\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - dD Geometry Kernel

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CGAL::Vector_d< Kernel > Class Template Reference
Kernel Objects

`#include <CGAL/Kernel_d/Vector_d.h>`

## Definition

An instance of data type `Vector_d<Kernel>` is a vector of Euclidean space in dimension \( d\).

A vector \( r = (r_0,\ldots,r_{ d - 1})\) can be represented in homogeneous coordinates \( (h_0,\ldots,h_d)\) of number type `RT`, such that \( r_i = h_i/h_d\) which is of type `FT`. We call the \( r_i\)'s the Cartesian coordinates of the vector. The homogenizing coordinate \( h_d\) is positive.

This data type is meant for use in computational geometry. It realizes free vectors as opposed to position vectors (type `Point_d`). The main difference between position vectors and free vectors is their behavior under affine transformations, e.g., free vectors are invariant under translations.

Downward compatibility

We provide all operations of the lower dimensional interface `x()`, `y()`, `z()`, `hx()`, `hy()`, `hz()`, `hw()`.

Implementation

Vectors are implemented by arrays of variables of type `RT`. All operations like creation, initialization, tests, vector arithmetic, input and output on a vector \( v\) take time \( O(v.dimension())\). coordinate access, `dimension()` and conversions take constant time. The space requirement of a vector is \( O(v.dimension())\).

## Types
typedef unspecified_type | LA
| the linear algebra layer.
typedef unspecified_type | Cartesian_const_iterator
| a read-only iterator for the Cartesian coordinates.
typedef unspecified_type | Homogeneous_const_iterator
| a read-only iterator for the homogeneous coordinates.
typedef unspecified_type | Base_vector
| construction tag.

## Creation
| Vector_d ()
| introduces a variable `v` of type `Vector_d<Kernel>`.
| Vector_d (int d, Null_vector)
| introduces the zero vector `v` of type `Vector_d<Kernel>` in \( d\)-dimensional space. More...
template<class InputIterator >
| Vector_d (int d, InputIterator first, InputIterator last)
| introduces a variable `v` of type `Vector_d<Kernel>` in dimension `d`. More...
template<class InputIterator >
| Vector_d (int d, InputIterator first, InputIterator last, RT D)
| introduces a variable `v` of type `Vector_d<Kernel>` in dimension `d` initialized to the vector with homogeneous coordinates as defined by `H = set [first,last)` and `D`: \( (\pmH[0], \pmH[1], \ldots, \pmH[d-1], \pmD)\). More...
| Vector_d (int d, Base_vector, int i)
| returns a variable `v` of type `Vector_d<Kernel>` initialized to the \( i\)-th base vector of dimension \( d\). More...
| Vector_d (RT x, RT y, RT w=1)
| introduces a variable `v` of type `Vector_d<Kernel>` in \( 2\)-dimensional space. More...
| Vector_d (RT x, RT y, RT z, RT w)
| introduces a variable `v` of type `Vector_d<Kernel>` in \( 3\)-dimensional space. More...

## Operations
int | dimension ()
| returns the dimension of `v`.
FT | cartesian (int i)
| returns the \( i\)-th Cartesian coordinate of `v`. More...
FT | operator[] (int i)
| returns the \( i\)-th Cartesian coordinate of `v`. More...
RT | homogeneous (int i)
| returns the \( i\)-th homogeneous coordinate of `v`. More...
FT | squared_length ()
| returns the square of the length of `v`.
Cartesian_const_iterator | cartesian_begin ()
| returns an iterator pointing to the zeroth Cartesian coordinate of `v`.
Cartesian_const_iterator | cartesian_end ()
| returns an iterator pointing beyond the last Cartesian coordinate of `v`.
Homogeneous_const_iterator | homogeneous_begin ()
| returns an iterator pointing to the zeroth homogeneous coordinate of `v`.
Homogeneous_const_iterator | homogeneous_end ()
| returns an iterator pointing beyond the last homogeneous coordinate of `v`.
Direction_d< Kernel > | direction ()
| returns the direction of `v`.
Vector_d< Kernel > | transform (const Aff_transformation_d< Kernel > &t)
| returns \( t(v)\).

## Arithmetic Operators, Tests and IO
Vector_d< Kernel > & | operator*= (const RT &n)
| multiplies all Cartesian coordinates by `n`.
Vector_d< Kernel > & | operator*= (const FT &r)
| multiplies all Cartesian coordinates by `r`.
Vector_d< Kernel > | operator/ (const RT &n)
| returns the vector with Cartesian coordinates \( v_i/n, 0 \leq i < d\).
Vector_d< Kernel > | operator/ (const FT &r)
| returns the vector with Cartesian coordinates \( v_i/r, 0 \leq i < d\).
Vector_d< Kernel > & | operator/= (const RT &n)
| divides all Cartesian coordinates by `n`.
Vector_d< Kernel > & | operator/= (const FT &r)
| divides all Cartesian coordinates by `r`.
FT | operator* (const Vector_d< Kernel > &w)
| inner product, i.e., \( \sum_{ 0 \le i < d } v_i w_i\), where \( v_i\) and \( w_i\) are the Cartesian coordinates of \( v\) and \( w\) respectively.
Vector_d< Kernel > | operator+ (const Vector_d< Kernel > &w)
| returns the vector with Cartesian coordinates \( v_i+w_i, 0 \leq i < d\).
Vector_d< Kernel > & | operator+= (const Vector_d< Kernel > &w)
| addition plus assignment.
Vector_d< Kernel > | operator- (const Vector_d< Kernel > &w)
| returns the vector with Cartesian coordinates \( v_i-w_i, 0 \leq i < d\).
Vector_d< Kernel > & | operator-= (const Vector_d< Kernel > &w)
| subtraction plus assignment.
Vector_d< Kernel > | operator- ()
| returns the vector in opposite direction.
bool | is_zero ()
| returns true if `v` is the zero vector.
Vector_d< Kernel > | operator* (const RT &n, const Vector_d< Kernel > &v)
| returns the vector with Cartesian coordinates \( n v_i\).
Vector_d< Kernel > | operator* (const FT &r, const Vector_d< Kernel > &v)
| returns the vector with Cartesian coordinates \( r v_i, 0 \leq i < d\).

## Constructor & Destructor Documentation

template<typename Kernel >

CGAL::Vector_d< Kernel >::Vector_d | ( | int | d,
|  | Null_vector |
| ) |  |

introduces the zero vector `v` of type `Vector_d<Kernel>` in \( d\)-dimensional space.

For the creation flag `CGAL::NULL_VECTOR` can be used.

template<typename Kernel >
template<class InputIterator >

CGAL::Vector_d< Kernel >::Vector_d | ( | int | d,
|  | InputIterator | first,
|  | InputIterator | last
| ) |  |

introduces a variable `v` of type `Vector_d<Kernel>` in dimension `d`.

If `size [first,last) == d` this creates a vector with Cartesian coordinates `set [first,last)`. If `size [first,last) == p+1` the range specifies the homogeneous coordinates \( H = set [first,last) = (\pm h_0, \pm h_1, \ldots, \pm h_d)\) where the sign chosen is the sign of \( h_d\).

Precondition
  `d` is nonnegative, `[first,last)` has `d` or `d+1` elements where the last has to be non-zero.

Template Parameters

  InputIterator | has `RT` as value type.

template<typename Kernel >
template<class InputIterator >

CGAL::Vector_d< Kernel >::Vector_d | ( | int | d,
|  | InputIterator | first,
|  | InputIterator | last,
|  | RT | D
| ) |  |

introduces a variable `v` of type `Vector_d<Kernel>` in dimension `d` initialized to the vector with homogeneous coordinates as defined by `H = set [first,last)` and `D`: \( (\pmH[0], \pmH[1], \ldots, \pmH[d-1], \pmD)\).

The sign chosen is the sign of \( D\).

Precondition
  `D` is non-zero, the iterator range defines a \( d\)-tuple of `RT`.

Template Parameters

  InputIterator | has `RT` as value type.

template<typename Kernel >

CGAL::Vector_d< Kernel >::Vector_d | ( | int | d,
|  | Base_vector | ,
|  | int | i
| ) |  |

returns a variable `v` of type `Vector_d<Kernel>` initialized to the \( i\)-th base vector of dimension \( d\).

Precondition
  \( 0 \leq i < d\).

template<typename Kernel >

CGAL::Vector_d< Kernel >::Vector_d | ( | RT | x,
|  | RT | y,
|  | RT | w = `1`
| ) |  |

introduces a variable `v` of type `Vector_d<Kernel>` in \( 2\)-dimensional space.

Precondition
  \( w \neq0\).

template<typename Kernel >

CGAL::Vector_d< Kernel >::Vector_d | ( | RT | x,
|  | RT | y,
|  | RT | z,
|  | RT | w
| ) |  |

introduces a variable `v` of type `Vector_d<Kernel>` in \( 3\)-dimensional space.

Precondition
  \( w \neq0\).

## Member Function Documentation

template<typename Kernel >

FT CGAL::Vector_d< Kernel >::cartesian | ( | int | i) |  |

returns the \( i\)-th Cartesian coordinate of `v`.

Precondition
  \( 0 \leq i < d\).

template<typename Kernel >

RT CGAL::Vector_d< Kernel >::homogeneous | ( | int | i) |  |

returns the \( i\)-th homogeneous coordinate of `v`.

Precondition
  \( 0 \leq i \leq d\).

template<typename Kernel >

FT CGAL::Vector_d< Kernel >::operator[] | ( | int | i) |  |

returns the \( i\)-th Cartesian coordinate of `v`.

Precondition
  \( 0 \leq i < d\).