CGAL::Vector_3<Kernel>

Definition

An object of the class Vector_3<Kernel> 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<Kernel> v ( Point_3<Kernel> a, Point_3<Kernel> b);
	introduces the vector $$b-a.
Vector_3<Kernel> v ( Segment_3<Kernel> s);
	introduces the vector $$s.target()-s.source().
Vector_3<Kernel> v ( Ray_3<Kernel> r);
	introduces a vector having the same direction as $$r.
Vector_3<Kernel> v ( Line_3<Kernel> l);
	introduces a vector having the same direction as $$l.
Vector_3<Kernel> v ( Null_vector NULL_VECTOR);
	introduces a null vector v.
Vector_3<Kernel> v ( Kernel::RT hx, Kernel::RT hy, Kernel::FT hz, Kernel::RT hw = RT(1));
	introduces a vector v initialized to $$(hx/hw, hy/hw, hz/hw).

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 kernel model.

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

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

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

The following operations are for convenience and for compatibility with higher dimensional vectors. Again they come in a Cartesian and homogeneous flavor.

Kernel::RT	v.homogeneous ( int i)
		returns the i'th homogeneous coordinate of v, starting with 0. Precondition: $$0 i 3.
Kernel::FT	v.cartesian ( int i)
		returns the i'th Cartesian coordinate of v, starting at 0. Precondition: $$0 i 2.
Kernel::FT	v.operator[] ( int i)
		returns cartesian(i). Precondition: $$0 i 2.
int	v.dimension ()	returns the dimension (the constant 3).
Vector_3<Kernel>	v.transform ( Aff_transformation_3<Kernel> t)
		returns the vector obtained by applying $$t on v.
Direction_3<Kernel>
	v.direction ()	returns the direction of v.

Operators

The following operations can be applied on vectors:

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

See Also