sphere_lebedev_rule, a FORTRAN90 code which computes a Lebedev quadrature rule for approximating integrals over the surface of the unit sphere in 3D.
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.
Each rule is defined as a list of N values of theta, phi, and w. Here:
Of course, each pair of values (theta_{i}, phi_{i}) has a corresponding Cartesian representation:
x_{i} = cos ( theta_{i} ) * sin ( phi_{i} )which may be more useful when evaluating integrands.
y_{i} = sin ( theta_{i} ) * sin ( phi_{i} )
z_{i} = cos ( phi_{i} )
The integral of a function f(x,y,z) over the surface of the unit sphere can be approximated by
integral f(x,y,z) = 4 * pi * sum ( 1 <= i <= N ) f(x_{i},y_{i},z_{i})
The computer code and data files described and made available on this web page are distributed under the MIT license
sphere_lebedev_rule is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
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PYRAMID_FELIPPA_RULE, a FORTRAN90 code which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.
SPHERE_CVT, a FORTRAN90 code which creates a mesh of well-separated points on a unit sphere using Centroidal Voronoi Tessellations.
SPHERE_DESIGN_RULE, a FORTRAN90 code 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.
SPHERE_GRID, a FORTRAN90 code 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 surface of the unit sphere in 3D.
SPHERE_INTEGRALS, a FORTRAN90 code which defines test functions for integration over the surface of the unit sphere in 3D.
SPHERE_LEBEDEV_RULE, a dataset directory which contains sets of points on a sphere which can be used for quadrature rules of a known precision;
SPHERE_LEBEDEV_RULE_DISPLAY, a MATLAB program which reads a file defining a Lebedev quadrature rule for the sphere and displays the point locations.
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SPHERE_TRIANGLE_QUAD, a FORTRAN90 code which estimates the integral of a function over a spherical triangle.
SPHERE_VORONOI, a FORTRAN90 code which computes and plots the Voronoi diagram of points over the surface of the unit sphere in 3D.
SPHERE_XYZ_DISPLAY_OPENGL, a C++ program which reads XYZ information defining points in 3D, and displays a unit sphere and the points, using OpenGL.
SPHERE_XYZF_DISPLAY, 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.
SQUARE_FELIPPA_RULE, a FORTRAN90 code which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.
TETRAHEDRON_FELIPPA_RULE, a FORTRAN90 code which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.
TRIANGLE_FELIPPA_RULE, a FORTRAN90 code which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.
WEDGE_FELIPPA_RULE, a FORTRAN90 code which returns quadratures rules for approximating integrals over the interior of the unit wedge in 3D.