## CGAL::Point_2<Kernel>

### Definition

An object of the class Point_2<Kernel> is a point in the two-dimensional Euclidean plane 2.

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_2<Kernel>::Cartesian_const_iterator
| An iterator for enumerating the Cartesian coordinates of a point.

### Creation

Point_2<Kernel> p ( Origin ORIGIN);
| introduces a variable p with Cartesian coordinates (0,0).
Point_2<Kernel> p ( Kernel::RT hx, Kernel::RT hy, Kernel::RT hw = RT(1));
| introduces a point p initialized to (hx/hw,hy/hw). Precondition: hw Kernel::RT(0)

### Operations

bool | p.operator== ( q)
|  | Test for equality. Two points are equal, iff their x and y coordinates are equal. The point can be compared with ORIGIN.
bool | p.operator!= ( q)
|  | Test for inequality. The point can be compared with ORIGIN.

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.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.

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 | p.homogeneous ( int i)
|  | returns the i'th homogeneous coordinate of p, starting with 0. Precondition: 0 i 2.
Kernel::FT | p.cartesian ( int i)
|  | returns the i'th Cartesian coordinate of p, starting with 0. Precondition: 0 i 1.
Kernel::FT | p.operator[] ( int i)
|  | returns cartesian(i). Precondition: 0 i 1.
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 2).
Bbox_2 | p.bbox () | returns a bounding box containing p. Note that bounding boxes are not parameterized with whatsoever.
Point_2<Kernel> | p.transform ( Aff_transformation_2<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, i.e. either if p.x() < q.x() or if p.x() == q.x() and p.y() < q.y().
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_2<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_2<Kernel> | operator+ ( p, Vector_2<Kernel> v)
|  | returns the point obtained by translating p by the vector v.
Point_2<Kernel> | operator- ( p, Vector_2<Kernel> v)
|  | returns the point obtained by translating p by the vector -v.

### Example

The following declaration creates two points with Cartesian double coordinates.

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

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;

### See Also

Kernel::Point_2