Natural Neighbor Interpolator fails at borders of problematic TIN

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.

[gwlucastrig]

In a related development, I noticed that the getArea() method for the SimpleTriangle class (see Triangle Collector Techniques ) was computing a small-magnitude negative area for triangle 5,8,0. This glitch violates the rule that all triangles produced by Tinfour are oriented counterclockwise and have a positive area.

This unfortunate result was due to numeric round-off issues arising from the nearly-degenerate geometry of the triangle. However, when I used the extended-precision logic from the GeometricOperations class, I confirmed that the triangle was, in fact, constructed correctly and did have a positive area.

I am looking at the best way to incorporate extended-precision logic into the SimpleTriangle class. The extended-precision logic is slower than simple floating-point operations but does produce more accurate results.

This change will be integrated into the upcoming 2.1.5 release of Tinfour. Once it is addressed, I will be closing this issue.

[gwlucastrig]

This issue in now fully addressed. Fixes are included in the 2.1.5 release which was issued 10 Jan 2021.