w) |  | const

Test for equality: two vectors are equal, iff their \( x\) and \( y\) coordinates are equal.

You can compare a vector with the `NULL_VECTOR`.

template<typename Kernel >

Kernel::FT CGAL::Vector_2< Kernel >::operator[] | ( | int | i) |  | const

returns `cartesian(i)`.

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