#include <CGAL/Point_3.h>

Definition

template<typename Kernel>
class CGAL::Point_3< Kernel >

An object of the class Point_3 is a point in the three-dimensional Euclidean space \( \E^3\).

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>.

Operators

The following operations can be applied on points:

Is model of: Kernel::Point_3
: 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_3< Kernel > &p, const Point_3< Kernel > &q)
	returns true iff `p` is lexicographically smaller than `q` (the lexicographical order being defined on the Cartesian coordinates).

bool	operator> (const Point_3< Kernel > &p, const Point_3< Kernel > &q)
	returns true iff `p` is lexicographically greater than `q`.

bool	operator<= (const Point_3< Kernel > &p, const Point_3< Kernel > &q)
	returns true iff `p` is lexicographically smaller or equal to `q`.

bool	operator>= (const Point_3< Kernel > &p, const Point_3< Kernel > &q)
	returns true iff `p` is lexicographically greater or equal to `q`.

Vector_3< Kernel >	operator- (const Point_3< Kernel > &p, const Point_3< 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_3 (const Origin &ORIGIN)
	introduces a point with Cartesian coordinates \( (0,0,0)\).

	Point_3 (int x, int y, int z)
	introduces a point `p` initialized to `(x,y,z)`.

	Point_3 (double x, double y, double z)
	introduces a point `p` initialized to `(x,y,z)` provided `RT` supports it.

	Point_3 (const Kernel::RT &hx, const Kernel::RT &hy, const Kernel::RT &hz, const Kernel::RT &hw=RT(1))
	introduces a point `p` initialized to `(hx/hw,hy/hw, hz/hw)`.

	Point_3 (const Kernel::FT &x, const Kernel::FT &y, const Kernel::FT &z)
	introduces a point `p` initialized to `(x,y,z)`.

	Point_3 (const Kernel::Weighted_point_3 &wp)
	introduces a point from a weighted point.

Operations
bool	operator== (const Point_3< Kernel > &q) const
	Test for equality: Two points are equal, iff their \( x\), \( y\) and \( z\) coordinates are equal.

bool	operator!= (const Point_3< Kernel > &q) const
	Test for inequality.

Point_3< Kernel > &	operator+= (const Vector_3< Kernel > &v)
	translates the point by the vector `v`.

Point_3< Kernel > &	operator-= (const Vector_3< 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	hz () const
	returns the homogeneous \( z\) 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()`.

Kernel::FT	z () const
	returns the Cartesian \( z\) coordinate, that is `hz()`/`hw()`.

Convenience Operators
The following operations are for convenience and for compatibility with code for 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 3).

Bbox_3	bbox () const
	returns a bounding box containing `p`.

Point_3< Kernel >	transform (const Aff_transformation_3< Kernel > &t) const
	returns the point obtained by applying `t` on `p`.

Point_3< Kernel >	operator+ (const Point_3< Kernel > &p, const Vector_3< Kernel > &v)
	returns the point obtained by translating `p` by the vector `v`.

Point_3< Kernel >	operator- (const Point_3< Kernel > &p, const Vector_3< Kernel > &v)
	returns the point obtained by translating `p` by the vector -`v`.

Constructor & Destructor Documentation

◆ Point_3() [1/5]

template<typename Kernel >

CGAL::Point_3< Kernel >::Point_3 ( const Origin & ORIGIN )

introduces a point with Cartesian coordinates \( (0,0,0)\).

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

◆ Point_3() [2/5]

template<typename Kernel >

CGAL::Point_3< Kernel >::Point_3	(	double	x,
		double	y,
		double	z
	)

introduces a point p initialized to (x,y,z) provided RT supports it.

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

◆ Point_3() [3/5]

template<typename Kernel >

CGAL::Point_3< Kernel >::Point_3	(	const Kernel::RT &	hx,
		const Kernel::RT &	hy,
		const Kernel::RT &	hz,
		const Kernel::RT &	hw = `RT(1)`
	)

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

Precondition: hw != 0.

◆ Point_3() [4/5]

template<typename Kernel >

CGAL::Point_3< Kernel >::Point_3	(	const Kernel::FT &	x,
		const Kernel::FT &	y,
		const Kernel::FT &	z
	)

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

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

◆ Point_3() [5/5]

template<typename Kernel >

CGAL::Point_3< Kernel >::Point_3 ( const Kernel::Weighted_point_3 & wp )

explicit

introduces a point from a weighted point.

Warning: The explicit keyword is used to avoid accidental implicit conversions between Point_3 and Weighted_point_3.

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

Member Function Documentation

◆ bbox()

template<typename Kernel >

Bbox_3 CGAL::Point_3< 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_3< Kernel >::cartesian ( int i ) const

returns the i-th Cartesian coordinate of p.

Precondition: 0 <= i <= 2.

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

◆ homogeneous()

template<typename Kernel >

Kernel::RT CGAL::Point_3< Kernel >::homogeneous ( int i ) const

returns the i-th homogeneous coordinate of p.

Precondition: 0 <= i <= 3.

◆ operator[]()

template<typename Kernel >

Kernel::FT CGAL::Point_3< Kernel >::operator[] ( int i ) const

returns cartesian(i).

Precondition: 0 <= i <= 2.

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

◆ x()

template<typename Kernel >

Kernel::FT CGAL::Point_3< 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_3< 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.

◆ z()

template<typename Kernel >

Kernel::FT CGAL::Point_3< Kernel >::z ( ) const

returns the Cartesian \( z\) coordinate, that is hz()/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_3< Kernel > operator-	(	const Point_3< Kernel > &	p,
		const Point_3< Kernel > &	q
	)

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

Definition

Related Functions

Types

Creation

Operations

Coordinate Access

Convenience Operators

Constructor & Destructor Documentation

◆ Point_3() [1/5]

◆ Point_3() [2/5]

◆ Point_3() [3/5]

◆ Point_3() [4/5]

◆ Point_3() [5/5]

Member Function Documentation

◆ bbox()

◆ cartesian()

◆ homogeneous()

◆ operator[]()

◆ x()

◆ y()

◆ z()

Friends And Related Function Documentation

◆ operator-()