Dubai Towers Perspective

Geometry Level 5

An aerial shot is made of three Dubai towers and their spires. Measurement of the photograph shows that the distances from the horizon line to the top of the spires and distances between the vertical lines through the spires are in the ratios as shown in this graphic

It is known from public records that relative to a surveyor's benchmark nearby, the coordinates of the top of the three spires are, in feet

\(\left( 554, 878, 1167 \right) \)
\(\left( 770, 690, 1005 \right) \)
\(\left( 1510, -630, 775 \right) \)

The shot was made horizontal, so that vertical lines in the landscape appear as vertical lines in the photograph. But the shot was made \(z\) feet above the ground. That is, at the time of the shot, the camera was at, in feet, relative to the same benchmark

\(\left( x,y,z \right) \)

Find \(z\), in feet

Note: Disregard the curvature of the Earth. Also, assume ideal pinhole optics and flat film plate. Not a "swept panorama shot". Also, solution is unique.


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