Interpolation:
Triangulation |
---|

The triangulation method uses Renka's algorithm (Renka, 1984)
to carry out a Delaunay triangulation (Okabe et al., 1992)
of the observation points.
The purpose is to identify a neighborhood of nearby observation points to be
used in the interpolation.

Two options are available to perform the interpolation. The simplest
method is a linear
interpolation of the vertices of the triangulation. In this case, the
barycentric coordinates of the
point are used to take a weighted average of the three observation
values (associated with the
three vertices of the triangle). The resulting interpolation function is
continuous, but not
differentiable across an edge.

The second triangulation method uses a polynomial fit within each
triangle in the triangulation
(Renka, 1984). This method involves an intermediate step of fitting a
cubic spline on each edge
of the triangulation. The interior values are taken as weighted averages
of the edge values,
based upon the barycentric coordinates of a point. In the intermediate
step of fitting the spline,
approximations to the gradient at each point must be computed. The user
has the option to
specify which of two gradient estimation methods is to be used. The
local method uses only a
few neighboring points, while the global method uses all points. The
latter method may not be
possible if the number of points is larger than a few hundred.

### References

Brown, J. L. (1994), Natural neighbor interpolation on the sphere,
in Wavelets, Images, and
Surface Fitting, P.-J. Laurent, A. Le Mehaute, and L. L. Schumaker eds.,
A K Peters, Wellesley,
MA, 67-74.

Okabe, A., B. Boots, and K. Sugihara (1992) Spatial Tessellations,
Wiley, 532 pp.
Renka, R. J. (1984), Interpolation of data on the surface of a
sphere, ACM Transactions on
Mathematical Software, 10, 417-436.

Watson, D. F. (1992) Contouring: A Guide to the Analysis and Display
of Spatial Data, Pergamon Press, 321 pp.