SPHERE_LEBEDEV_RULE, a MATLAB library which computes a Lebedev quadrature rule 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 GNU LGPL license.
SPHERE_LEBEDEV_RULE is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
ANNULUS_RULE, a MATLAB library which computes a quadrature rule for estimating integrals of a function over the interior of a circular annulus in 2D.
CUBE_FELIPPA_RULE, a MATLAB library which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.
PYRAMID_FELIPPA_RULE, a MATLAB library which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.
SPHERE_CVT, a MATLAB library which creates a mesh of well-separated points using Centroidal Voronoi Tessellations over the surface of the unit sphere in 3D.
SPHERE_DESIGN_RULE, 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.
SPHERE_GRID, 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.
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.
SPHERE_QUAD, a MATLAB library which approximates an integral over the surface of the unit sphere by applying a triangulation to the surface;
SPHERE_VORONOI, a MATLAB program which computes and plots the Voronoi diagram of points over the surface of the unit sphere in 3D.
SPHERE_XYZ_DISPLAY, 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.
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 MATLAB library which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.
TETRAHEDRON_FELIPPA_RULE, a MATLAB library which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.
TRIANGLE_FEKETE_RULE, a MATLAB library which defines Fekete rules for quadrature or interpolation over the interior of a triangle in 2D.
TRIANGLE_FELIPPA_RULE, a MATLAB library which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.
WEDGE_FELIPPA_RULE, a MATLAB library which returns quadratures rules for approximating integrals over the interior of the unit wedge in 3D.