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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 C++ 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 = "../../c_src/sphere_triangle_quad/sphere_triangle_quad.html">a C version</a> and
<a href = "../../cpp_src/sphere_triangle_quad/sphere_triangle_quad.html">a C++ version</a> and
<a href = "../../f77_src/sphere_triangle_quad/sphere_triangle_quad.html">a FORTRAN77 version</a> and
<a href = "../../f_src/sphere_triangle_quad/sphere_triangle_quad.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sphere_triangle_quad/sphere_triangle_quad.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/random_data/random_data.html">
RANDOM_DATA</a>,
a C++ library which
generates sample points for
various probability distributions, spatial dimensions, and geometries;
</p>
<p>
<a href = "../../cpp_src/sphere_exactness/sphere_exactness.html">
SPHERE_EXACTNESS</a>,
a C++ program which
tests the polynomial exactness of a quadrature rule for the unit sphere;
</p>
<p>
<a href = "../../cpp_src/sphere_grid/sphere_grid.html">
SPHERE_GRID</a>,
a C++ 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 = "../../cpp_src/sphere_lebedev_rule/sphere_lebedev_rule.html">
SPHERE_LEBEDEV_RULE</a>,
a C++ library which
computes Lebedev quadrature rules for the unit sphere;
</p>
<p>
<a href = "../../cpp_src/sphere_monte_carlo/sphere_monte_carlo.html">
SPHERE_MONTE_CARLO</a>,
a C++ 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 = "../../cpp_src/sphere_quad/sphere_quad.html">
SPHERE_QUAD</a>,
a C++ 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_triangle_monte_carlo/sphere_triangle_monte_carlo.html">
SPHERE_TRIANGLE_MONTE_CARLO</a>,
a C++ library which
applies a Monte Carlo method to estimate the integral of a function
over a spherical triangle on the surface of the unit sphere in 3D;
</p>
<p>
<a href = "../../cpp_src/stroud/stroud.html">
STROUD</a>,
a C++ 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 = "../../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.cpp">sphere_triangle_quad.cpp</a>, the source code.
</li>
<li>
<a href = "sphere_triangle_quad.hpp">sphere_triangle_quad.hpp</a>,
the include file.
</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.cpp">sphere_triangle_quad_prb.cpp</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>ARC_SINE</b> computes the arc sine function, with argument truncation.
</li>
<li>
<b>ATAN4</b> computes the inverse tangent of the ratio Y / X.
</li>
<li>
<b>I4_POWER</b> returns the value of I^J.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_MAX</b> returns the maximum of two R8's.
</li>
<li>
<b>R8_MIN</b> returns the minimum of two R8's.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R8VEC_DOT_PRODUCT</b> computes the dot product of a pair of R8VEC's.
</li>
<li>
<b>R8VEC_NORM</b> returns the L2 norm of an R8VEC.
</li>
<li>
<b>R8VEC_POLARIZE</b> decomposes an R8VEC into normal and parallel components.
</li>
<li>
<b>SPHERE01_DISTANCE_XYZ</b> computes great circle distances on a unit sphere.
</li>
<li>
<b>SPHERE01_SAMPLE</b> picks random points on a unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_ANGLES_TO_AREA:</b> area of spherical triangle on unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_PROJECT</b> projects from plane to spherical triangle.
</li>
<li>
<b>SPHERE01_TRIANGLE_PROJECT2</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_QUAD_ICOS1M:</b> midside rule, subdivide then project.
</li>
<li>
<b>SPHERE01_TRIANGLE_QUAD_ICOS1V:</b> vertex rule, subdivide then project.
</li>
<li>
<b>SPHERE01_TRIANGLE_QUAD_ICOS2V:</b> vertex rule, subdivide then project.
</li>
<li>
<b>SPHERE01_TRIANGLE_SAMPLE:</b> sample points from triangle on unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_SIDES_TO_ANGLES:</b> angles of spherical triangle on unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_VERTICES_TO_AREA:</b> area of spherical triangle on unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_VERTICES_TO_CENTROID:</b> centroid of spherical triangle on unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_VERTICES_TO_MIDPOINTS</b> gets the midsides of a spherical triangle.
</li>
<li>
<b>SPHERE01_TRIANGLE_VERTICES_TO_SIDES_3D:</b> sides of spherical 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 = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 29 September 2010.
</i>
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