SPHERE_DESIGN_RULE
Hardin and Sloane Spherical Designs

SPHERE_DESIGN_RULE is a FORTRAN90 library which implements a number of Hardin and Sloane "spherical designs", which can be used for numerical quadrature over the surface of a sphere in 3D.

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 X^AY^BZ^C is A+B+C.

Licensing:

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

Languages:

SPHERE_DESIGN_RULE is available in a FORTRAN90 version.

Related Data and Programs:

CUBE_FELIPPA_RULE, a FORTRAN90 library which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

PYRAMID_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

SPHERE_CVT, a FORTRAN90 library which creates a mesh of well-separated points on a unit sphere using Centroidal Voronoi Tessellations.

SPHERE_DELAUNAY, a MATLAB program which computes the Delaunay triangulation of points on a sphere.

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_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of a quadrature rule over the surface of the unit sphere in 3D.

SPHERE_GRID, a dataset directory which contains grids of points, lines, triangles or quadrilaterals on a sphere;

SPHERE_INTEGRALS, a FORTRAN90 library which defines test functions for integration over the surface of the unit sphere in 3D.

SPHERE_LEBEDEV_RULE, a FORTRAN90 library which computes Lebedev quadrature rules for the unit sphere;

SPHERE_MONTE_CARLO, a FORTRAN90 library which applies a Monte Carlo method to estimate the integral of a function over the surface of the sphere in 3D;

SPHERE_QUAD, a FORTRAN90 library which approximates an integral over the surface of the unit sphere by applying a triangulation to the surface;

SPHERE_STEREOGRAPH, a FORTRAN90 library which computes the stereographic mapping between points on the unit sphere and points on the plane Z = 1; a generalized mapping is also available.

SPHERE_TRIANGLE_QUAD, a FORTRAN90 library which estimates the integral of a function over a spherical triangle.

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_VOLUME_QUAD, a FORTRAN90 program which applies a quadrature rule to estimate the volume of the unit 6D sphere;

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.

SQUARE_FELIPPA_RULE, a FORTRAN90 library which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.

STRIPACK, a FORTRAN90 library which computes the Delaunay triangulation or Voronoi diagram of points on a sphere.

STRIPACK_DELAUNAY, a FORTRAN90 program which reads a set of points on the unit sphere, computes the Delaunay triangulation, and writes it to a file.

STROUD, a FORTRAN90 library which defines quadrature rules for various geometric shapes.

SXYZ_DELAUNAY, a FORTRAN90 program which computes and plots the Delaunay triangulation of points over the surface of the unit sphere in 3D.

SXYZ_VORONOI, a FORTRAN90 library which computes and plots Delaunay triangulations and Voronoi diagrams of points on the sphere.

TETRAHEDRON_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TRIANGLE_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.

WEDGE_FELIPPA_RULE, a FORTRAN90 library which returns quadratures rules for approximating integrals over the interior of the unit wedge in 3D.

Reference:

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

Source Code:

• sphere_design_rule.f90, the source code.

Examples and Tests:

• sphere_design_rule_test.f90, a sample problem.
• sphere_design_rule_test.txt, the output file.
• sphere_design_rule_18.txt, a file written out by the program, containing sphere design #18.
• sphere_design_rule_18.png, a PNG image of sphere design #18 created by SPHERE_XYZ_DISPLAY.

List of Routines:

• DESIGN_00_001_3D returns a 0-design of 1 point in 3D.
• DESIGN_01_002_3D returns a 1-design of 2 points in 3D.
• DESIGN_02_004_3D returns a 2-design of 4 points in 3D.
• DESIGN_03_006_3D returns a 3-design of 6 points in 3D.
• DESIGN_04_014_3D returns a 4-design of 14 points in 3D.
• DESIGN_05_012_3D returns a 5-design of 12 points in 3D.
• DESIGN_06_026_3D returns a 6-design of 26 points in 3D.
• DESIGN_07_024_3D returns a 7-design of 24 points in 3D.
• DESIGN_08_036_3D returns an 8-design of 36 points in 3D.
• DESIGN_09_048_3D returns a 9-design of 48 points in 3D.
• DESIGN_10_060_3D returns a 10-design of 60 points in 3D.
• DESIGN_11_070_3D returns an 11-design of 70 points in 3D.
• DESIGN_12_084_3D returns a 12-design of 84 points in 3D.
• DESIGN_13_094_3D returns a 13-design of 94 points in 3D.
• DESIGN_14_108_3D returns a 14-design of 108 points in 3D.
• DESIGN_15_120_3D returns a 15-design of 120 points in 3D.
• DESIGN_16_144_3D returns a 16-design of 144 points in 3D.
• DESIGN_17_156_3D returns a 17-design of 156 points in 3D.
• DESIGN_18_180_3D returns an 18-design of 180 points in 3D.
• DESIGN_19_204_3D returns a 19-design of 204 points in 3D.
• DESIGN_20_216_3D returns a 20-design of 216 points in 3D.
• DESIGN_21_240_3D returns a 21-design of 240 points in 3D.
• DESIGN_ORDER_MAX returns the maximum design order.
• DESIGN_POINTS returns the points for a given 3D spherical design.
• DESIGN_QUAD approximates an integral with a design of a given order.
• DESIGN_SIZE returns the size of a 3D spherical design of a given order.
• GET_UNIT returns a free FORTRAN unit number.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8MAT_WRITE writes an R8MAT file.
• S_CAT concatenates two strings to make a third string.
• SPHERE_AREA_3D computes the surface area of a sphere in 3D.
• SPHERE_MONOMIAL_INT_3D integrates a monomial on a sphere in 3D.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.