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

CGAL 4.3 - 2D and 3D Linear Geometry Kernel

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CGAL::Point_3< Kernel > Class Template Reference
3D Kernel Objects

`#include <CGAL/Point_3.h>`

## Definition

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<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:

See Also
  `Kernel::Point_3`

## 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`. More...
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`.

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

## 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.

## 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 | 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`, 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 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`.

## Constructor & Destructor Documentation

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` \( \neq\) 0.

## Member Function Documentation

template<typename Kernel >

Kernel::FT CGAL::Point_3< Kernel >::cartesian | ( | int | i) |  | const

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

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

template<typename Kernel >

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

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

Precondition
  \( 0\leq i \leq3\).

template<typename Kernel >

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

returns `cartesian(i)`.

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

## Friends And Related Function Documentation

template<typename Kernel >

Vector_3< Kernel > operator- | ( | const Point_3< Kernel > & | p,
|  | const Point_3< Kernel > & | q
| ) |  |
| related

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

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