in reply to Re: OT:Math problem: Grids and conical sections.
in thread OT:Math problem: Grids and conical sections.

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.
  • Comment on Re^2: OT:Math problem: Grids and conical sections.

Replies are listed 'Best First'.
Re^3: OT:Math problem: Grids and conical sections.
by jeffguy (Sexton) on Nov 25, 2005 at 17:18 UTC
    A solution is obviously correct if it gives an x, y, and z of the peak such that when you use any of the four measured heights to find the slope, that slope finishes the equation:

    (x-x')^2+(y-y')^2=m^2(z-z')^2
    (or maybe it's 1/m^2 instead of m^2 -- irrelavent details)

    and the other three measurements all fit into that equation with the given x, y, z, and m.

    So, when given a solution, it will be clear that it is a solution. Is it clear that there's only one solution for the non-degenerate case? Not yet. But the goal is the same as for the colored points last week that we wanted to separate with (n-1)-dimensional hyperplanes -- can A solution (or even ALL solutions) be found? How?
[reply]

In Section Seekers of Perl Wisdom