SPHERE_GRID
Grids on a Sphere

SPHERE_GRID, a MATLAB library which constructs a variety of sets of points over the surface of a 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.

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:

BALL_GRID, a MATLAB library which computes a grid of points over the interior of a ball in 3D.

CIRCLE_ARC_GRID, a MATLAB program which computes points equally spaced along a circular arc;

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

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

ELLIPSE_GRID, a MATLAB library which computes a grid of points over the interior of an ellipse in 2D.

ELLIPSOID_GRID, a MATLAB library which computes a grid of points over the interior of an ellipsoid in 3D.

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

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

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

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

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

SPHERE_CVT, a MATLAB library which creates a mesh of well-separated points using Centroidal Voronoi Tessellations over the surface of the unit sphere in 3D.

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

SPHERE_FIBONACCI_GRID, a MATLAB library which computes a grid of points using the Fibonacci spiral over the surface of a sphere in 3D.

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

sphere_grid_test

SPHERE_LEBEDEV_RULE, a MATLAB library which computes Lebedev quadrature rules over the surface of the unit sphere in 3D.

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

SPHERE_STEREOGRAPH, a MATLAB library which computes the stereographic mapping between points on the unit sphere and points on the plane Z = 1.

SPHERE_LLQ_GRID, a MATLAB library which computes a grid of quadrilaterals bounded by latitude and longitude lines over the surface of a sphere in 3D.

SPHERE_LLT_GRID, a MATLAB library which computes a grid of triangles bounded by latitude and longitude lines over the surface of a sphere in 3D.

SPHERE_VORONOI, a MATLAB program which computes the Voronoi diagram of points on a sphere.

SQUARE_GRID, a MATLAB library which computes a grid of points over the interior of a square in 2D.

TETRAHEDRON_GRID, a MATLAB library which computes a grid of points over the interior of a tetrahedron in 3D.

TRIANGLE_GRID, a MATLAB library which computes a grid of points over the interior of a triangle in 2D.

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

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:

• arc_cosine.m, computes the arc cosine function, with argument truncation.
• arc_sine.m, computes the arc sine function, with argument truncation.
• atan4.m, computes the inverse tangent of the ratio Y / X.
• i4mat_transpose_print.m, prints an I4MAT, transposed.
• i4mat_transpose_print_some.m, prints some of the transpose of an I4MAT.
• icos_shape.m, describes an icosahedron.
• icos_num.m, gives "sizes" for an icosahedron in 3D.
• r8_uniform_01.m, returns a unit pseudorandom R8.
• r8mat_transpose_print.m, prints a R8MAT, transposed.
• r8mat_transpose_print_some.m, prints some of an R8MAT, transposed.
• r8vec_diff_norm.m, returns the L2 norm of the difference of R8VEC's.
• r8vec_norm.m, returns the L2 norm of an R8VEC.
• r8vec_polarize.m, decomposes an R8VEC into normal and parallel components.
• r8vec_print.m, prints an R8VEC.
• sphere_cubed_ijk_to_xyz.m, converts cubed sphere I,J,K coordinates to X,Y,Z coordinates.
• sphere_cubed_line_num.m, counts the lines in a cubed sphere grid.
• sphere_cubed_lines.m, computes the lines in a cubed sphere grid.
• sphere_cubed_points.m, computes the points in a cubed sphere grid.
• sphere_cubed_points_face.m, computes points on one face of a cubed sphere grid.
• sphere_cubed_point_num.m, counts the points in a cubed sphere grid.
• sphere_distance_xyz.m, computes great circle distances on a sphere.
• sphere_grid_q4.m, rectangular grid on a sphere.
• sphere_grid_t3.m, produces a triangle grid on a sphere.
• sphere_icos_edge_num.m, counts edges on an icosahedral grid on a sphere.
• sphere_icos_face_num.m, counts faces on an icosahedral grid on a sphere.
• sphere_icos_point_num.m, counts points on an icosahedral grid on a sphere.
• sphere_icos1_points.m, returns icosahedral grid points on a sphere.
• sphere_icos2_points.m, returns icosahedral grid points on a sphere.
• sphere_line_project.m, projects a line onto a sphere.
• sphere_ll_lines.m, produces lines on a latitude/longitude grid.
• sphere_ll_line_num.m, counts lines on a latitude/longitude grid.
• sphere_ll_points.m, produces points on a latitude/longitude grid.
• sphere_ll_point_num.m, counts points on a latitude/longitude grid.
• sphere_llq_lines.m, produces lines on a latitude/longitude quadrilateral grid.
• sphere_llq_line_num.m, counts lines on a latitude/longitude quadrilateral grid.
• sphere_spiralpoints.m, spiral points on a sphere.
• sphere_unit_sample.m, picks a random point on the unit sphere.
• timestamp.m, prints the current YMDHMS date as a time stamp.