Construction of Triangular Grids

Triangular grids are constructed by recursive (or iterative) refinement of an initial set triangles. This inital set is usually given by the faces of a body that is delimited by triangles. There are only three bodies for which all the triangle edges have the same length (implying that all other convex bodies made of triangles will have edges with varying lengths, and this will also be true for LOCSMITH grids). These three bodies are the tetrahedron, the octahedron, and the icosahedron:

The three basic bodies
Tetrahedron
4 faces, 6 edges, 4 nodes
Octahedron
8 faces, 12 edges, 6 nodes
Icosahedron
20 faces, 30 edges, 12 nodes

Note that the three bodies have different symmetries.

The refinement method used in LOCSMITH is the introduction of new nodes at the middle of each edge. Since the Grid is supposed to be spherical, edges are defined as the great circle arcs between nodes, and the mid points are chosen on these great circle arcs. But the concept may be illustrated in the plane.

In one iteration step, each edge is divided in two halves and new edges are introduced. The size of the original triangle does not change.

Iteration Scheme
0th iteration 1st iteration 2nd iteration

In 3D, the coordinates of a new node N on an edge AB are constructed from the cartesian coordinates of the end points A and B of the edge in three simple steps:

  1. average the individual coordinates to get the coordinates of N. The x coordinate, for example, is
    Nx=0.5(Ax +Bx)
  2. transform the cartesian coordinates Nx, Ny, Nz of N into spherical coordinates Nlon, Nlat, Nr.
  3. Set the radius coordinate Nr to the sphere (or layer) radius. It is this modification of a spherical coordinate which introduces small fluctuations into the edge lengths. But without it, the iterated faces would remain flat.

This procedure produces new nodes in a way that they are on the shorter part of the great circle through A and B, dividing it into two halves.

Since the layer radius of triangular grids in LOCSMITH is stored only once for all nodes of the layer, the radius coordinate of N can simply be omitted. Only its longitude and latitude are stored.

After subdivision of all faces of one of the three starting bodies, the number of edges meeting at each node will also vary, depending on the initial body:

Initial Body Edges per node
Tetrahedron 3 or 6
Octahedron 4 or 6
Icosahedron 5 or 6

eof.