The Natural Neighbor interpolation can fail near the borders of a mesh with problematic positioning of vertices near the borders of the TIN. For example, the following test set was contributed by a user who was having problems with interpolation. It gives rise to a mesh which has very thin triangles located along three edges of the mesh. Those triangles lead to the computation of circumcircles with very large radius values. In these cases, numerical issues lead to incorrect interpolations.
index, x,y,z
0, 120.83333333333333, 398.6666666666667, 28.0
1, 204.16666666666666, 351.3333333333333, 23.0
2, 292.49999999999994, 398.66666666666663, 18.0
3, 375.8333333333333, 351.3333333333333, 4.0
4, 464.16666666666663, 398.66666666666663, 0.0
5, 204.16666666666666, 256.6666666666667, 12.0
6, 287.5, 209.33333333333331, 5.0
7, 375.8333333333333, 256.66666666666663, 9.0
8, 287.5, 114.66666666666666, 0.0
The following images show the layout of the vertices and the resulting interpolation. Vertices 2 and 5 are just inside the line segments that would be formed by Vertices 0,4 and Vertices 0,8. The resulting triangles 2,4,0 and 5,8,0 have very small areas and are almost degenerate. In the picture below, the triangles are so flattened that their three points appear to be lying on a single line. Numerical issues result.
Here are the color-coded results from interpolation. Numerical issues lead to erratic behavior.