Hardin and Sloane Spherical Designs

SPHERE_DESIGN_RULE is a dataset directory which contains files defining a number of Hardin and Sloane "spherical designs", which can be used for numerical quadrature.

A set of N points on the surface of a 3D sphere is called a spherical T-design if the integral of any polynomial p(x,y,z) of degree at most T over the surface of the sphere is equal to the average value of the polynomial evaluated at the set of points.

Note that the degree of a polynomial in several variables is the highest degree of any of its terms, and that the degree of a term like XAYBZC is A+B+C.


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 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_LEBEDEV_RULE, a dataset directory which contains sets of Lebedev points on a sphere, which can be used for quadrature rules of a known precision;

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.

STRI_QUAD, a FORTRAN90 library which estimates the integral of a function defined on the sphere.


  1. Charles Colbourn, Jeffrey Dinitz,
    The CRC Handbook of Combinatorial Designs,
    CRC Press, 1996.
  2. Ronald Hardin, Neil Sloane,
    McLaren's Improved Snub Cube and Other New Spherical Designs in Three Dimensions,
    Discrete and Computational Geometry,
    Volume 15, 1996, pages 429-441.
  3. Sloane's Spherical Designs Web Page

Sample files

You can go up one level to the DATASETS page.

Last revised on 10 September 2010.