## CGAL::Point_3<Kernel>

### Definition

An object of the class Point_3<Kernel> is a point in the three-dimensional Euclidean space 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>.

### Types

Point_3<Kernel>::Cartesian_const_iterator
| An iterator for enumerating the Cartesian coordinates of a point.

### Creation

Point_3<Kernel> p ( Origin ORIGIN);
| introduces a point with Cartesian coordinates(0,0,0).

Point_3<Kernel> p ( | Kernel::RT hx, Kernel::RT hy, Kernel::RT hz, Kernel::RT hw = RT(1));

| introduces a point p initialized to (hx/hw,hy/hw, hz/hw). Precondition: hw 0.

### Operations

bool | p.operator== ( q)
|  | Test for equality: Two points are equal, iff their x, y and z coordinates are equal.
bool | p.operator!= ( q)
|  | Test for inequality.

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.

Kernel::RT | p.hx () | returns the homogeneous x coordinate.
Kernel::RT | p.hy () | returns the homogeneous y coordinate.
Kernel::RT | p.hz () | returns the homogeneous z coordinate.
Kernel::RT | p.hw () | returns the homogenizing coordinate.

Note that you do not loose information with the homogeneous representation, because the FieldNumberType is a quotient.

Kernel::FT | p.x () | returns the Cartesian x coordinate, that is hx/hw.
Kernel::FT | p.y () | returns the Cartesian y coordinate, that is hy/hw.
Kernel::FT | p.z () | returns the Cartesian z coordinate, that is hz/hw.

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 | p.homogeneous ( int i)
|  | returns the i'th homogeneous coordinate of p, starting with 0. Precondition: 0 i 3.
Kernel::FT | p.cartesian ( int i)
|  | returns the i'th Cartesian coordinate of p, starting with 0. Precondition: 0 i 2.
Kernel::FT | p.operator[] ( int i)
|  | returns cartesian(i). Precondition: 0 i 2.
Cartesian_const_iterator
| p.cartesian_begin ()
|  | returns an iterator to the Cartesian coordinates of p, starting with the 0th coordinate.
Cartesian_const_iterator
| p.cartesian_end () | returns an off the end iterator to the Cartesian coordinates of p.
int | p.dimension () | returns the dimension (the constant 3).
Bbox_3 | p.bbox () | returns a bounding box containing p.
Point_3<Kernel> | p.transform ( Aff_transformation_3<Kernel> t)
|  | returns the point obtained by applying t on p.

### Operators

The following operations can be applied on points:

bool | operator< ( p, q)
|  | returns true iff p is lexicographically smaller than q (the lexicographical order being defined on the Cartesian coordinates).
bool | operator> ( p, q)
|  | returns true iff p is lexicographically greater than q.
bool | operator<= ( p, q)
|  | returns true iff p is lexicographically smaller or equal to q.
bool | operator>= ( p, q)
|  | returns true iff p is lexicographically greater or equal to q.
Vector_3<Kernel> | operator- ( p, q)
|  | returns the difference vector between q and p. You can substitute ORIGIN for either p or q, but not for both.
Point_3<Kernel> | operator+ ( p, Vector_3<Kernel> v)
|  | returns the point obtained by translating p by the vector v.
Point_3<Kernel> | operator- ( p, Vector_3<Kernel> v)
|  | returns the point obtained by translating p by the vector -v.

### See Also

Kernel::Point_3