Vector_3<Kernel>

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CGAL::Vector_3<Kernel>

Definition

An object of the class Vector_3<Kernel> is a vector in the three-dimensional vector space ℝ³. 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.

Types

Vector_3<Kernel>::Cartesian_const_iterator
	An iterator for enumerating the Cartesian coordinates of a vector.

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 ( int x, int y, int z);
	introduces a vector v initialized to (x, y, z).

Vector_3<Kernel> v ( double x, double y, double z);
	introduces a vector v initialized to (x, y, z).

Vector_3<Kernel> v ( Kernel::RT hx, Kernel::RT hy, Kernel::RT hz, Kernel::RT hw = RT(1));
	introduces a vector v initialized to (hx/hw, hy/hw, hz/hw).

Vector_3<Kernel> v ( Kernel::FT x, Kernel::FT y, Kernel::FT z);
	introduces a vector v initialized to (x, y, z).

Operations

bool	v.operator== ( w) const	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) const	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 () const	returns the homogeneous x coordinate.
Kernel::RT	v.hy () const	returns the homogeneous y coordinate.
Kernel::RT	v.hz () const	returns the homogeneous z coordinate.
Kernel::RT	v.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	v.x () const	returns the x-coordinate of v, that is hx/hw.
Kernel::FT	v.y () const	returns the y-coordinate of v, that is hy/hw.
Kernel::FT	v.z () const	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) const

returns the i'th homogeneous coordinate of v, starting with 0.

Precondition:

0 ≤ i ≤ 3.

Kernel::FT

v.cartesian ( int i) const

returns the i'th Cartesian coordinate of v, starting at 0.

Precondition:

0 ≤ i ≤ 2.

Kernel::FT

v.operator[] ( int i) const

returns cartesian(i).

Precondition:

0 ≤ i ≤ 2.

Cartesian_const_iterator

v.cartesian_begin () const

returns an iterator to the Cartesian coordinates of v, starting with the 0th coordinate.

Cartesian_const_iterator

v.cartesian_end () const

returns an off the end iterator to the Cartesian coordinates of v.

int

v.dimension () const

returns the dimension (the constant 3).

Vector_3<Kernel>

v.transform ( Aff_transformation_3<Kernel> t) const

returns the vector obtained by applying t on v.

Direction_3<Kernel>

v.direction () const

returns the direction of v.

Operators

The following operations can be applied on vectors:

Vector_3<Kernel>	v.operator+ ( w) const	Addition.

Vector_3<Kernel>	v.operator- ( w) const	Subtraction.

Vector_3<Kernel>	v.operator- () const	Returns the opposite vector.

Kernel::FT	v.operator ( w) const*	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) const
		Division by a scalar.

Kernel::FT	v.squared_length () const	returns the squared length of v.

CGAL::Vector_3<Kernel>

Definition

Types

Creation

Operations

Operators

See Also