## 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::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 ( int x, int y, int z);
introduces a point p initialized to (x,y,z).

Point_3<Kernel> p ( double x, double y, double z);
introduces a point p initialized to (x,y,z) provided RT supports it.

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.

Point_3<Kernel> p ( Kernel::FT x, Kernel::FT y, Kernel::FT z);
introduces a point p initialized to (x,y,z).

### Operations

 bool p.operator== ( q) const Test for equality: Two points are equal, iff their x, y and z coordinates are equal. bool p.operator!= ( q) const 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 () const returns the homogeneous x coordinate. Kernel::RT p.hy () const returns the homogeneous y coordinate. Kernel::RT p.hz () const returns the homogeneous z coordinate. Kernel::RT p.hw () const 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 () const returns the Cartesian x coordinate, that is hx/hw. Kernel::FT p.y () const returns the Cartesian y coordinate, that is hy/hw. Kernel::FT p.z () const 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) const returns the i'th homogeneous coordinate of p, starting with 0.
 Precondition: 0 ≤ i ≤ 3.

Kernel::FT p.cartesian ( int i) const returns the i'th Cartesian coordinate of p, starting with 0.
 Precondition: 0 ≤ i ≤ 2.

Kernel::FT p.operator[] ( int i) const returns cartesian(i).
 Precondition: 0 ≤ i ≤ 2.

Cartesian_const_iterator p.cartesian_begin () const returns an iterator to the Cartesian coordinates of p, starting with the 0th coordinate.

Cartesian_const_iterator p.cartesian_end () const returns an off the end iterator to the Cartesian coordinates of p.

int p.dimension () const returns the dimension (the constant 3).

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

Point_3<Kernel> p.transform ( Aff_transformation_3<Kernel> t) const
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 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 operator+ ( p, Vector_3 v) returns the point obtained by translating p by the vector v. Point_3 operator- ( p, Vector_3 v) returns the point obtained by translating p by the vector -v.