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Unit square overlap II (Posted on 2023-05-18) Difficulty: 4 of 5
Two unit squares overlap with an area 1/16. The shape of this overlap may be a triangle or a quadrilateral.

Find the minimum distance between their centers.

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soln | Comment 1 of 2
Although this is under the Calculus heading, it was unclear to me
how to use calculus to solve it. The overlap is governed by three 
parameters of the overlapping square: its rotation around its 
center, its delta_x, and delta_y wrt to the fixed square. 
I solved this problem using a grid search.  Delta-y was ranged between 0 and (1+sqrt(2))/2 and theta from 0 to 45 deg. For each pair, I found
the delta_x that gave the required overlap and  using these solutions, then 
found the minimum separation out of all the cases produced. 
A rough grid is shown here

The trough for the minimum separation is broad and shallow and centered 
on theta =  9 +/- 2 degrees, delta x=0.921 and delta y=0.16 
with x and y uncertainty of about 0.02. 

The rough grid has some uncertainty in its resulting parameters. But more 
refined grids gave consistently a minimum separation of d=0.934. 
The (0.16, 0.921, 9 deg, d=0.935) configuration looks like the this. 

The solution overlap was a three sided shape, but many overlaps that came close had four sides such as this. (0.10, 0.935, 12 deg, d= 0.943)

Edited on May 31, 2023, 10:05 pm
  Posted by Steven Lord on 2023-05-31 17:44:01

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