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<html>
<head>
<title>
SPHERE_DESIGN_RULE - Hardin and Sloane Spherical Designs
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPHERE_DESIGN_RULE <br> Hardin and Sloane Spherical Designs
</h1>
<hr>
<p>
<b>SPHERE_DESIGN_RULE</b>
is a FORTRAN90 library which
implements a number of Hardin and Sloane "spherical designs", which
can be used for numerical quadrature.
</p>
<p>
A set of <b>N</b> points on the surface of a 3D sphere is called a
<i>spherical T-design</i> if the integral of any polynomial <b>p(x,y,z)</b>
of degree at most <b>T</b> over the surface of the sphere is equal to
the average value of the polynomial evaluated at the set of points.
</p>
<p>
Note that the degree of a polynomial in several variables is the
highest degree of any of its terms, and that the degree of a term like
<b>X<sup>A</sup>Y<sup>B</sup>Z<sup>C</sup></b> is <b>A+B+C</b>.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/sphere_cvt/sphere_cvt.html">
SPHERE_CVT</a>,
a FORTRAN90 library which
creates a mesh of well-separated points on a unit sphere using Centroidal Voronoi
Tessellations.
</p>
<p>
<a href = "../../m_src/sphere_delaunay/sphere_delaunay.html">
SPHERE_DELAUNAY</a>,
a MATLAB program which
computes the Delaunay triangulation of points on a sphere.
</p>
<p>
<a href = "../../datasets/sphere_design_rule/sphere_design_rule.html">
SPHERE_DESIGN_RULE</a>,
a dataset directory which
contains files defining point sets on the surface of the unit sphere,
known as "designs", which can be useful for estimating integrals
on the surface, among other uses.
</p>
<p>
<a href = "../../f_src/sphere_exactness/sphere_exactness.html">
SPHERE_EXACTNESS</a>,
a FORTRAN90 program which
tests the polynomial exactness of a quadrature rule for the unit sphere;
</p>
<p>
<a href = "../../datasets/sphere_grid/sphere_grid.html">
SPHERE_GRID</a>,
a dataset directory which
contains grids of points, lines, triangles or quadrilaterals on a sphere;
</p>
<p>
<a href = "../../f_src/sphere_lebedev_rule/sphere_lebedev_rule.html">
SPHERE_LEBEDEV_RULE</a>,
a FORTRAN90 library which
computes Lebedev quadrature rules for the unit sphere;
</p>
<p>
<a href = "../../f_src/sphere_monte_carlo/sphere_monte_carlo.html">
SPHERE_MONTE_CARLO</a>,
a FORTRAN90 library which
applies a Monte Carlo method to estimate the integral of a function
over the surface of the sphere in 3D;
</p>
<p>
<a href = "../../f_src/sphere_quad/sphere_quad.html">
SPHERE_QUAD</a>,
a FORTRAN90 library which
approximates an integral over the surface of the unit sphere
by applying a triangulation to the surface;
</p>
<p>
<a href = "../../f_src/sphere_stereograph/sphere_stereograph.html">
SPHERE_STEREOGRAPH</a>,
a FORTRAN90 library which
computes the stereographic mapping between points on the unit sphere
and points on the plane Z = 1; a generalized mapping is also available.
</p>
<p>
<a href = "../../f_src/sphere_triangle_quad/sphere_triangle_quad.html">
SPHERE_TRIANGLE_QUAD</a>,
a FORTRAN90 library which
estimates the integral of a function over a spherical triangle.
</p>
<p>
<a href = "../../m_src/sphere_voronoi/sphere_voronoi.html">
SPHERE_VORONOI</a>,
a MATLAB program which
computes the Voronoi diagram of points on a sphere.
</p>
<p>
<a href = "../../cpp_src/sphere_voronoi_display_opengl/sphere_voronoi_display_opengl.html">
SPHERE_VORONOI_DISPLAY_OPENGL</a>,
a C++ program which
displays a sphere and randomly selected generator points, and then
gradually colors in points in the sphere that are closest to each generator.
</p>
<p>
<a href = "../../f_src/sphere_volume_quad/sphere_volume_quad.html">
SPHERE_VOLUME_QUAD</a>,
a FORTRAN90 program which
applies a quadrature rule to estimate the volume of the unit 6D sphere;
</p>
<p>
<a href = "../../m_src/sphere_xyz_display/sphere_xyz_display.html">
SPHERE_XYZ_DISPLAY</a>,
a MATLAB program which
reads XYZ information defining points in 3D,
and displays a unit sphere and the points in the MATLAB graphics window.
</p>
<p>
<a href = "../../f_src/stripack/stripack.html">
STRIPACK</a>,
a FORTRAN90 library which
computes the Delaunay triangulation or Voronoi diagram
of points on a sphere.
</p>
<p>
<a href = "../../f_src/stripack_interactive/stripack_interactive.html">
STRIPACK_INTERACTIVE</a>,
a FORTRAN90 program which
reads a set of points on the unit sphere, computes the Delaunay triangulation,
and writes it to a file.
</p>
<p>
<a href = "../../f_src/stroud/stroud.html">
STROUD</a>,
a FORTRAN90 library which
defines quadrature rules for
various geometric shapes.
</p>
<p>
<a href = "../../f_src/sxyz_delaunay/sxyz_delaunay.html">
SXYZ_DELAUNAY</a>,
a FORTRAN90 program which
computes and plots the Delaunay triangulation of points on the unit sphere.
</p>
<p>
<a href = "../../f_src/sxyz_voronoi/sxyz_voronoi.html">
SXYZ_VORONOI</a>,
a FORTRAN90 library which
computes and plots Delaunay triangulations and Voronoi diagrams
of points on the sphere.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Charles Colbourn, Jeffrey Dinitz,<br>
The CRC Handbook of Combinatorial Designs,<br>
CRC Press, 1996.
</li>
<li>
Ronald Hardin, Neil Sloane,<br>
McLaren's Improved Snub Cube and Other New Spherical Designs
in Three Dimensions,<br>
Discrete and Computational Geometry,<br>
Volume 15, 1996, pages 429-441.
</li>
<li>
<a href = "http://www.research.att.com/~njas/sphdesigns/">
Sloane's Spherical Designs Web Page</a>
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sphere_design_rule.f90">sphere_design_rule.f90</a>, the source code.
</li>
<li>
<a href = "sphere_design_rule.sh">sphere_design_rule_prb.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "sphere_design_rule_prb.f90">sphere_design_rule_prb.f90</a>, a sample problem.
</li>
<li>
<a href = "sphere_design_rule_prb.sh">sphere_design_rule_prb.sh</a>,
commands to compile, link and run the sample problem.
</li>
<li>
<a href = "sphere_design_rule_prb_output.txt">sphere_design_rule_prb_output.txt</a>,
the output file.
</li>
<li>
<a href = "sphere_design_rule_18.txt">sphere_design_rule_18.txt</a>,
a file written out by the program, containing sphere design #18.
</li>
<li>
<a href = "sphere_design_rule_18.png">sphere_design_rule_18.png</a>,
a PNG image of sphere design #18 created by SPHERE_XYZ_DISPLAY.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>DESIGN_00_001_3D</b> returns a 0-design of 1 point in 3D.
</li>
<li>
<b>DESIGN_01_002_3D</b> returns a 1-design of 2 points in 3D.
</li>
<li>
<b>DESIGN_02_004_3D</b> returns a 2-design of 4 points in 3D.
</li>
<li>
<b>DESIGN_03_006_3D</b> returns a 3-design of 6 points in 3D.
</li>
<li>
<b>DESIGN_04_014_3D</b> returns a 4-design of 14 points in 3D.
</li>
<li>
<b>DESIGN_05_012_3D</b> returns a 5-design of 12 points in 3D.
</li>
<li>
<b>DESIGN_06_026_3D</b> returns a 6-design of 26 points in 3D.
</li>
<li>
<b>DESIGN_07_024_3D</b> returns a 7-design of 24 points in 3D.
</li>
<li>
<b>DESIGN_08_036_3D</b> returns an 8-design of 36 points in 3D.
</li>
<li>
<b>DESIGN_09_048_3D</b> returns a 9-design of 48 points in 3D.
</li>
<li>
<b>DESIGN_10_060_3D</b> returns a 10-design of 60 points in 3D.
</li>
<li>
<b>DESIGN_11_070_3D</b> returns an 11-design of 70 points in 3D.
</li>
<li>
<b>DESIGN_12_084_3D</b> returns a 12-design of 84 points in 3D.
</li>
<li>
<b>DESIGN_13_094_3D</b> returns a 13-design of 94 points in 3D.
</li>
<li>
<b>DESIGN_14_108_3D</b> returns a 14-design of 108 points in 3D.
</li>
<li>
<b>DESIGN_15_120_3D</b> returns a 15-design of 120 points in 3D.
</li>
<li>
<b>DESIGN_16_144_3D</b> returns a 16-design of 144 points in 3D.
</li>
<li>
<b>DESIGN_17_156_3D</b> returns a 17-design of 156 points in 3D.
</li>
<li>
<b>DESIGN_18_180_3D</b> returns an 18-design of 180 points in 3D.
</li>
<li>
<b>DESIGN_19_204_3D</b> returns a 19-design of 204 points in 3D.
</li>
<li>
<b>DESIGN_20_216_3D</b> returns a 20-design of 216 points in 3D.
</li>
<li>
<b>DESIGN_21_240_3D</b> returns a 21-design of 240 points in 3D.
</li>
<li>
<b>DESIGN_ORDER_MAX</b> returns the maximum design order.
</li>
<li>
<b>DESIGN_POINTS</b> returns the points for a given 3D spherical design.
</li>
<li>
<b>DESIGN_QUAD</b> approximates an integral with a design of a given order.
</li>
<li>
<b>DESIGN_SIZE</b> returns the size of a 3D spherical design of a given order.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>R8_GAMMA</b> evaluates Gamma(X) for a real argument.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>S_CAT</b> concatenates two strings to make a third string.
</li>
<li>
<b>SPHERE_AREA_3D</b> computes the surface area of a sphere in 3D.
</li>
<li>
<b>SPHERE_MONOMIAL_INT_3D</b> integrates a monomial on a sphere in 3D.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 16 February 2010.
</i>
<!-- John Burkardt -->
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