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<html>
<head>
<title>
SPHERE_VORONOI - Voronoi Diagram of Points on the Unit Sphere
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPHERE_VORONOI <br> Voronoi Diagram of Points on the Unit Sphere
</h1>
<hr>
<p>
<b>SPHERE_VORONOI</b>
is a MATLAB library which
computes the Voronoi diagram of points on the unit sphere.
</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>
<p>
<b>SPHERE_VORONOI</b> uses this approach, by calling MATLAB's <b>convhulln</b>
function to generate the convex hull. The information defining the convex hull
is actually the Delaunay triangulation of the points. From here, it is possible
to compute the Voronoi vertices, and to determine how these vertices are
joined to form the Voronoi polygons.
</p>
<p>
The code, as presented here, is quite preliminary. In particular, the process
of converting the Delaunay information into information about the Voronoi polygons
is inefficient. I suspect, though, that I can compute the centroids almost
immediately, without having to go through the tedious process of determining
the ordering of the Voronoi vertices that constitute each Voronoi polygon.
If I can clear that up, then it should be possible to apply this simple algorithm
to systems with hundreds of points.
</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_VORONOI</b> is available in
<a href = "../../f_src/sphere_voronoi/sphere_voronoi.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sphere_voronoi/sphere_voronoi.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/geometry/geometry.html">
GEOMETRY</a>,
a MATLAB library which
computes various geometric quantities, including grids on spheres.
</p>
<p>
<a href = "../../m_src/sphere_cvt/sphere_cvt.html">
SPHERE_CVT</a>,
a MATLAB library which
creates a mesh of well-separated points on a unit sphere by applying the
Centroidal Voronoi Tessellation (CVT) iteration.
</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 = "../../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 = "../../m_src/sphere_grid/sphere_grid.html">
SPHERE_GRID</a>,
a MATLAB library which
provides a number of ways of generating grids of points, or of
points and lines, or of points and lines and faces, over the unit sphere.
</p>
<p>
<a href = "../../m_src/sphere_quad/sphere_quad.html">
SPHERE_QUAD</a>,
a MATLAB library which
approximates an integral over the surface of the unit sphere
by applying a triangulation to the surface;
</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 = "../../m_src/sphere_xyzf_display/sphere_xyzf_display.html">
SPHERE_XYZF_DISPLAY</a>,
a MATLAB program which
reads XYZF information defining points and faces,
and displays a unit sphere, the points, and the faces,
in the MATLAB 3D graphics window. This can be used, for instance, to
display Voronoi diagrams or Delaunay triangulations on the unit sphere.
</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 unit sphere.
</p>
<p>
<a href = "../../f_src/stripack_voronoi/stripack_voronoi.html">
STRIPACK_VORONOI</a>,
a FORTRAN90 program which
reads an XYZ file of 3D points on
the unit sphere, computes the Voronoi diagram, and writes it
to a file.
</p>
<p>
<a href = "../../f77_src/toms772/toms772.html">
TOMS772</a>,
a FORTRAN77 library which
is the original text of the STRIPACK program.
</p>
<p>
<a href = "../../m_src/voronoi_plot/voronoi_plot.html">
VORONOI_PLOT</a>,
a MATLAB program which
plots the Voronoi neighborhoods of points using L1, L2, LInfinity
or arbitrary LP norms;
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<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>
Robert Renka,<br>
Algorithm 772: <br>
STRIPACK:
Delaunay Triangulation and Voronoi Diagram on the Surface
of a Sphere,<br>
ACM Transactions on Mathematical Software,<br>
Volume 23, Number 3, September 1997, pages 416-434.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "arc_cosine.m">arc_cosine.m</a>,
computes the inverse cosine function;
</li>
<li>
<a href = "i4col_compare.m">i4col_compare.m</a>,
compares two columns of an I4COL;
</li>
<li>
<a href = "i4col_sort_a.m">i4col_sort_a.m</a>,
ascending sorts the columns of an I4COL;
</li>
<li>
<a href = "i4col_swap.m">
i4col_swap.m</a>,
swaps two columns of an I4COL;
</li>
<li>
<a href = "i4list_print.m">i4list_print.m</a>,
prints an I4LIST;
</li>
<li>
<a href = "i4mat_transpose_print.m">i4mat_transpose_print.m</a>,
prints an I4MAT, transposed;
</li>
<li>
<a href = "i4mat_transpose_print_some.m">
i4mat_transpose_print_some.m</a>,
prints some of an I4MAT, transposed;
</li>
<li>
<a href = "i4vec_print.m">i4vec_print.m</a>,
prints an I4VEC.
</li>
<li>
<a href = "r8mat_transpose_print.m">r8mat_transpose_print.m</a>,
prints the transpose of an R8MAT;
</li>
<li>
<a href = "r8mat_transpose_print_some.m">
r8mat_transpose_print_some.m</a>,
prints some of the transpose of an R8MAT;
</li>
<li>
<a href = "r8vec_normal_01.m">r8vec_normal_01.m</a>,
returns unit pseudonormal R8VEC.
</li>
<li>
<a href = "r8vec_print.m">r8vec_print.m</a>,
prints an R8VEC.
</li>
<li>
<a href = "r8vec_uniform_01.m">r8vec_uniform_01.m</a>,
returns a unit pseudorandom R8VEC.
</li>
<li>
<a href = "sort_heap_external.m">sort_heap_external.m</a>,
external sorts a list of values into ascending order;
</li>
<li>
<a href = "sphere_delaunay.m">
sphere_delaunay.m</a>,
returns the Delaunay triangulation of points on the unit sphere.
</li>
<li>
<a href = "stri_angles_to_area.m">
stri_angles_to_area.m</a>,
computes the area of a spherical triangle;
</li>
<li>
<a href = "stri_sides_to_angles.m">
stri_sides_to_angles.m</a>,
computes the angles of a spherical triangle from its sides;
</li>
<li>
<a href = "stri_vertices_to_area.m">
stri_vertices_to_area.m</a>,
computes the area of a spherical triangle from its vertices;
</li>
<li>
<a href = "stri_vertices_to_centroid.m">
stri_vertices_to_centroid.m</a>,
computes the centroid of a spherical triangle from its vertices;
</li>
<li>
<a href = "stri_vertices_to_orientation.m">
stri_vertices_to_orientation.m</a>,
attempts to define an orientation for a spherical triangle from its vertices;
</li>
<li>
<a href = "stri_vertices_to_sides.m">
stri_vertices_to_sides.m</a>,
computes the sides of a spherical triangle from its sides;
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the current YMDHMS date as a timestamp;
</li>
<li>
<a href = "triangulation_neighbor_triangles.m">
triangulation_neighhbor_triangles.m</a>,
determines triangle neighbors in a triangulation;
</li>
<li>
<a href = "uniform_on_sphere01_map.m">
uniform_on_sphere01_map.m</a>,
returns uniform random points on the unit sphere.
</li>
<li>
<a href = "voronoi_areas.m">
voronoi_areas.m</a>,
determines the area of each Voronoi polygon, given the
Delaunay triangulation of points on the unit sphere and
the location of the Voronoi vertices and
the explicit description of the Voronoi polygons.
</li>
<li>
<a href = "voronoi_areas_direct.m">
voronoi_areas_direct.m</a>,
determines the area of each Voronoi polygon, given the
Delaunay triangulation of points on the unit sphere and
the location of the Voronoi vertices.
</li>
<li>
<a href = "voronoi_centroids.m">
voronoi_centroids.m</a>,
computes the centroids of each polygon in a Voronoi diagram.
</li>
<li>
<a href = "voronoi_data.m">
voronoi_data.m</a>,
computes Voronoi areas and centroids directly.
</li>
<li>
<a href = "voronoi_order.m">
voronoi_order.m</a>,
determines the order of each Voronoi polygon, given the
Delaunay triangulation of points on the unit sphere.
</li>
<li>
<a href = "voronoi_polygons.m">
voronoi_polygons.m</a>,
determines the Voronoi polygons, given the
Delaunay triangulation of points on the unit sphere.
</li>
<li>
<a href = "voronoi_vertices.m">
voronoi_vertices.m</a>,
determines the location of Voronoi vertices, given the
Delaunay triangulation of points on the unit sphere.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "sphere_voronoi_test.m">sphere_voronoi_test.m</a>,
a sample calling program.
</li>
<li>
<a href = "sphere_voronoi_test01.m">sphere_voronoi_test01.m</a>,
demonstrates some of the Voronoi calculations.
</li>
<li>
<a href = "sphere_voronoi_test02.m">sphere_voronoi_test02.m</a>,
displays the Voronoi diagram on a sphere.
</li>
<li>
<a href = "sphere_voronoi_test03.m">sphere_voronoi_test03.m</a>,
compares naive and direct approaches to computing the area of
the Voronoi polygons.
</li>
<li>
<a href = "sphere_voronoi_test_output.txt">sphere_voronoi_test_output.txt</a>,
the output file.
</li>
<li>
<a href = "figure_01.png">figure_01.png</a>,
an image of the Voronoi diagram made using PATCH.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 12 February 2011.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>