Failure of Sibson gradient fitting: when and why?

[stla]

Hello,

The blue points are given in the unit circle. To each blue point (x,y) I associate a number z in [0,1], the squared distance from the center of the circle to (x,y).

Then I take the other points (xnew, ynew), and I predict znew with the Sibson interpolation. I color a point (xnew,ynew) with the gray color rgb(znew,znew,znew) if znew is available, or in red if the gradient fitting has failed (so I don't have znew).

I'm wondering whether someone could be able to "guess" the red points: "hmm... I think the gradient fitting will fail here".

More seriously, is there a precise condition for the success of the gradient fitting?