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<html>
<head>
<title>
SPHERE_CVT - Spherical Centroidal Voronoi Tesselation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPHERE_CVT <br> Spherical Centroidal Voronoi Tesselation
</h1>
<hr>
<p>
<b>SPHERE_CVT</b>
is a FORTRAN90 program which
seeks to determine N well-separated sites on the unit sphere in 3D,
using centroidal Voronoi tessellation (CVT) techniques.
</p>
<p>
The code assumes that good separation will follow automatically
if the points are the centroids of their Voronoi regions.
Thus, the code actually places N points at random on the sphere,
and then applies probabilistic centroidal Voronoi tessellation
techniques in an attempt to force the the CVT condition to be satisfied.
The output of the program is an
<a href = "../../data/xyz/xyz.html">XYZ file</a>
containing the coordinates of the points.
</p>
<p>
According to Steven Fortune, it is possible to compute the Delaunay
triangulation of points on a sphere by computing their convex hull.
If the sphere is the unit sphere at the origin, the facet normals are
the Voronoi vertices.
</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">
Languages:
</h3>
<p>
<b>SPHERE_CVT</b> is available in
<a href = "../../f_src/sphere_cvt/sphere_cvt.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sphere_cvt/sphere_cvt.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/sphere_delaunay/sphere_delaunay.html">
SPHERE_DELAUNAY</a>,
a FORTRAN90 program which
computes the Delaunay triangulation of points on a sphere.
</p>
<p>
<a href = "../../f_src/sphere_design_rule/sphere_design_rule.html">
SPHERE_DESIGN_RULE</a>,
a FORTRAN90 library which
returns 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 = "../../datasets/sphere_lebedev_rule/sphere_lebedev_rule.html">
SPHERE_LEBEDEV_RULE</a>,
a dataset directory which
contains sets of points on a sphere which can be used for
quadrature rules of a known precision;
</p>
<p>
<a href = "../../f_src/sphere_quad/sphere_quad.html">
SPHERE_QUAD</a>,
a FORTRAN90 library which
estimates the integral of a function defined on the sphere.
</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_voronoi/sphere_voronoi.html">
SPHERE_VORONOI</a>,
a FORTRAN90 program which
computes and plots the Voronoi diagram of points on the unit 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 = "../../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
can determine the Voronoi diagram or Delaunay triangulation of
a given set of points on the sphere.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Franz Aurenhammer,<br>
Voronoi diagrams -
a study of a fundamental geometric data structure,<br>
ACM Computing Surveys,<br>
Volume 23, pages 345-405, September 1991.
</li>
<li>
Qiang Du, Vance Faber, Max Gunzburger,<br>
Centroidal Voronoi Tesselations: Applications and Algorithms,<br>
SIAM Review, Volume 41, 1999, pages 637-676.
</li>
<li>
Jacob Goodman, Joseph ORourke, editors,<br>
Handbook of Discrete and Computational Geometry,<br>
Second Edition,<br>
CRC/Chapman and Hall, 2004,<br>
ISBN: 1-58488-301-4,<br>
LC: QA167.H36.
</li>
<li>
Douglas Hardin, Edward Saff,<br>
Discretizing Manifolds via Minimum Energy Points, <br>
Notices of the American Mathematical Society,<br>
Volume 51, Number 10, November 2004, pages 1186-1194.
</li>
<li>
Edward Saff, Arno Kuijlaars,<br>
Distributing Many Points on a Sphere, <br>
The Mathematical Intelligencer,<br>
Volume 19, Number 1, 1997, pages 5-11.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sphere_cvt.f90">sphere_cvt.f90</a>, the source code.
</li>
<li>
<a href = "sphere_cvt.sh">sphere_cvt.sh</a>,
commands to compile and load the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "random32/random32.html">32 random points.</a>
</li>
<li>
<a href = "refine_del/refine_del.html">Delaunay refinement</a>,
start with soccer ball centers, add midpoints of Delaunay sides;
</li>
<li>
<a href = "refine_vor/refine_vor.html">Voronoi refinement</a>,
start with soccer ball centers, add Voronoi vertices;
</li>
<li>
<a href = "soccer32/soccer32.html">32 soccer ball centers.</a>
</li>
</ul>
</p>
<p>
<b>SPIRAL32</b> carries out a calculation that begins with 32
points on a spiral.
<ul>
<li>
<a href = "spiral32_output.txt">spiral32_output.txt</a>,
the output file.
</li>
<li>
<a href = "spiral32_init.xyz">spiral32_init.xyz</a>,
the initial dataset.
</li>
<li>
<a href = "spiral32_init_del.png">spiral32_init_del.png</a>,
a PNG image of the Delaunay triangulation of the initial dataset.
</li>
<li>
<a href = "spiral32_init_vor.png">spiral32_init_vor.png</a>,
a PNG image of the Voronoi diagram of the initial dataset;
</li>
<li>
<a href = "spiral32_final.xyz">spiral32_final.xyz</a>,
the final dataset.
</li>
<li>
<a href = "spiral32_final_del.png">spiral32_final_del.png</a>,
a PNG image of the Delaunay triangulation of the final dataset;
</li>
<li>
<a href = "spiral32_final_vor.png">spiral32_final_vor.png</a>,
a PNG image of the Voronoi diagram of the final dataseet;
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for SPHERE_CVT.
</li>
<li>
<b>DELAUNAY_MIDPOINTS</b> returns the midpoints of a Delaunay triangulation.
</li>
<li>
<b>FIND_CLOSEST</b> finds the nearest R point to each S point.
</li>
<li>
<b>GENERATOR_INITIALIZE</b> sets initial values for the generators.
</li>
<li>
<b>HALTON_MEMORY</b> sets or returns quantities associated with the Halton sequence.
</li>
<li>
<b>HALTON_VECTOR_SEQUENCE</b> computes the next N elements in the vector Halton sequence.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of integer division.
</li>
<li>
<b>I4_TO_HALTON_VECTOR_SEQUENCE</b> computes N elements of a vector Halton sequence.
</li>
<li>
<b>I4_WRAP</b> forces an integer to lie between given limits by wrapping.
</li>
<li>
<b>I4VEC_INDICATOR</b> sets an I4VEC to the indicator vector A(I)=I.
</li>
<li>
<b>MOTION</b> computes the "motion" between two sets of points on the sphere.
</li>
<li>
<b>PRIME</b> returns any of the first PRIME_MAX prime numbers.
</li>
<li>
<b>R83VEC_UNIT_L2</b> makes each R83 vector in an R83VEC have unit L2 norm.
</li>
<li>
<b>RANDOM_INITIALIZE</b> initializes the FORTRAN 90 random number seed.
</li>
<li>
<b>SPHERE_CVT_CENTROID</b> computes the centroids of the regions.
</li>
<li>
<b>SOCCER_CENTERS</b> returns the centers of the truncated icosahedron in 3D.
</li>
<li>
<b>SOCCER_VERTICES</b> returns the vertices of the truncated icosahedron in 3D.
</li>
<li>
<b>SPHERE_UNIT_HALTONS_3D</b> picks a Halton point on the unit sphere in 3D.
</li>
<li>
<b>SPHERE_UNIT_SAMPLES_3D</b> picks a random point on the unit sphere in 3D.
</li>
<li>
<b>SPHERE_UNIT_SPIRALPOINTS_3D</b> produces spiral points on the unit sphere in 3D.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TIMESTRING</b> writes the current YMDHMS date into a string.
</li>
<li>
<b>VORONOI_VERTICES</b> returns the vertices of a Voronoi diagram.
</li>
<li>
<b>XYZ_PRINT</b> prints out a set of XYZ points.
</li>
<li>
<b>XYZ_WRITE</b> writes out a set of XYZ points to a file.
</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 26 August 2010.
</i>
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