CGAL 5.6.3 - 2D Periodic Hyperbolic Triangulations

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CGAL::Hyperbolic_octagon_translation< FT > Class Template Reference
2D Periodic Hyperbolic Triangulations Reference » Hyperbolic Translations Classes

`#include <CGAL/Hyperbolic_octagon_translation.h>`

## Definition

template<typename FT = CORE::Expr>
class CGAL::Hyperbolic_octagon_translation< FT >

The class `Hyperbolic_octagon_translation` defines an object to represent a hyperbolic translation of the fundamental group of the Bolza surface \(\mathcal M\).

It accepts one template parameter:

Template Parameters

  FT | Field number type. Must provide exact computations with algebraic numbers, notably with nested square roots. The default value for this parameter is `CORE::Expr`.

A translation \(g\) in \(\mathcal G\) is a mapping acting on the hyperbolic plane \(\mathbb H^2\). It has the form

\[ g(z) = \frac{ \alpha\cdot z + \beta }{ \overline{\beta}\cdot z + \overline{\alpha} }, \qquad \alpha,\beta \in \mathbb C, \qquad z \in \mathbb H^2, \qquad |\alpha|^2 - |\beta|^2 = 1, \]

where \(\overline{\alpha}\) and \(\overline{\beta}\) are the complex conjugates of \(\alpha\) and \(\beta\) respectively. In this implementation, the translation \(g\) is uniquely defined by its coefficients \(\alpha\) and \(\beta\).

Considering the set of generators \(\mathcal A = [a, \overline{b}, c, \overline{d}, \overline{a}, b, \overline{c}, d]\) as an alphabet, a translation \(g\) in \(\mathcal G\) can be seen as a word on the alphabet \(\mathcal A\). Each letter of this alphabet is represented as an unsigned integer from from 0 to 7, and each word (i.e., translation) is a sequence of letters.

## Types
enum | Generator : Word_letter { A = 0 , B_BAR = 1 , C = 2 , D_BAR = 3 , A_BAR = 4 , B = 5 , C_BAR = 6 , D = 7 }
| Enumeration type for the alphabet \(\mathcal A\). More...
typedef unspecified_type | FT
| Field number type of the coefficients of \(\alpha\) and \(\beta\).
typedef unsigned short int | Word_letter
| Represents a single letter of the alphabet \(\mathcal A\).
typedef std::vector< Word_letter > | Word
| Represents a word on the alphabet \(\mathcal A\).

## Creation
| Hyperbolic_octagon_translation ()
| Default constructor. Creates the identity translation of the group \(\mathcal G\).
| Hyperbolic_octagon_translation (Word w)
| Creates the translation described by the word `w`.
| Hyperbolic_octagon_translation (Word_letter l)
| Creates the translation described by the one-letter word `l`.
| Hyperbolic_octagon_translation (Word_letter l, Word_letter m)
| Creates the translation described by the two-letter word `lm`.
| Hyperbolic_octagon_translation (Word_letter l, Word_letter m, Word_letter n)
| Creates the translation described by the three-letter word `lmn`.
| Hyperbolic_octagon_translation (Word_letter l, Word_letter m, Word_letter n, Word_letter o)
| Creates the translation described by the four-letter word `lmno`.

## Operations
Hyperbolic_octagon_translation | operator* (const Hyperbolic_octagon_translation &other) const
| Multiplication operator; composes the current translation with `other`.
Hyperbolic_octagon_translation | operator- (const Hyperbolic_octagon_translation &other) const
| Difference operator; the difference of two translations \(v\) and \(w\) is defined as \(v * w^{-1}\).
Hyperbolic_octagon_translation & | operator= (const Hyperbolic_octagon_translation &other)
| Assignment operator; modifying the translation after the assignment leaves `other` unaffected.
bool | operator== (const Hyperbolic_octagon_translation< FT > &other) const
| Equality operator.
bool | operator!= (const Hyperbolic_octagon_translation< FT > &other) const
| Inequality operator.
bool | operator< (const Hyperbolic_octagon_translation< FT > &other) const
| Comparison operator.
Hyperbolic_octagon_translation | inverse () const
| Returns the inverse of the current translation.

## Access Functions
std::pair< FT, FT > | alpha () const
| Returns the coefficient \(\alpha\) of the translation.
std::pair< FT, FT > | beta () const
| Returns the coefficient \(\beta\) of the translation.
bool | is_identity ()
| Returns `true` if the current translation represents the identity element of the group \(\mathcal G\).

## Utility Functions
std::string | to_string () const
| Returns a string representation of the translation, containing its `Word_letter`s.

## Static Access Functions
static Self | generator (const Word_letter wl)
| Return the generator `wl` of the group \(\mathcal G\).
static void | generators (std::vector< Hyperbolic_octagon_translation< FT > > &gens)
| Returns the set of generators of \(\mathcal G\) and their inverses in a `std::vector`.

## Member Typedef Documentation

## ◆ FT

template<typename FT = CORE::Expr>

typedef unspecified_type CGAL::Hyperbolic_octagon_translation< FT >::FT

Field number type of the coefficients of \(\alpha\) and \(\beta\).

This number type must be the same as the field number type `FT` of the class `Periodic_4_hyperbolic_Delaunay_triangulation_2_traits`.

See also
  `alpha()`
  `beta()`

## ◆ Word

template<typename FT = CORE::Expr>

typedef std::vector<Word_letter> CGAL::Hyperbolic_octagon_translation< FT >::Word

Represents a word on the alphabet \(\mathcal A\).

By extension, represents a hyperbolic translation in the group \(\mathcal A\).

## ◆ Word_letter

template<typename FT = CORE::Expr>

typedef unsigned short int CGAL::Hyperbolic_octagon_translation< FT >::Word_letter

Represents a single letter of the alphabet \(\mathcal A\).

By extension, represents a generator of the group \(\mathcal G\).

## Member Enumeration Documentation

## ◆ Generator

template<typename FT = CORE::Expr>

enum CGAL::Hyperbolic_octagon_translation::Generator : Word_letter

Enumeration type for the alphabet \(\mathcal A\).

This enumeration can be used to recover the generators of the group \(\mathcal G\).

See also
  `generator()`

Enumerator
A | translation \(a\)
B_BAR | translation \(\overline{b}\)
C | translation \(c\)
D_BAR | translation \(\overline{d}\)
A_BAR | translation \(\overline{a}\)
B | translation \(b\)
C_BAR | translation \(\overline{c}\)
D | translation \(d\)

## Member Function Documentation

## ◆ alpha()

template<typename FT = CORE::Expr>

std::pair< FT, FT > CGAL::Hyperbolic_octagon_translation< FT >::alpha | ( |  | ) | const

Returns the coefficient \(\alpha\) of the translation.

The first element of the returned `pair` contains the real part of \(\alpha\). The second element contains its imaginary part.

## ◆ beta()

template<typename FT = CORE::Expr>

std::pair< FT, FT > CGAL::Hyperbolic_octagon_translation< FT >::beta | ( |  | ) | const

Returns the coefficient \(\beta\) of the translation.

The first element of the returned `pair` contains the real part of \(\beta\). The second element contains its imaginary part.

## ◆ generator()

template<typename FT = CORE::Expr>

static Self CGAL::Hyperbolic_octagon_translation< FT >::generator | ( | const Word_letter | wl | ) |
| static

Return the generator `wl` of the group \(\mathcal G\).

Note that `wl` can be an element of the enumeration set Generator. The calls `generator(0)` and `generator(A)` will both return the translation \(a\).

## ◆ generators()

template<typename FT = CORE::Expr>

static void CGAL::Hyperbolic_octagon_translation< FT >::generators | ( | std::vector< Hyperbolic_octagon_translation< FT > > & | gens | ) |
| static

Returns the set of generators of \(\mathcal G\) and their inverses in a `std::vector`.

The generators are given in the order: \( [a, \overline{b}, c, \overline{d}, \overline{a}, b, \overline{c}, d].\)

## ◆ operator<()

template<typename FT = CORE::Expr>

bool CGAL::Hyperbolic_octagon_translation< FT >::operator< | ( | const Hyperbolic_octagon_translation< FT > & | other | ) | const

Comparison operator.

Each translation \(g\) of \(\mathcal G\), when applied to the octagon \(\mathcal D_O\), produces a copy of \(\mathcal D_O\) labeled by the translation \(g\). The copies of \(\mathcal D_O\) incident to \(\mathcal D_O\) are naturally ordered counter-clockwise around \(\mathcal D_O\). The comparison operator compares two translations based on the ordering of the copies of \(\mathcal D_O\) that they produce. For more details, see Section Representation of the User manual.

## ◆ to_string()

template<typename FT = CORE::Expr>

std::string CGAL::Hyperbolic_octagon_translation< FT >::to_string | ( |  | ) | const

Returns a string representation of the translation, containing its `Word_letter`s.

This function is given as utility for printing/debugging purposes. For example, for the translation \(abcd\), this function returns `0527`, and for the identity it returns `_`.