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.
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