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:

DESIGN is a FORTRAN90 library which returns point sets on the sphere that constitute "designs".

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

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

SPHERE_VORONOI_DISPLAY_OPEN_GL, 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.

STRI_QUAD is a FORTRAN90 library which estimates the integral of a function defined on the sphere.

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

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

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

TOMS772 is the original FORTRAN77 text of the STRIPACK program.

TRI_MESH_DISPLAY is 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

SPARSE_GRID_ICOS1_F*_XYZ 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.

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. Therefore, if we projected the new subtriangles, they will not be congruent.

The point grids available include:

• sphere_grid_icos1_f1_xyz.txt, the 12 node grid.
• 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.txt, the 42 node grid.
• 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.txt, the 92 node grid.
• 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.txt, the 162 node grid.
• 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.txt, the 252 node grid.
• 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.txt, the 362 node grid.
• 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.txt, the 492 node grid.
• sphere_grid_icos1_f7_triangulation.txt, the triangulation.
• sphere_grid_icos1_f7_triangulation.png, a PNG image of the triangulation.

You can go up one level to the DATASETS page.