More about interpolation


In the two hemispheres, octants of the computational grid look like this: 
                  North               * * * * * * *
                    *                  * * * * * *
                   * *                  * * * * *
                  * * *                  * * * *
                 * * * *                  * * *
                * * * * *                  * *
               * * * * * *                  *
              * * * * * * *               South
Each face has upright and inverted equilateral triangles, and after the patches have been stitched (here), every triangle is surrounded by others on which known data is specified at vertices.

The new naming of vertices is: 
            upright                        inverted

 
          nw      ne                   nw      o      ne 



       o       n       o            ww      w       e      ee  



   ww      w       e      ee            o       s       o 



       sw      o      se                    sw      se

This time, the names of the vertices suggest directions (NEWS) on the sphere, but it is just a convenient suggestion. The map of the line 'w-e' is generally close to a line of constant latitude, but the map of a line through the vertex 'n' or 's' and perpendicular to 'w-e' is not a meridian (constant longitude) unless the triangle is situated on the vertical centerline of one of the faces of the octahedron. Otherwise the inverse map of a meridian that contains the map of 'n' or 's' varies from 'w-n' or 'w-s' on western edges of the faces and 'e-n' or 'e-s' on eastern edges of the faces. 
  •  The homogeneous coordinates are e, v & w where
    
              v is n or s and e+v+w=1.
    

  •  w=1 at the node 'w' and w=0 on the line 'e-v'.

  •  e=1 at the node 'e' and e=0 on the line 'w-v'.

  •  v=1 at the node 'v' and v=0 on the line 'w-e'.

  •  Data at vertices marked 'o' is also used.

    • Some of the members of the four sets of coordinate lines are: 
                                             e=0         w=0
                \     /                    \         /
            -----x---x-----n=2           ---x---o---x---s=-1
                  \ /                        \     /
            ---o---n---o---n=1           -x---w---e---x-s=0 
                  / \                          \ /
            -x---w---e---x-n=0           ---o---s---o---s=1
                /     \                        / \
            ---x---o---x---n=-1          -----x---x-----s=2
              /         \                    /     \
            e=0         w=0
    The linear interpolations of f(e,v,w) on a central triangle are 
    	    L(e,v,w)=ef(e)+vf(v)+wf(w)  (v is n or s),

where f(e)=f(1,0,0), f(v)=f(0,1,0) and f(w)=f(0,0,1).

    To correct for the systematic error in L(e,v,w), introduce a function N(e,v,w) which is not linear, but N(e)=N(v)=N(w)=0.
    An alternate way of naming the vertices is with three-character symbols which are the values of the coordinates, evw, in that order. Note the character, '!' stands for -1. Then 'w' is '001' f(w)=f(001)=f(0,0,1), f(n)=f(s)=f(v)=f(010), f(e)=f(100), etc. The diagram of vertices for the upright triangles is 
    
 
                      !20     02!           



                  !11     010     11!      



              !02     001     100     20!  



                  0!2     1!1     2!0

    With this, the evaluations of quadratic, cubic and quartic polynomials that figure in the construction of N(e,n,w) becomes relatively easy. Note, there are nine independent values of N(enw) in the diagrams (N(e)=N(n)=N(w)=0), the number of independent quadratic functions is three (e.g. ee, ew, ww), and the number of independent cubic functions is four (e.g. eee, eew, eww, www). That leaves two quartic functions, but to maintain the symmetry of the triangles, three quartics whose sum is a cubic will be introduced.

    For now, this is continued here.