That was my gut reaction when I first encountered the problem, but a combination of tilly's analysis, and my own research and sketches, lead me to have doubts.

The most illustrative example is the case of all four corners being the same height. Yes, it fixes the x,y of the center, but you could poke a cone of any slope from flat to infinitely pointy into the 4 points and it would touch all four points if it has sufficient base diameter.

There appears to me to be no way to fix the height in that case without a 5th reading.

In the non-degenerate cases with 3 or 4 different heights, I am not yet convinced either way that there is a unique solution. Or if there are multiple solutions, that one of them will be obviously correct.

Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
Lingua non convalesco, consenesco et abolesco. -- Rule 1 has a caveat! -- Who broke the cabal?
"Science is about questioning the status quo. Questioning authority".
In the absence of evidence, opinion is indistinguishable from prejudice.

In reply to Re^2: OT:Math problem: Grids and conical sections. by BrowserUk
in thread OT:Math problem: Grids and conical sections. by BrowserUk

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