CGAL::Vector_3<R>

Definition

An object of the class Vector_3<R> is a vector in the three-dimensional vector space $$ ³. Geometrically spoken a vector is the difference of two points $$p₂, $$p₁ and denotes the direction and the distance from $$p₁ to $$p₂.

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

Creation

Vector_3<R> v ( Point_3<R> a, Point_3<R> b);
	introduces the vector $$b-a.
Vector_3<R> v ( Null_vector NULL_VECTOR);
	introduces a null vector v.
Vector_3<R> v ( R::RT hx, R::RT hy, R::FT hz, R::RT hw = R::RT(1));
	introduces a vector v initialized to $$(hx/hw, hy/hw, hz/hw). If the third argument is not explicitly given it defaults to R::RT(1).

Operations

bool	v.operator== ( w)
		Test for equality: two vectors are equal, iff their $$x, $$y and $$z coordinates are equal. You can compare a vector with the NULL_VECTOR.
bool	v.operator!= ( w)
		Test for inequality. You can compare a vector with the 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.hz ()	returns the homogeneous $$z 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.
R::FT	v.z ()	returns the z coordinate of v, that is $$hz/hw.

The following operations are for convenience and for making the class Vector_3<R> 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: $$0 i 3.
R::FT	v.cartesian ( int i)
		returns the i'th Cartesian coordinate of v, starting at 0. Precondition: $$0 i 2.
R::FT	v.operator[] ( int i)
		returns cartesian(i). Precondition: $$0 i 2.
int	v.dimension ()	returns the dimension (the constant 3).
Vector_3<R>	v.transform ( Aff_transformation_3<R> t)
		returns the vector obtained by applying $$t on v.
Direction_3<R>	v.direction ()	returns the direction of v.

Operators

The following operations can be applied on vectors:

Vector_3<R>	v.operator+ ( w)	Addition.
Vector_3<R>	v.operator- ( w)	Subtraction.
Vector_3<R>	v.operator- ()	Negation.
R::FT	v.operator* ( w)	returns the scalar product (= inner product) of the two vectors.
Vector_3<R>	v.operator* ( 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.)
Vector_3<R>	v.operator* ( Quotient<RT> s)
		Multiplication with a scalar from the right.
Vector_3<R>	v.operator/ ( R::RT s)
		Division by a scalar.

See Also