SPHERE_GRID
Grids on a Sphere

SPHERE_GRID is a FORTRAN90 library which constructs a variety of sets of points over the surface of the unit sphere in 3D.

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.

The test program uses GNUPLOT to make images of some of the grids.

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 FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CUBE_GRID, a FORTRAN90 library which computes a grid of points over the interior of a cube in 3D.

DISK_GRID, a FORTRAN90 library which computes a grid of points over the interior of a disk in 2D.

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.

GNUPLOT, FORTRAN90 programs which illustrate how a program can write data and command files so that gnuplot can create plots of the program results.

HYPERCUBE_GRID, a FORTRAN90 library which computes a grid of points over the interior of a hypercube in M dimensions.

LINE_GRID, a FORTRAN90 library which computes a grid of points over the interior of a line segment in 1D.

POLYGON_GRID, a FORTRAN90 library which generates a grid of points over the interior of a polygon in 2D.

PYRAMID_GRID, a FORTRAN90 library which computes a grid of points over the interior of the unit pyramid in 3D;

SIMPLEX_GRID, a FORTRAN90 library which generates a regular grid of points over the interior of a simplex in M dimensions.

SPHERE_CUBED_GRID, a FORTRAN90 library which uses the projection of a cube to create grids of points, lines, and quadrilaterals on the surface of the unit sphere in 3D.

SPHERE_FIBONACCI_GRID, a FORTRAN90 library which uses a Fibonacci spiral to create a grid of points on the surface of the unit sphere in 3D.

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, on the surface of the unit sphere in 3D.

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_LLQ_GRID, a FORTRAN90 library which uses longitudes and latitudes to create grids of points, lines, and quadrilaterals on the surface of the unit sphere in 3D.

SPHERE_LLT_GRID, a FORTRAN90 library which uses longitudes and latitudes to create grids of points, lines, and triangles on the surface of the unit sphere in 3D.

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 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 over the surface of the unit sphere in 3D.

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.

WEDGE_GRID, a FORTRAN90 library which computes a grid of points over the interior of the unit wedge in 3D.

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.
2. Richard Swinbank, James Purser,
   Fibonacci grids: A novel approach to global modelling,
   Quarterly Journal of the Royal Meteorological Society,
   Volume 132, Number 619, July 2006 Part B, pages 1769-1793.

Source Code:

• sphere_grid.f90, the source code.

Examples and Tests:

• sphere_grid_test.f90, a sample calling program.
• sphere_grid_test.txt, the output file.
• sphere_grid_test14_commands.txt, GNUPLOT commands
• sphere_grid_test14_data.txt, graphics data.
• sphere_grid_test14.png, a graphics image.

To see the XYZ 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_NUM gives "sizes" for an icosahedron in 3D.
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8MAT_TRANSPOSE_PRINT prints a 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_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_IJK_TO_XYZ: cubed sphere IJK to XYZ coordinates.
• SPHERE_CUBED_LINE_NUM counts lines on a cubed sphere grid.
• 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_POINT_NUM counts the points on a cubed sphere grid.
• SPHERE_DISTANCE_XYZ computes great circle distances on a sphere.
• SPHERE_FIBONACCI_POINTS computes sphere points on a Fibonacci spiral.
• SPHERE_GRID_Q4: rectangular grid on a sphere.
• SPHERE_GRID_T3 produces a triangle grid on a sphere.
• SPHERE_ICOS_EDGE_NUM sizes an icosahedral grid on a sphere.
• SPHERE_ICOS_FACE_NUM sizes an icosahedral grid on a sphere.
• SPHERE_ICOS_POINT_NUM sizes an icosahedral grid on a sphere.
• SPHERE_ICOS1_POINTS returns icosahedral grid points on a sphere.
• SPHERE_ICOS2_POINTS 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 for 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 the current YMDHMS date as a time stamp.

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