Information concerning undocumented function "Region`Mesh`MeshNearestCellIndex[r ]"

Question

I'm looking for further information of the undocumented function Region`Mesh`MeshNearestCellIndex[r] (Thanks to Henrik Schumacher!)

The function considers a meshregion r and detects the nearest cell to a given point. Trying to understand I look at a very simple triangle mesh in space:

pi={{0., 0., 0.303}, {1., -0.5, 0.09}, {0.7, 0.8, -0.233}, {0.,1., -0.584}, {-0.8, -0.7, -0.734}} 
Δi = {{1, 2, 3}, {1, 3, 4}, {1, 4, 5}, {1, 5, 2}};
r = MeshRegion[pi , Triangle[Δi]];
HighlightMesh[r, {Labeled[0, "Index"], Labeled[2, "Index"]}]

Now I want to evaluate the nearest cells of point 1

 Region`Mesh`MeshNearestCellIndex[r , pi[[1]] ]
 (*{2, 1}*)

Expecting four possible cells as nearest neighbors MMA returns element #1.

My question: How does MMA evaluate the priority of the possible cells? Thanks!

Answer 1

If I had to make a guess, I'd say Mathematica breaks ties by choosing the cell with the smallest index:

Δi = {{1, 3, 4}, {1, 2, 3}, {1, 4, 5}, {1, 5, 2}};
Table[
 r = MeshRegion[pi, Triangle[Δi[[perm]]]];
 Region`Mesh`MeshNearestCellIndex[r, pi[[1]]],
 {perm, PermutationList /@ Permutations[Range[4]]}
 ]

{{2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {2, 1}}

By the way, I just found out that you can also obtain the $n$ nearest cells as follows:

r = MeshRegion[pi, Triangle[Δi]];
Region`Mesh`MeshNearestCellIndex[r, pi[[1]], 10]

{{2, 1}, {2, 2}, {2, 3}, {2, 4}, {2, 0}, {2, 0}, {2, 0}, {2, 0}, {2, 0}, {2, 0}}

Apparently, superfluous cells obtain the index 0...

Oh, an apparently, there is also an operator version of the Region`Mesh`MeshNearestCellIndex, similarly as for Nearest:

cellfun = Region`Mesh`MeshNearestCellIndex[r];

cellfun[pi[[1]]]

{2, 2}