SPHERE_GRID
Points, Lines, Faces on a Sphere

SPHERE_GRID is a dataset directory which contains "grids" on a sphere. For simplicity, the sphere is always assumed to have a radius of 1, and center at the origin.

The grid may be

• a point grid, simply a set of points.
• a line grid, a set of points, and a list of pairs of points that form line segments.
• a triangle grid, a set of points, and a list of triples of points that form triangles.
• a quadrilateral grid, a set of points, and a list of quartets of points that form quadrilaterals.

Licensing:

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, is a FORTRAN90 library which computes various geometric quantities, including grids on spheres.

SPHERE_CVT, a FORTRAN90 library which can find a set of well separated points on a sphere using Centroidal Voronoi Tessellations.

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

STRIPACK, a FORTRAN90 library which includes routines to compute the Voronoi diagram or Delaunay triangulation of pointsets on a sphere.

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

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.

TOMS772, a FORTRAN77 library which carries out computational geometry on the unit sphere; this library is commonly called STRIPACK; this is ACM TOMS algorithm 772.

TRI_SURFACE_DISPLAY, a MATLAB program which can read a node file and a triangle file and display the triangular mesh surface.

Reference:

1. Thomas Ericson, Victor Zinoviev,
   Codes on Euclidean Spheres,
   Elsevier, 2001,
   ISBN: 0444503293,
   LC: QA166.7E75
2. Gerald Folland,
   How to Integrate a Polynomial Over a Sphere,
   American Mathematical Monthly,
   Volume 108, Number 5, May 2001, pages 446-448.
3. AD McLaren,
   Optimal Numerical Integration on a Sphere,
   Mathematics of Computation,
   Volume 17, Number 84, October 1963, pages 361-383.
4. Robert Renka,
   Algorithm 772:
   STRIPACK: Delaunay Triangulation and Voronoi Diagram on the Surface of a Sphere,
   ACM Transactions on Mathematical Software,
   Volume 23, Number 3, September 1997, pages 416-434.
5. Edward Saff, Arno Kuijlaars,
   Distributing Many Points on a Sphere,
   The Mathematical Intelligencer,
   Volume 19, Number 1, 1997, pages 5-11.

Sample files

SPHERE_GRID_ICOS1_F* is a family of point grids. The first grid is formed by the 12 vertices of an icosahedron The second grid replaces each triangular face of the icosahedron by 2*2 triangles, and projects all the new vertices onto the sphere. The third grid refines each triangle to 9 subtriangles, and so on.

The points were computed by the SPHERE_GRID program.

The triangulations were computed by the STRIPACK_DELAUNAY program (which calls STRIPACK).

These grids are produced by linear interpolation on the (straight) edges of the icosahedron, instead of on the (curved) edges of the image of the icosahedron on the sphere.

The point grids available include:

• sphere_grid_icos1_f1.xyz, the 12 node grid.
• sphere_grid_icos1_f1.png, a PNG image of the nodes.
• sphere_grid_icos1_f1_triangulation.txt, the triangulation.
• sphere_grid_icos1_f1_triangulation.png, a PNG image of the triangulation.
• sphere_grid_icos1_f2.xyz, the 42 node grid.
• sphere_grid_icos1_f2.png, a PNG image of the nodes.
• sphere_grid_icos1_f2_triangulation.txt, the triangulation.
• sphere_grid_icos1_f2_triangulation.png, a PNG image of the triangulation.
• sphere_grid_icos1_f3.xyz, the 92 node grid.
• sphere_grid_icos1_f3.png, a PNG image of the nodes.
• sphere_grid_icos1_f3_triangulation.txt, the triangulation.
• sphere_grid_icos1_f3_triangulation.png, a PNG image of the triangulation.
• sphere_grid_icos1_f4.xyz, the 162 node grid.
• sphere_grid_icos1_f4.png, a PNG image of the nodes.
• sphere_grid_icos1_f4_triangulation.txt, the triangulation.
• sphere_grid_icos1_f4_triangulation.png, a PNG image of the triangulation.
• sphere_grid_icos1_f5.xyz, the 252 node grid.
• sphere_grid_icos1_f5.png, a PNG image of the nodes.
• sphere_grid_icos1_f5_triangulation.txt, the triangulation.
• sphere_grid_icos1_f5_triangulation.png, a PNG image of the triangulation.
• sphere_grid_icos1_f6.xyz, the 362 node grid.
• sphere_grid_icos1_f6.png, a PNG image of the nodes.
• sphere_grid_icos1_f6_triangulation.txt, the triangulation.
• sphere_grid_icos1_f6_triangulation.png, a PNG image of the triangulation.
• sphere_grid_icos1_f7.xyz, the 492 node grid.
• sphere_grid_icos1_f7_triangulation.txt, the triangulation.
• sphere_grid_icos1_f7_triangulation.png, a PNG image of the triangulation.

SPHERE_GRID_ICOS2_F* is a family of point grids. The first grid is formed by the 12 vertices of an icosahedron The second grid replaces each triangular face of the icosahedron by 2*2 spherical triangles, and so on.

These grids are produced by interpolation on the spherical angles of each face of the icosahedron.

The point grids available include:

• sphere_grid_icos2_f1.xyz, the 12 node grid.
• sphere_grid_icos2_f1.png, a PNG image of the nodes.
• sphere_grid_icos2_f2.xyz, the 42 node grid.
• sphere_grid_icos2_f2.png, a PNG image of the nodes.
• sphere_grid_icos2_f3.xyz, the 92 node grid.
• sphere_grid_icos2_f3.png, a PNG image of the nodes.
• sphere_grid_icos2_f4.xyz, the 162 node grid.
• sphere_grid_icos2_f4.png, a PNG image of the nodes.
• sphere_grid_icos2_f5.xyz, the 252 node grid.
• sphere_grid_icos2_f5.png, a PNG image of the nodes.
• sphere_grid_icos2_f6.xyz, the 362 node grid.
• sphere_grid_icos2_f6.png, a PNG image of the nodes.

SPHERE_GRID_SPIRAL_N* is a family of point grids created by mapping a spiral onto the sphere.

The point grids available include:

• sphere_grid_spiral_n0020.xyz, the 20 node grid.
• sphere_grid_spiral_n0020.png, an image of the nodes.
• sphere_grid_spiral_n0500.xyz, the 500 node grid.
• sphere_grid_spiral_n0500.png, an image of the nodes.

SPHERE_SAMPLE_N* is a family of point grids created by randomly sampling points on the unit sphere.

The point grids available include:

• sphere_sample_n001000.xyz, a 1000 node sample.
• sphere_sample_n001000.png, an image of the nodes.

You can go up one level to the DATASETS page.