SPHERE_LEBEDEV_RULE
Quadrature Rules for the Sphere


SPHERE_LEBEDEV_RULE is a dataset directory which contains files defining Lebedev rules on the unit sphere, which can be used for quadrature, and have a known precision.

A Lebedev rule of precision p can be used to correctly integrate any polynomial for which the highest degree term xiyjzk satisfies i+j+k <= p.

The approximation to the integral of f(x) has the form Integral f(x,y,z) = 4 * pi * sum ( 1 <= i < n ) wi * f(xi,yi,zi) where

        xi = cos ( thetai ) * sin ( phii )
        yi = sin ( thetai ) * sin ( phii )
        zi =                  cos ( phii )
      

The data file for an n point rule includes n lines, where the i-th line lists the values of

        thetai phii wi
      
The angles are measured in degrees, and chosen so that:
        - 180 <= thetai <= + 180
            0 <= phii <= + 180
      
and the weights wi should sum to 1.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Related Data and Programs:

GEOMETRY, a FORTRAN90 library which computes various geometric quantities, including grids on spheres.

SCVT, a FORTRAN90 library which can find a set of well separated points on a sphere using Centroidal Voronoi Tessellations.

SPHERE_DESIGN_RULE, a dataset directory which contains files defining 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_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 dataset directory which contains grids of points, lines, triangles or quadrilaterals on a sphere;

SPHERE_GRID, a FORTRAN90 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_VORONOI, a MATLAB program which computes the Voronoi diagram of points on a sphere.

SPHERE_VORONOI_DISPLAY_OPENGL, a C++ program which displays a sphere and randomly selected generator points, and then gradually colors in points in the sphere that are closest to each generator.

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 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.

Reference:

  1. Thomas Ericson, Victor Zinoviev,
    Codes on Euclidean Spheres,
    Elsevier, 2001,
    ISBN: 0444503293,
    LC: QA166.7E75
  2. Gerald Folland,
    How to Integrate a Polynomial Over a Sphere,
    American Mathematical Monthly,
    Volume 108, Number 5, May 2001, pages 446-448.
  3. Vyacheslav Lebedev, Dmitri Laikov,
    A quadrature formula for the sphere of the 131st algebraic order of accuracy,
    Russian Academy of Sciences Doklady Mathematics,
    Volume 59, Number 3, 1999, pages 477-481.
  4. Vyacheslav Lebedev,
    A quadrature formula for the sphere of 59th algebraic order of accuracy,
    Russian Academy of Sciences Doklady Mathematics,
    Volume 50, 1995, pages 283-286.
  5. Vyacheslav Lebedev, A.L. Skorokhodov,
    Quadrature formulas of orders 41, 47, and 53 for the sphere,
    Russian Academy of Sciences Doklady Mathematics,
    Volume 45, 1992, pages 587-592.
  6. Vyacheslav Lebedev,
    Spherical quadrature formulas exact to orders 25-29,
    Siberian Mathematical Journal,
    Volume 18, 1977, pages 99-107.
  7. Vyacheslav Lebedev,
    Quadratures on a sphere,
    Computational Mathematics and Mathematical Physics,
    Volume 16, 1976, pages 10-24.
  8. 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.
  9. AD McLaren,
    Optimal Numerical Integration on a Sphere,
    Mathematics of Computation,
    Volume 17, Number 84, October 1963, pages 361-383.
  10. Edward Saff, Arno Kuijlaars,
    Distributing Many Points on a Sphere,
    The Mathematical Intelligencer,
    Volume 19, Number 1, 1997, pages 5-11.

Sample files

You can go up one level to the DATASETS page.


Last revised on 09 September 2010.