2D Vector (CGAL_Vector_2)

Definition

CGAL defines a symbolic constant CGAL_NULL_VECTOR. We will explicitly state where you can pass this constant as an argument instead of a vector initialized with zeros.

#include <CGAL/Vector_2.h>

Creation

CGAL_Vector_2<R> v ( CGAL_Null_vector CGAL_NULL_VECTOR);
	introduces a null vector v.
CGAL_Vector_2<R> v ( R::RT hx, R::RT hy, R::RT hw = R::RT(1));
	introduces a vector v initialized to $(hx/hw,hy/hw)$. If the third argument is not explicitly given, it defaults to R::RT(1). Precondition: hw not equal to 0

Operations

bool	v == w	Test for equality: two vectors are equal, iff their $x$ and $y$ coordinates are equal. You can compare a vector with the CGAL_NULL_VECTOR.
bool	v != w	Test for inequality. You can compare a vector with the CGAL_NULL_VECTOR.

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 representation type R.

R::RT	v.hx ()	returns the homogeneous $x$ coordinate.
R::RT	v.hy ()	returns the homogeneous $y$ coordinate.
R::RT	v.hw ()	returns the homogenizing coordinate.

Here come the Cartesian access functions. Note that you do not loose information with the homogeneous representation, because then the field type is a quotient.

R::FT	v.x ()	returns the x-coordinate of v, that is $hx/hw$.
R::FT	v.y ()	returns the y-coordinate of v, that is $hy/hw$.

The following operations are for convenience and for making the class CGAL_Vector_2 compatible with code for higher dimensional vectors. Again they come in a Cartesian and homogeneous flavor.

R::RT	v.homogeneous ( int i)
		returns the i'th homogeneous coordinate of v, starting with 0. Precondition: .
R::FT	v.cartesian ( int i)
		returns the i'th Cartesian coordinate of v, starting at 0. Precondition: .
R::FT	v [ int i]	returns coordinate(i). Precondition: .
int	v.dimension ()	returns the dimension (the constant 2).

Miscellaneous

CGAL_Direction_2<R>
	v.direction ()	returns the direction which passes through v.
CGAL_Vector_2<R>	v.transform ( CGAL_Aff_transformation_2<R> t)
		returns the vector obtained by applying $t$ on v.
CGAL_Vector_2<R>	v.perpendicular ( CGAL_Orientation o)
		returns the vector perpendicular to v in clockwise or counterclockwise orientation.

Operations

CGAL_Vector_2<R>	v + w	Addition.
CGAL_Vector_2<R>	v - w	Subtraction.
CGAL_Vector_2<R>	- v	returns the opposite vector.
R::FT	v * w	returns the scalar product (= inner product) of the two vectors.
CGAL_Vector_2<R>	v * R::RT s	Multiplication with a scalar from the right. Although it would be more natural, CGAL does not offer a multiplication with a scalar from the left. (This is due to problems of some compilers.)
CGAL_Vector_2<R>	v * CGAL_Quotient<RT> s
		Multiplication with a scalar from the right.
CGAL_Vector_2<R>	v / R::RT s	Division by a scalar.