If you take 4 vertical sections through the mountain along the axes between the peak and each corner of the grid, you end up with 4 similar triangles. Whilst the angles aren't known, the proportion between the sides is fixed by the spot heights at the corners.

If you lay any two of these sections together so that the tip of the mountain and the axis through it are co-incident, and with the slopes running in opposite directions (like the sails of childs yacht): 

     .|.
   .  |  .  
 .    |    .
------|      .
      |--------

[download]

Then all the dimensions of the two triangles will be proportional in the same ratio as the spot heights? And the difference between the two spot heights is known.

With this, I think you can form a pair of simultaneous equations that will solve the height; in height units.

Using two such pairings, with one "sail" in common, and knowing the ratio between the sides of the grid, assume 1:1, and their absolute lengths in grid units, it should be possible to to solve the two sets of simultaneous equations in concert without recourse to iteration, and find the ratio between the grid coordinates and the heights.

I think! Constructing the two sets of simultaneous equations is eluding me.

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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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