ldsgen::Sphere3Hopf Class Reference

S(3) sequence generator by Hopf fibration. More...

#include <lds.hpp>

Inheritance diagram for ldsgen::Sphere3Hopf:

Collaboration diagram for ldsgen::Sphere3Hopf:

Public Member Functions
	Sphere3Hopf (const unsigned long base0, const unsigned long base1, const unsigned long base2)
	Construct a new Sphere 3 Hopf object.
auto	value_at (unsigned long n) const -> std::array< double, 4 >
	Evaluate the point on the 3-sphere at a given index (pure, no state change)
Public Member Functions inherited from ldsgen::GeneratorIterable< Sphere3Hopf, std::array< double, 4 > >
auto	begin () -> GeneratorIterator< Sphere3Hopf, std::array< double, 4 > >
	Get iterator to beginning.
auto	end () const -> GeneratorIterator< Sphere3Hopf, std::array< double, 4 > >
	Get iterator to end (infinite sequence)
Public Member Functions inherited from ldsgen::GeneratorBase< Derived, Value >
auto	pop () -> Value
	Generate the next value in the sequence (advances state).
auto	peek () -> Value
	Peek at the next value without advancing state.
auto	skip (unsigned int n) -> void
	Skip n values in the sequence.
auto	reseed (const unsigned long &seed) -> void
	Reset the generator to a specific seed value.
auto	get_index () const -> unsigned long
	Get current index in the sequence.

Additional Inherited Members
Protected Attributes inherited from ldsgen::GeneratorBase< Derived, Value >
std::atomic< unsigned long >	count_ {0}
	Current sequence index (single source of state)

Detailed Description

S(3) sequence generator by Hopf fibration.

The Sphere3Hopf class is a sequence generator that generates points on a 3-sphere using the Hopf fibration. It uses three instances of the VdCorput class to generate the sequence values and maps them to points on the 3-sphere. The pop() method returns the next point on the 3-sphere as a std::array<double, 4>, where the first three elements represent the x, y, and z coordinates of the point, and the fourth element represents the w coordinate. The reseed() method is used to reset the state of the sequence generator to a specific seed value.

*     3-Sphere (S3) visualization:
*     A 4D hypersphere where points (x,y,z,w)
*     satisfy x²+y²+z²+w² = 1
*
*         4D hypersurface
*            _____
*         .-'     '-.
*       ,'           ',
*      /               \
*     |        O        |  (4D analog of sphere)
*      \               /
*       '.           .'
*         '-.....-'
* 

Constructor & Destructor Documentation

Sphere3Hopf()

ldsgen::Sphere3Hopf::Sphere3Hopf ( const unsigned long  base0, const unsigned long  base1, const unsigned long  base2  )

inline

Construct a new Sphere 3 Hopf object.

Constructs a 3-sphere sequence generator using the Hopf fibration with the specified bases.

Parameters

[in]	base0	the base for the first Van der Corput generator (phi coordinate)
[in]	base1	the base for the second Van der Corput generator (psi coordinate)
[in]	base2	the base for the third Van der Corput generator (eta coordinate)

Member Function Documentation

value_at()

auto ldsgen::Sphere3Hopf::value_at ( unsigned long  n ) const -> std::array<double, 4>

inline

Evaluate the point on the 3-sphere at a given index (pure, no state change)

Uses the Hopf fibration parametrization:

\[ \begin{aligned} x &= \cos\eta\cos\psi \\ y &= \cos\eta\sin\psi \\ z &= \sin\eta\cos(\phi + \psi) \\ w &= \sin\eta\sin(\phi + \psi) \end{aligned} \]

where \( \phi = 2\pi\phi_{b_0}(n) \), \( \psi = 2\pi\phi_{b_1}(n) \), and \( \eta = \arccos\sqrt{\phi_{b_2}(n)} \).

Parameters

[in]

n

The sequence index.

Returns

The point on the 3-sphere for index n.

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The documentation for this class was generated from the following file:

• ldsgen/lds.hpp