#include <CGAL/Vector_2.h>

Definition

template<typename Kernel>
class CGAL::Vector_2< Kernel >

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.

Is model of: Kernel::Vector_2
: Hashable if Kernel is a Cartesian kernel and if Kernel::FT is Hashable

Public Member Functions
Kernel::FT	squared_length () const
	returns the squared length of `v`.

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)`.

	Vector_2 (const Kernel::FT &x, const Kernel::FT &y)
	introduces a vector `v` initialized to `(x,y)`.

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 lose 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`.

Kernel::FT	cartesian (int i) const
	returns the i-th Cartesian coordinate of `v`.

Kernel::FT	operator[] (int i) const
	returns `cartesian(i)`.

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
bool	operator== (const Vector_2< Kernel > &w) const
	Test for equality: two vectors are equal, iff their \( x\) and \( y\) coordinates are equal.

bool	operator!= (const Vector_2< Kernel > &w) const
	Test for inequality.

Vector_2< Kernel >	operator+ (const Vector_2< Kernel > &w) const
	Addition.

Vector_2< Kernel > &	operator+= (const Vector_2< Kernel > &w)
	Addition.

Vector_2< Kernel >	operator- (const Vector_2< Kernel > &w) const
	Subtraction.

Vector_2< Kernel > &	operator-= (const Vector_2< Kernel > &w)
	Subtraction.

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.

Vector_2< Kernel > &	operator/= (const Kernel::RT &s)
	Division by a scalar.

Vector_2< Kernel > &	*operator=** (const Kernel::RT &s)
	Multiplication by a scalar.

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.

Constructor & Destructor Documentation

◆ Vector_2() [1/3]

template<typename Kernel >

CGAL::Vector_2< Kernel >::Vector_2	(	double	x,
		double	y
	)

introduces a vector v initialized to (x,y).

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

◆ Vector_2() [2/3]

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 != 0.

◆ Vector_2() [3/3]

template<typename Kernel >

CGAL::Vector_2< Kernel >::Vector_2	(	const Kernel::FT &	x,
		const Kernel::FT &	y
	)

introduces a vector v initialized to (x,y).

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

Member Function Documentation

◆ cartesian()

template<typename Kernel >

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

returns the i-th Cartesian coordinate of v.

Precondition: 0 <= i <= 1.

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

◆ direction()

template<typename Kernel >

Direction_2< Kernel > CGAL::Vector_2< Kernel >::direction ( ) const

returns the direction which passes through v.

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

◆ homogeneous()

template<typename Kernel >

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

returns the i-th homogeneous coordinate of v.

Precondition: 0 <= i <= 2.

◆ operator!=()

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.

◆ operator-()

template<typename Kernel >

Vector_2< Kernel > CGAL::Vector_2< Kernel >::operator- ( ) const

returns the opposite vector.

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

◆ operator==()

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.

◆ operator[]()

template<typename Kernel >

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

returns cartesian(i).

Precondition: 0 <= i <= 1.

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

◆ x()

template<typename Kernel >

Kernel::FT CGAL::Vector_2< Kernel >::x ( ) const

returns the x-coordinate of v, that is hx()/hw().

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

◆ y()

template<typename Kernel >

Kernel::FT CGAL::Vector_2< Kernel >::y ( ) const

returns the y-coordinate of v, that is hy()/hw().

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

Definition

Public Member Functions

Types

Creation

Coordinate Access

Convenience Operators

Operators

Constructor & Destructor Documentation

◆ Vector_2() [1/3]

◆ Vector_2() [2/3]

◆ Vector_2() [3/3]

Member Function Documentation

◆ cartesian()

◆ direction()

◆ homogeneous()

◆ operator!=()

◆ operator-()

◆ operator==()

◆ operator[]()

◆ x()

◆ y()