Recursive Zonal Equal Area Sphere Partitioning Toolbox
SourceForge.net project page
Download the current version of the Recursive Zonal Equal Area (EQ) Sphere Partitioning Toolbox from the
EQ Sphere Partitions project page
hosted at:
FAQ  from the README
What is the Recursive Zonal Equal Area (EQ) Sphere Partitioning Toolbox?
The Recursive Zonal Equal Area (EQ) Sphere Partitioning Toolbox is a suite of
Matlab functions. These functions are intended for use in exploring different
aspects of EQ sphere partitioning.
The functions are grouped into the following groups of tasks:
 Create EQ partitions
 Find properties of EQ partitions
 Find properties of EQ point sets
 Produce illustrations
 Test the toolbox
 Perform some utility function
What is an EQ partition?
An EQ partition is a partition of S^dim [the unit sphere in the dim+1 Euclidean
space R^(dim+1)] into a finite number of regions of equal area. The area of
each region is defined using the Lebesgue measure inherited from R^(dim+1).
The diameter of a region is the sup of the Euclidean distance between any two
points of the region. The regions of an EQ partition have been proven to have
small diameter, in the sense that there exists a constant C(dim) such that the
maximum diameter of the regions of an N region EQ partition of S^dim is bounded
above by C(dim)*N^(1/dim).
What is an EQ point set?
An EQ point set is the set of center points of the regions of an EQ partition.
Each region is defined as a product of intervals in spherical polar coordinates.
The center point of a region is defined via the center points of each interval,
with the exception of spherical caps and their descendants, where the center
point is defined using the center of the spherical cap.
Which versions of Matlab can I use?
This toolbox has been tested with Matlab versions 6.5 and 7.0.1 on Linux,
and 6.5.1 on Windows.
How do I install the Recursive Zonal Equal Area Sphere Partitioning Toolbox?
This toolbox is organized into a number of directories. To use it effectively,
these directories need to be on your Matlab path every time you start Matlab.
You will therefore need to install the toolbox before using it.
To do this,

Unzip the file eqspx.y.zip into the directory where you
want the toolbox to reside. This will create the subdirectory
eq_sphere_partitions.

Run Matlab, change directory to eq_sphere_partitions and then run
install_eq_toolbox.
For more information, see INSTALL.txt.
What documentation is available?
The user documentation includes the help comments in each Matlab M file,
plus the extra files Contents.m, AUTHORS, CHANGELOG, COPYING, INSTALL.txt
and README.txt.
The extra files contain the following:
 Contents.m:
 A brief description of those functions in the toolbox which are
visible to end users.
 AUTHORS:
 Authors, acknowledgements and references.
 CHANGELOG:
 Revision history.
 COPYING:
 Software license terms.
 INSTALL.txt:
 Installation instructions in DOS CRLF text format.
 README.txt:
 Information for firsttime users in DOS CRLF text format.
How do I get help?
To see a brief description of the functions in the toolbox, enter the command
HELP EQ_SPHERE_PARTITIONS (in lower case).
The command WHAT lists all the functions in your current directory.
For each function, the command HELP FUNCTION, where FUNCTION is
the name of the function, will give the help for the function.
What is the input? What is the output?
The help for each function briefly describes the input and the output, as per
the example for partsphere, above.
Which file to begin with?
You need to find a function which does what you want to do. Examples:

Create EQ partitions

Create an array in Cartesian coordinates representing the `center' points
of an EQ partition of S^dim into N regions:
points_x = eq_point_set(dim,N);

Create an array in spherical polar coordinates representing the `center'
points of an EQ partition of S^dim into N regions:
points_s = (eq_point_set_polar(dim,N);

Create an array in polar coordinates representing the regions of an EQ
partition of S^dim into N regions:
regions = eq_regions(dim,N);

Find properties of EQ partitions

Find the (perpartition) maximum diameter bound of the EQ partition of S^dim
into N regions:
diam_bound = eq_diam_bound(dim,N);

Find properties of EQ point sets

Find the r^(s) energy and min distance of the EQ `center' point sets of
S^dim for N points:
[energy,dist] = eq_energy_dist(dim,N,s);

Produce an illustration

Use projection to illustrate the EQ partition of S^2 into N regions:
project_s2_partition(N);
Is the toolbox for use with S^2 and S^3 only? What is the maximum dimension?
In principle, any function which has dim as a parameter will work for any
integer dim >= 1. In practice, for large d, the function may be slow,
or may consume large amounts of memory.
What is the range of the number of points, N?
In principle, any function which takes N as an argument will work with any
positive integer value of N. In practice, for very large N, the function may
be slow, or may consume large amounts of memory.
What are the options for visualizing points or equal area regions?
A number of different illustrations are available:

Use a 3D plot to illustrate the EQ partition of S^2 into N regions:
show_s2_partition(N);

Use projection to illustrate the EQ partition of S^2 into N regions:
project_s2_partition(N);

Use projection to illustrate the EQ partition of S^3 into N regions.
project_s3_partition(N);

Illustrate the EQ algorithm for the partition of S^dim into N regions.
illustrate_eq_algorithm(dim,N);
See the help for these functions for details.
Presentations

A partition of the unit sphere into regions of equal area and small diameter, 2004.

Partitions of the unit sphere into regions of equal area and small diameter, 2005.
 Movie of the partition EQ(3,99)
[MS MPEG4 V2 AVI file],
[MPEG4 file],
[MPEG4 Quicktime MOV file], 2006.

The Riesz energy of point sets on the unit sphere under weakstar convergence, 2005.
 Spherical codes with good separation, discrepancy and energy,
HDA07,
ICIAM07,
AustMS07,
2007.
HDA07 talk handout, 2007.
Publications and preprints
PhD Thesis
Distributing points on the sphere: Partitions, separation, quadrature and energy
, UNSW, 2007.
Accompanying
Thesis/Dissertation Sheet.
Authors
Origin
Maple and Matlab code is based on Ed Saff [SafSP], [Saf03]
and Ian Sloan [Slo03].
References
 [Dahl78]:
 B. E. J. Dahlberg,
"On the distribution of Fekete points",
Duke Math. J. 45 (1978), no. 3, pp. 537542.
 [KuiS98]:
 A. B. J. Kuijlaars, E. B. Saff,
"Asymptotics for minimal discrete energy on the sphere",
Transactions of the American Mathematical Society, v. 350 no. 2 (Feb 1998)
pp. 523538.
 [KuiSS04]:
 A. B. J. Kuijlaars, E. B. Saff, X. Sun,
"Minimum separation f the minimal energy points on spheres in Euclidean
spaces", (preprint) 20041130.
 [LeGS01]:
 T. Le Gia, I. H. Sloan,
"The uniform norm of hyperinterpolation on the unit sphere in an arbitrary
number of dimensions", Constructive Approximation (2001) 17: p249265.
 [Mue98]:
 C. Mueller, "Analysis of spherical symmetries in Euclidean spaces",
Springer, 1998.
 [RakSZ94]:
 E. A. Rakhmanov, E. B. Saff, Y. M. Zhou,
"Minimal discrete energy on the sphere",
Mathematics Research Letters, 1 (1994), pp. 647662.
 [RakSZ95]:
 E. A. Rakhmanov, E. B. Saff, Y. M. Zhou,
"Electrons on the sphere",
Computational methods and function theory 1994 (Penang), pp. 293309,
Ser. Approx. Decompos., 5, World Sci. Publishing, River Edge, NJ, 1995.
 [SafK97]:
 E. B. Saff, A. B. J. Kuijlaars,
"Distributing many points on a sphere",
Mathematical Intelligencer, v19 no1 (1997), pp. 511.
 [SafSP]:
 E. B. Saff, Points on Spheres and Manifolds  Equal Area Points.
 [Saf03]:
 Ed Saff, "Equalarea partitions of sphere",
Presentation at UNSW, 20030728.
 [Slo03]:
 Ian Sloan, "Equal area partition of S^3", Notes, 20030729.
 [WeiMW]:
 E. W. Weisstein,
Hypersphere.
From MathWorldA Wolfram Web Resource.
 [Zho95]:
 Y. M. Zhou, "Arrangement of points on the sphere",
Thesis, University of South Florida, 1995.
 [Zho98]:
 Y. M. Zhou, "Equidistribution and extremal energy of N points on
the sphere", Modelling and computation for applications in mathematics,
science, and engineering (Evanston, IL, 1996), pp. 3957,
Numer. Math. Sci. Comput., Oxford Univ. Press, New York, 1998.
Installation and Utilities
Toolbox Installer 2.2, 20030722 by B. Rasmus Anthin.
Files modified and relicenced with permission of B. Rasmus Anthin:
./private:
install.m uninstall.m
Home page
Page Generated: Friday 9 November 2007