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<html>
<head>
<title>
SPHERE_LEBEDEV_RULE - Quadrature Rules for the Unit Sphere
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPHERE_LEBEDEV_RULE <br> Quadrature Rules for the Unit Sphere
</h1>
<hr>
<p>
<b>SPHERE_LEBEDEV_RULE</b>
is a C++ library which
computes Lebedev quadrature rules
over the surface of the unit sphere in 3D.
</p>
<p>
Vyacheslav Lebedev determined a family of 65 quadrature rules for the
unit sphere, increasing in precision from 3 to 131, by 2 each time.
This software library computes any one of a subset of 32 of these rules.
</p>
<p>
Each rule is defined as a list of <b>N</b> values of <b>theta</b>,
<b>phi</b>, and <b>w</b>.
Here:
<ul>
<li>
<b>theta</b> is a longitudinal angle, measured in degrees,
and ranging from -180 to +180.
</li>
<li>
<b>phi</b> is a latitudinal angle, measured in degrees,
and ranging from 0 to 180.
</li>
<li>
<b>w</b> is a weight.
</li>
</ul>
</p>
<p>
Of course, each pair of values
(<b>theta<sub>i</sub></b>, <b>phi<sub>i</sub></b>) has a corresponding
Cartesian representation:
<blockquote>
<b>x<sub>i</sub></b> = cos ( <b>theta<sub>i</sub></b> ) * sin ( <b>phi<sub>i</sub></b> )<br>
<b>y<sub>i</sub></b> = sin ( <b>theta<sub>i</sub></b> ) * sin ( <b>phi<sub>i</sub></b> )<br>
<b>z<sub>i</sub></b> = cos ( <b>phi<sub>i</sub></b> )<br>
</blockquote>
which may be more useful when evaluating integrands.
</p>
<p>
The integral of a function <b>f(x,y,z)</b> over the surface of the
unit sphere can be approximated by
<blockquote>
integral <b>f(x,y,z)</b> = 4 * pi * sum ( 1 <= i <= <b>N</b> )
f(<b>x<sub>i</sub>,y<sub>i</sub>,z<sub>i</sub></b>)
</blockquote>
</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_LEBEDEV_RULE</b> is available in
<a href = "../../c_src/sphere_lebedev_rule/sphere_lebedev_rule.html">a C version</a> and
<a href = "../../cpp_src/sphere_lebedev_rule/sphere_lebedev_rule.html">a C++ version</a> and
<a href = "../../f77_src/sphere_lebedev_rule/sphere_lebedev_rule.html">a FORTRAN77 version</a> and
<a href = "../../f_src/sphere_lebedev_rule/sphere_lebedev_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sphere_lebedev_rule/sphere_lebedev_rule.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Programs:
</h3>
<p>
<a href = "../../cpp_src/circle_rule/circle_rule.html">
CIRCLE_RULE</a>,
a C++ library which
computes quadrature rules
over the circumference of the unit circle in 2D.
</p>
<p>
<a href = "../../cpp_src/cube_felippa_rule/cube_felippa_rule.html">
CUBE_FELIPPA_RULE</a>,
a C++ library which
returns the points and weights of a Felippa quadrature rule
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../cpp_src/pyramid_felippa_rule/pyramid_felippa_rule.html">
PYRAMID_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a pyramid in 3D.
</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 using Centroidal Voronoi Tessellations,
over the surface of the unit sphere in 3D.
</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 = "../../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 = "../../datasets/sphere_lebedev_rule/sphere_lebedev_rule.html">
SPHERE_LEBEDEV_RULE</a>,
a dataset directory which
contains sets of points on a unit sphere which can be used for
quadrature rules of a known precision;
</p>
<p>
<a href = "../../m_src/sphere_lebedev_rule_display/sphere_lebedev_rule_display.html">
SPHERE_LEBEDEV_RULE_DISPLAY</a>,
a MATLAB program which
reads a file defining a Lebedev quadrature rule for the unit sphere and
displays the point locations.
</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 unit sphere in 3D;
</p>
<p>
<a href = "../../cpp_src/sphere_quad/sphere_quad.html">
SPHERE_QUAD</a>,
a C++ library which
approximates an integral by applying a triangulation
over the surface of the unit sphere in 3D;
</p>
<p>
<a href = "../../cpp_src/sphere_triangle_quad/sphere_triangle_quad.html">
SPHERE_TRIANGLE_QUAD</a>,
a C++ library which
estimates the integral of a function over a spherical triangle.
</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 3D graphics window.
</p>
<p>
<a href = "../../cpp_src/sphere_xyz_display_opengl/sphere_xyz_display_opengl.html">
SPHERE_XYZ_DISPLAY_OPENGL</a>,
a C++ program which
reads XYZ information defining points in 3D,
and displays a unit sphere and the points, using OpenGL.
</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 = "../../cpp_src/square_felippa_rule/square_felippa_rule.html">
SQUARE_FELIPPA_RULE</a>,
a C++ library which
returns the points and weights of a Felippa quadrature rule
over the interior of a square in 2D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_felippa_rule/tetrahedron_felippa_rule.html">
TETRAHEDRON_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/triangle_fekete_rule/triangle_fekete_rule.html">
TRIANGLE_FEKETE_RULE</a>,
a C++ library which
defines Fekete rules for interpolation or quadrature
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_felippa_rule/triangle_felippa_rule.html">
TRIANGLE_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/wedge_felippa_rule/wedge_felippa_rule.html">
WEDGE_FELIPPA_RULE</a>,
a C++ library which
returns quadratures rules for approximating integrals
over the interior of the unit wedge in 3D.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Axel Becke,<br>
A multicenter numerical integration scheme for polyatomic molecules,<br>
Journal of Chemical Physics,<br>
Volume 88, Number 4, 15 February 1988, pages 2547-2553.
</li>
<li>
Vyacheslav Lebedev, Dmitri Laikov,<br>
A quadrature formula for the sphere of the 131st
algebraic order of accuracy,<br>
Russian Academy of Sciences Doklady Mathematics,<br>
Volume 59, Number 3, 1999, pages 477-481.
</li>
<li>
Vyacheslav Lebedev,<br>
A quadrature formula for the sphere of 59th algebraic
order of accuracy,<br>
Russian Academy of Sciences Doklady Mathematics, <br>
Volume 50, 1995, pages 283-286.
</li>
<li>
Vyacheslav Lebedev, A.L. Skorokhodov,<br>
Quadrature formulas of orders 41, 47, and 53 for the sphere,<br>
Russian Academy of Sciences Doklady Mathematics, <br>
Volume 45, 1992, pages 587-592.
</li>
<li>
Vyacheslav Lebedev,<br>
Spherical quadrature formulas exact to orders 25-29,<br>
Siberian Mathematical Journal, <br>
Volume 18, 1977, pages 99-107.
</li>
<li>
Vyacheslav Lebedev,<br>
Quadratures on a sphere,<br>
Computational Mathematics and Mathematical Physics, <br>
Volume 16, 1976, pages 10-24.
</li>
<li>
Vyacheslav Lebedev,<br>
Values of the nodes and weights of ninth to seventeenth
order Gauss-Markov quadrature formulae invariant under the
octahedron group with inversion,<br>
Computational Mathematics and Mathematical Physics,<br>
Volume 15, 1975, pages 44-51.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sphere_lebedev_rule.cpp">sphere_lebedev_rule.cpp</a>, the source code;
</li>
<li>
<a href = "sphere_lebedev_rule.hpp">sphere_lebedev_rule.hpp</a>, the include file.
</li>
<li>
<a href = "sphere_lebedev_rule.sh">sphere_lebedev_rule.sh</a>,
commands to compile the source code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "sphere_lebedev_rule_prb.cpp">sphere_lebedev_rule_prb.cpp</a>, the calling program;
</li>
<li>
<a href = "sphere_lebedev_rule_prb.sh">sphere_lebedev_rule_prb.sh</a>,
commands to compile, link and run the calling program;
</li>
<li>
<a href = "sphere_lebedev_rule_prb_output.txt">sphere_lebedev_rule_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>AVAILABLE_TABLE</b> returns the availability of a Lebedev rule.
</li>
<li>
<b>GEN_OH</b> generates points under OH symmetry.
</li>
<li>
<b>LD_BY_ORDER</b> returns a Lebedev angular grid given its order.
</li>
<li>
<b>LD0006</b> computes the 6 point Lebedev angular grid.
</li>
<li>
<b>LD0014</b> computes the 14 point Lebedev angular grid.
</li>
<li>
<b>LD0026</b> computes the 26 point Lebedev angular grid.
</li>
<li>
<b>LD0038</b> computes the 38 point Lebedev angular grid.
</li>
<li>
<b>LD0050</b> computes the 50 point Lebedev angular grid.
</li>
<li>
<b>LD0074</b> computes the 74 point Lebedev angular grid.
</li>
<li>
<b>LD0086</b> computes the 86 point Lebedev angular grid.
</li>
<li>
<b>LD0110</b> computes the 110 point Lebedev angular grid.
</li>
<li>
<b>LD0146</b> computes the 146 point Lebedev angular grid.
</li>
<li>
<b>LD0170</b> computes the 170 point Lebedev angular grid.
</li>
<li>
<b>LD0194</b> computes the 194 point Lebedev angular grid.
</li>
<li>
<b>LD0230</b> computes the 230 point Lebedev angular grid.
</li>
<li>
<b>LD0266</b> computes the 266 point Lebedev angular grid.
</li>
<li>
<b>LD0302</b> computes the 302 point Lebedev angular grid.
</li>
<li>
<b>LD0350</b> computes the 350 point Lebedev angular grid.
</li>
<li>
<b>LD0434</b> computes the 434 point Lebedev angular grid.
</li>
<li>
<b>LD0590</b> computes the 590 point Lebedev angular grid.
</li>
<li>
<b>LD0770</b> computes the 770 point Lebedev angular grid.
</li>
<li>
<b>LD0974</b> computes the 974 point Lebedev angular grid.
</li>
<li>
<b>LD1202</b> computes the 1202 point Lebedev angular grid.
</li>
<li>
<b>LD1454</b> computes the 1454 point Lebedev angular grid.
</li>
<li>
<b>LD1730</b> computes the 1730 point Lebedev angular grid.
</li>
<li>
<b>LD2030</b> computes the 2030 point Lebedev angular grid.
</li>
<li>
<b>LD2354</b> computes the 2354 point Lebedev angular grid.
</li>
<li>
<b>LD2702</b> computes the 2702 point Lebedev angular grid.
</li>
<li>
<b>LD3074</b> computes the 3074 point Lebedev angular grid.
</li>
<li>
<b>LD3470</b> computes the 3470 point Lebedev angular grid.
</li>
<li>
<b>LD3890</b> computes the 3890 point Lebedev angular grid.
</li>
<li>
<b>LD4334</b> computes the 4334 point Lebedev angular grid.
</li>
<li>
<b>LD4802</b> computes the 4802 point Lebedev angular grid.
</li>
<li>
<b>LD5294</b> computes the 5294 point Lebedev angular grid.
</li>
<li>
<b>LD5810</b> computes the 5810 point Lebedev angular grid.
</li>
<li>
<b>ORDER_TABLE</b> returns the order of a Lebedev rule.
</li>
<li>
<b>PRECISION_TABLE</b> returns the precision of a Lebedev rule.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>XYZ_TO_TP</b> converts (X,Y,Z) to (Theta,Phi) coordinates on the unit sphere.
</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 13 September 2010.
</i>
<!-- John Burkardt -->
</body>
</html>