SPHERE_LEBEDEV_RULE Quadrature Rules for the Unit Sphere

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:

• theta is a longitudinal angle, measured in degrees, and ranging from -180 to +180.
• phi is a latitudinal angle, measured in degrees, and ranging from 0 to 180.
• w is a weight.

Of course, each pair of values (thetai, phii) has a corresponding Cartesian representation:

xi = cos ( thetai ) * sin ( phii )
yi = sin ( thetai ) * sin ( phii )
zi = cos ( phii )
which may be more useful when evaluating integrands.

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(xi,yi,zi)

Languages:

SPHERE_LEBEDEV_RULE is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

Related Programs:

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.

Reference:

1. Axel Becke,
A multicenter numerical integration scheme for polyatomic molecules,
Journal of Chemical Physics,
Volume 88, Number 4, 15 February 1988, pages 2547-2553.
2. Vyacheslav Lebedev, Dmitri Laikov,
A quadrature formula for the sphere of the 131st algebraic order of accuracy,
Volume 59, Number 3, 1999, pages 477-481.
3. Vyacheslav Lebedev,
A quadrature formula for the sphere of 59th algebraic order of accuracy,
Volume 50, 1995, pages 283-286.
4. Vyacheslav Lebedev, A.L. Skorokhodov,
Quadrature formulas of orders 41, 47, and 53 for the sphere,
Volume 45, 1992, pages 587-592.
5. Vyacheslav Lebedev,
Spherical quadrature formulas exact to orders 25-29,
Siberian Mathematical Journal,
Volume 18, 1977, pages 99-107.
6. Vyacheslav Lebedev,
Computational Mathematics and Mathematical Physics,
Volume 16, 1976, pages 10-24.
7. Vyacheslav Lebedev,
Values of the nodes and weights of ninth to seventeenth order Gauss-Markov quadrature formulae invariant under the octahedron group with inversion,
Computational Mathematics and Mathematical Physics,
Volume 15, 1975, pages 44-51.

Source Code:

• available_table.m, returns the availability of a Lebedev rule.
• gen_oh.m, generates points under OH symmetry.
• ld_by_order.m, returns a Lebedev angular grid given its order.
• ld0006.m, computes the 6 point Lebedev angular grid.
• ld0014.m, computes the 14 point Lebedev angular grid.
• ld0026.m, computes the 26 point Lebedev angular grid.
• ld0038.m, computes the 38 point Lebedev angular grid.
• ld0050.m, computes the 50 point Lebedev angular grid.
• ld0074.m, computes the 74 point Lebedev angular grid.
• ld0086.m, computes the 86 point Lebedev angular grid.
• ld0110.m, computes the 110 point Lebedev angular grid.
• ld0146.m, computes the 146 point Lebedev angular grid.
• ld0170.m, computes the 170 point Lebedev angular grid.
• ld0194.m, computes the 194 point Lebedev angular grid.
• ld0230.m, computes the 230 point Lebedev angular grid.
• ld0266.m, computes the 266 point Lebedev angular grid.
• ld0302.m, computes the 302 point Lebedev angular grid.
• ld0350.m, computes the 350 point Lebedev angular grid.
• ld0434.m, computes the 434 point Lebedev angular grid.
• ld0590.m, computes the 590 point Lebedev angular grid.
• ld0770.m, computes the 770 point Lebedev angular grid.
• ld0974.m, computes the 974 point Lebedev angular grid.
• ld1202.m, computes the 1202 point Lebedev angular grid.
• ld1454.m, computes the 1454 point Lebedev angular grid.
• ld1730.m, computes the 1730 point Lebedev angular grid.
• ld2030.m, computes the 2030 point Lebedev angular grid.
• ld2354.m, computes the 2354 point Lebedev angular grid.
• ld2702.m, computes the 2702 point Lebedev angular grid.
• ld3074.m, computes the 3074 point Lebedev angular grid.
• ld3470.m, computes the 3470 point Lebedev angular grid.
• ld3890.m, computes the 3890 point Lebedev angular grid.
• ld4334.m, computes the 4334 point Lebedev angular grid.
• ld4802.m, computes the 4802 point Lebedev angular grid.
• ld5294.m, computes the 5294 point Lebedev angular grid.
• ld5810.m, computes the 5810 point Lebedev angular grid.
• order_table.m, returns the order of a Lebedev rule.
• precision_table.m, returns the precision of a Lebedev rule.
• timestamp.m, prints out the current YMDHMS date as a timestamp.
• xyz_to_tp.m, converts (X,Y,Z) to (Theta,Phi) coordinates on the unit sphere.

Last revised on 30 January 2019.