#include <CGAL/Point_2.h>

Definition

template<typename Kernel>
class CGAL::Point_2< Kernel >

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<NT>, the two types are the same. For the kernel model Homogeneous<NT>, Kernel::RT is equal to NT, and Kernel::FT is equal to Quotient<NT>.

Example

The following declaration creates two points with Cartesian double coordinates.

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

CGAL::Point_2

An object p of the class Point_2 is a point in the two-dimensional Euclidean plane .

Definition: Point_2.h:37

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;

Is model of: Kernel::Point_2
: Hashable if Kernel is a cartesian kernel and if Kernel::FT is Hashable

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`.

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)`.

	Point_2 (const Kernel::FT &x, const Kernel::FT &y)
	introduces a point `p` initialized to `(x,y)`.

	Point_2 (const Kernel::Weighted_point_2 &wp)
	introduces a point from a weighted point.

Operations
bool	operator== (const Point_2< Kernel > &q) const
	Test for equality.

bool	operator!= (const Point_2< Kernel > &q) const
	Test for inequality.

Point_2< Kernel > &	operator+= (const Vector_2< Kernel > &v)
	translates the point by the vector `v`.

Point_2< Kernel > &	operator-= (const Vector_2< Kernel > &v)
	translates the point by the vector -`v`.

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 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`.

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

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

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`.

Point_2< Kernel >	transform (const Aff_transformation_2< Kernel > &t) const
	returns the point obtained by applying `t` on `p`.

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`.

Constructor & Destructor Documentation

◆ Point_2() [1/5]

template<typename Kernel >

CGAL::Point_2< Kernel >::Point_2 ( const Origin & ORIGIN )

introduces a variable p with Cartesian coordinates \( (0,0)\).

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

◆ Point_2() [2/5]

template<typename Kernel >

CGAL::Point_2< Kernel >::Point_2	(	double	x,
		double	y
	)

introduces a point p initialized to (x,y) provided RT supports construction from double.

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

◆ Point_2() [3/5]

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 != Kernel::RT(0).

◆ Point_2() [4/5]

template<typename Kernel >

CGAL::Point_2< Kernel >::Point_2	(	const Kernel::FT &	x,
		const Kernel::FT &	y
	)

introduces a point p initialized to (x,y).

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

◆ Point_2() [5/5]

template<typename Kernel >

CGAL::Point_2< Kernel >::Point_2 ( const Kernel::Weighted_point_2 & wp )

explicit

introduces a point from a weighted point.

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

Warning: The explicit keyword is used to avoid accidental implicit conversions between Point_2 and Weighted_point_2.

Member Function Documentation

◆ bbox()

template<typename Kernel >

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

returns a bounding box containing p.

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

◆ cartesian()

template<typename Kernel >

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

returns the i-th Cartesian coordinate of p.

Precondition: 0 <= i <= 1.

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

◆ homogeneous()

template<typename Kernel >

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

returns the i-th homogeneous coordinate of p.

Precondition: 0 <= i <= 2.

◆ operator!=()

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.

◆ operator==()

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.

◆ operator[]()

template<typename Kernel >

Kernel::FT CGAL::Point_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::Point_2< Kernel >::x ( ) const

returns the Cartesian \( x\) coordinate, 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::Point_2< Kernel >::y ( ) const

returns the Cartesian \( y\) coordinate, that is hy()/hw().

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

Friends And Related Function Documentation

◆ operator-()

template<typename Kernel >

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

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

Definition

Related Functions

Types

Creation

Operations

Coordinate Access

Convenience Operations

Constructor & Destructor Documentation

◆ Point_2() [1/5]

◆ Point_2() [2/5]

◆ Point_2() [3/5]

◆ Point_2() [4/5]

◆ Point_2() [5/5]

Member Function Documentation

◆ bbox()

◆ cartesian()

◆ homogeneous()

◆ operator!=()

◆ operator==()

◆ operator[]()

◆ x()

◆ y()

Friends And Related Function Documentation

◆ operator-()