HyperSphere Class Reference

A hyper-sphere of K dimensions. More...

#include <HyperSphere.hpp>

Public Types
typedef rw::core::Ptr< const HyperSphere >	Ptr
	Smart pointer type for HyperSphere.

Public Member Functions
	HyperSphere (unsigned int dimensions)
	Construct a hyper-sphere of unit size. More...
virtual	~HyperSphere ()
	Destructor.
std::vector< Eigen::VectorXd >	uniformDistributionCartesian (double delta) const
	Create a uniform distribution in Cartesian coordinates. More...
std::vector< Eigen::VectorXd >	uniformDistributionSpherical (double delta) const
	Create a uniform distribution in spherical coordinates. More...
unsigned int	getDimensions () const
	Get the number of dimensions of the hyper-sphere. More...
double	area () const
	Calculate the surface area of a hyper-sphere. More...
double	volume () const
	The volume of a hyper-sphere. More...

Detailed Description

A hyper-sphere of K dimensions.

Functions are provided to create (almost) uniform distribution of points on a hyper-sphere as shown in [1].

The distribution of points is illustrated below for 2 and 3 dimensional hyper-spheres. Notice that the tessellation is best when \( \delta\) is small.

Distribution of points for K=2 and K=3.

[1] Lovisolo, L., and E. A. B. Da Silva. "Uniform distribution of points on a hyper-sphere with applications to vector bit-plane encoding." IEE Proceedings-Vision, Image and Signal Processing 148.3 (2001): 187-193.

Constructor & Destructor Documentation

HyperSphere()

HyperSphere

(

unsigned int

dimensions

)

Construct a hyper-sphere of unit size.

Parameters

dimensions

[in] the number of dimensions.

Member Function Documentation

area()

double area

(

)

const

Calculate the surface area of a hyper-sphere.

Calculated for even dimensionality as \( \frac{K \pi^{K/2}}{(K/2)!}\)

Calculated for odd dimensionality as \( \frac{K 2^K \pi^{(K-1)/2}}{K!}\)

Returns

the surface area.

getDimensions()

unsigned int getDimensions

(

)

const

Get the number of dimensions of the hyper-sphere.

Returns

the number of dimensions, \( 2 \leq K \leq 6\) .

uniformDistributionCartesian()

std::vector<Eigen::VectorXd> uniformDistributionCartesian

(

double

delta

)

const

Create a uniform distribution in Cartesian coordinates.

This uses uniformDistributionSpherical and maps the spherical coordinates to Cartesian coordinates. The mapping is documented in [1], section 2.1.

Parameters

delta

[in] the resolution.

Returns

unit vectors, \( [x_1 x_2 \dots x_K]^T\) , in Cartesian coordinates with dimension K.

Note

This function is only implemented for \( 2 \leq K \leq 6\) .

uniformDistributionSpherical()

std::vector<Eigen::VectorXd> uniformDistributionSpherical

(

double

delta

)

const

Create a uniform distribution in spherical coordinates.

This implements the algorithm in [1], section 2.1, for dimensions \( 2 \leq K \leq 6\).

Parameters

delta

[in] the resolution.

Returns

list of vectors, \( [\theta_1 \theta_2 \dots \theta_{K-1}]^T\) , in spherical coordinates with dimension K-1.

Note

This function is only implemented for \( 2 \leq K \leq 6\) .

volume()

double volume

(

)

const

The volume of a hyper-sphere.

Calculated for even dimensionality as \( \frac{\pi^{K/2}}{(K/2)!}\)

Calculated for odd dimensionality as \( \frac{2 (2 \pi)^{(K-1)/2}}{K!!}\) where the double factorial for odd K means \( 1 \cdot 3 \cdot 5 \dots K\)

Returns

the volume.

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

• HyperSphere.hpp