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<html>
<head>
<title>
SPHERE_TRIANGLE_QUAD - Estimate Integrals over Spherical Triangles
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPHERE_TRIANGLE_QUAD <br> Estimate Integrals over Spherical Triangles
</h1>
<hr>
<p>
<b>SPHERE_TRIANGLE_QUAD</b>
is a FORTRAN90 library which
estimates the integral of a scalar function
F(X,Y,Z) over a spherical triangle on the unit sphere.
</p>
<p>
Three methods of estimation are very crude and cannot be improved:
<ul>
<li>
the centroid rule, based on a single function value.
</li>
<li>
the vertex rule, which averages the vertex values.
</li>
<li>
the midside rule, which averages the midside values.
</li>
</ul>
</p>
<p>
One method of estimation uses random sampling, the Monte Carlo method,
whose accuracy can be improved by increasing the number of sample points.
</p>
<p>
Another method is based on the centroid rule, but allows the
user to decompose the original spherical triangle into collection
of smaller triangles. The accuracy of the estimate should improve
as the size of these triangles decreases.
</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_TRIANGLE_QUAD</b> is available in
<a href = "../../cpp_src/sphere_triangle_quad/sphere_triangle_quad.html">a C++ version</a> and
<a href = "../../f_src/sphere_triangle_quad/sphere_triangle_quad.html">a FORTRAN90 version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/random_data/random_data.html">
RANDOM_DATA</a>,
a FORTRAN90 library which
generates sample points for
various probability distributions, spatial dimensions, and geometries;
</p>
<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 = "../../f_src/sphere_delaunay/sphere_delaunay.html">
SPHERE_DELAUNAY</a>,
a FORTRAN90 program which
computes and plots the Delaunay triangulation of points on the unit 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 = "../../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 = "../../f_src/sphere_grid/sphere_grid.html">
SPHERE_GRID</a>,
a FORTRAN90 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 = "../../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_triangle_monte_carlo/sphere_triangle_monte_carlo.html">
SPHERE_TRIANGLE_MONTE_CARLO</a>,
a FORTRAN90 library which
estimates the integral of a function over a spherical triangle using the Monte Carlo method.
</p>
<p>
<a href = "../../f_src/sphere_voronoi/sphere_voronoi.html">
SPHERE_VORONOI</a>,
a FORTRAN90 program which
computes the Voronoi diagram of points on a sphere.
</p>
<p>
<a href = "../../f_src/stripack/stripack.html">
STRIPACK</a>,
a FORTRAN90 library which
computes the Voronoi diagram or Delaunay
triangulation of pointsets on a sphere.
</p>
<p>
<a href = "../../f_src/stroud/stroud.html">
STROUD</a>,
a FORTRAN90 library which
approximates the integral of a function on the surface or in the interior
of a variety of geometric shapes.
</p>
<p>
<a href = "../../m_src/xyz_display/xyz_display.html">
XYZ_DISPLAY</a>,
a MATLAB program which
reads XYZ information defining points in 3D,
and displays an image in the MATLAB graphics window.
</p>
<p>
<a href = "../../cpp_src/xyz_display_opengl/xyz_display_opengl.html">
XYZ_DISPLAY_OPENGL</a>,
a C++ program which
reads XYZ information defining points in 3D,
and displays an image using OpenGL.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ul>
<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>
</ul>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sphere_triangle_quad.f90">sphere_triangle_quad.f90</a>, the source code.
</li>
<li>
<a href = "sphere_triangle_quad.sh">sphere_triangle_quad.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "sphere_triangle_quad_prb.f90">sphere_triangle_quad_prb.f90</a>,
a sample problem.
</li>
<li>
<a href = "sphere_triangle_quad_prb.sh">sphere_triangle_quad_prb.sh</a>,
commands to compile and run the problem.
</li>
<li>
<a href = "sphere_triangle_quad_prb_output.txt">sphere_triangle_quad_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>ARC_COSINE</b> computes the arc cosine function, with argument truncation.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R8VEC_NORM</b> returns the L2 norm of an R8VEC.
</li>
<li>
<b>S_CAT</b> concatenates two strings to make a third string.
</li>
<li>
<b>SPHERE01_SAMPLE</b> picks random points on the unit sphere in 3D.
</li>
<li>
<b>SPHERE01_TRIANGLE_ANGLES_TO_AREA:</b> area of a triangle on the unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_PROJECT</b> projects from plane to spherical triangle.
</li>
<li>
<b>SPHERE01_TRIANGLE_QUAD_00:</b> quadrature over a triangle on the unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_QUAD_01:</b> quadrature over a triangle on the unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_QUAD_02:</b> quadrature over a triangle on the unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_QUAD_03:</b> quadrature over a triangle on the unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_QUAD_ICOS1C:</b> centroid rule, subdivide then project.
</li>
<li>
<b>SPHERE01_TRIANGLE_SAMPLE:</b> sample spherical triangle on unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_SIDES_TO_ANGLES:</b> angles of triangle on unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_VERTICES_TO_AREA:</b> area of triangle on unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_VERTICES_TO_CENTROID:</b> centroid of triangle on unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_VERTICES_TO_MIDPOINTS:</b> midsides of triangle on unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_VERTICES_TO_SIDES_3D:</b> sides of triangle on unit sphere.
</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 27 September 2010.
</i>
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