SPHERE_GRID
Grids on a Sphere

SPHERE_GRID is a FORTRAN77 library which constructs a variety of sets of points on the surface of the unit sphere.

A grid on a sphere may mean a set of points, or a set of points and lines that connect them, or a set of points, lines that connect them, and the faces that are bounded by those lines.

A grid may be desired which simply organizes areas. In that case, something like the latitude and longitude lines on a globe may be sufficient, even though "evenly spaced" latitude and longitude lines result in grid cells that are close to rectangular near the equator, but become more asymmetric near the poles.

A grid may also be desired for sampling, that is, for choosing a set of points that are well spread across the sphere. A simple Monte Carlo approach can be used, although this means that the data is only well spread out in the long view; there may be local clusters and gaps.

Other grids are generated by drawing a spiral on the surface of the sphere, and choosing points at regular spacings along that line, or by projecting an icosahedron onto the surface of the sphere, which divides the surface into 20 congruent spherical triangles, and then dealing with the simpler issue of choosing points from the triangles.

Licensing:

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

Languages:

SPHERE_GRID is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

DISTANCE_TO_POSITION_SPHERE, a MATLAB program which estimates the positions of cities on a sphere (such as the earth) based on a city-to-city distance table.

GEOMETRY, a FORTRAN77 library which performs geometric calculations in 2, 3 and N dimensional space.

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

SPHERE_DELAUNAY, a FORTRAN90 program which computes and plots the Delaunay triangulation of points on the unit sphere.

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_LEBEDEV_RULE, a FORTRAN77 library which computes Lebedev quadrature rules for the unit sphere;

SPHERE_QUAD, a FORTRAN77 library which approximates an integral over the surface of the unit sphere by applying a triangulation to the surface;

SPHERE_VORONOI, a FORTRAN90 program which computes and plots the Voronoi diagram of points on the unit 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.

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.

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

STRIPACK_INTERACTIVE, 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 FORTRAN77 library which defines quadrature rules for a variety of multidimensional regions.

XYZ_DISPLAY, a MATLAB program which reads XYZ information defining points in 3D, and displays an image in the MATLAB graphics window.

XYZ_DISPLAY_OPENGL, a C++ program which reads XYZ information defining points in 3D, and displays an image using OpenGL.

Reference:

1. Edward Saff, Arno Kuijlaars,
   Distributing Many Points on a Sphere,
   The Mathematical Intelligencer,
   Volume 19, Number 1, 1997, pages 5-11.

Source Code:

• sphere_grid.f, the source code.
• sphere_grid.sh, BASH commands to compile the source code.

Examples and Tests:

• sphere_grid_prb.f, a sample calling program.
• sphere_grid_prb.sh, BASH commands to compile and run the sample program.
• sphere_grid_prb_output.txt, the output file.

To see data files and images of the sphere grids created by this example program, go to the SPHERE_GRID dataset directory.

List of Routines:

• ARC_COSINE computes the arc cosine function, with argument truncation.
• ARC_SINE computes the arc sine function, with argument truncation.
• ATAN4 computes the inverse tangent of the ratio Y / X.
• I4MAT_TRANSPOSE_PRINT prints an I4MAT, transposed.
• I4MAT_TRANSPOSE_PRINT_SOME prints some of the transpose of an I4MAT.
• ICOS_SHAPE describes an icosahedron.
• ICOS_SIZE gives "sizes" for an icosahedron in 3D.
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT transposed.
• R8VEC_DIFF_NORM returns the L2 norm of the difference of R8VEC's.
• R8VEC_DOT_PRODUCT finds the dot product of a pair of R8VEC's.
• R8VEC_NORM returns the L2 norm of an R8VEC.
• R8VEC_POLARIZE decomposes an R8VEC into normal and parallel components.
• R8VEC_PRINT prints an R8VEC.
• SPHERE_CUBED_POINTS computes the points on a cubed sphere grid.
• SPHERE_CUBED_POINTS_FACE: points on one face of a cubed sphere grid.
• SPHERE_CUBED_POINTS_SIZE counts the points on a cubed sphere grid.
• SPHERE_DISTANCE_XYZ computes great circle distances on a sphere.
• SPHERE_GRID_ICOS_SIZE sizes an icosahedral grid on a sphere.
• SPHERE_GRID_Q4: rectangular grid on a sphere.
• SPHERE_GRID_T3 produces a triangle grid on a sphere.
• SPHERE_GRIDPOINTS_ICOS1 returns icosahedral grid points on a sphere.
• SPHERE_GRIDPOINTS_ICOS2 returns icosahedral grid points on a sphere.
• SPHERE_LINE_PROJECT projects a line onto a sphere.
• SPHERE_LL_LINES produces lines for a latitude/longitude grid.
• SPHERE_LL_LINE_NUM counts lines for a latitude/longitude grid.
• SPHERE_LL_POINTS produces points on a latitude/longitude grid.
• SPHERE_LL_POINT_NUM counts points for a latitude/longitude grid.
• SPHERE_LLQ_LINES: latitude/longitude quadrilateral grid lines.
• SPHERE_LLQ_LINE_NUM counts lines for a latitude/longitude quadrilateral grid.
• SPHERE_SPIRALPOINTS: spiral points on a sphere.
• SPHERE_UNIT_SAMPLE picks a random point on the unit sphere.
• TIMESTAMP prints out the current YMDHMS date as a timestamp.

You can go up one level to the FORTRAN77 source codes.