CGAL::Hyperbolic_octagon_translation< FT > Class Template Reference

#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_letters.

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_letters.

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 _.