Moving circle on a square with fixed overlap

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turtleturtleturtle
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Moving circle on a square with fixed overlap

Postby turtleturtleturtle » Thu May 17, 2012 12:04 am UTC

Given: There is a square, ABCD, with side lengths l, a constant. Point E lies somewhere on side CD. Points B and E both lie on the perimeter of a circle, with center at point F, and with radius y. The length of line segment CE is labeled x. Finally, the space of overlap between circle F and square ABCD has a fixed area, equal to Z.
The goal: Find a function y=f(x) which gives the radius of circle F based upon the length of line segment CE. Use in the function, if necessary, the positive real constants l and Z.


For any curious, I was staring at some interesting tiling while in a bathroom at a university, and, being very bored, made up this problem. I'm not sure, however, how to solve it, and thought maybe someone else would like to.

Twelfthroot
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Re: Moving circle on a square with fixed overlap

Postby Twelfthroot » Thu May 17, 2012 8:10 pm UTC

Unless I'm mistaken, we need a bit more information, because there are infinitely many circles with points B and E on their perimeters. Do you mean the one such that BE is a diameter? Or the one such that F lies on BC? Or some other circle?

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gmalivuk
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Re: Moving circle on a square with fixed overlap

Postby gmalivuk » Thu May 17, 2012 8:45 pm UTC

There are, but that only means that the function will depend on more than just x. There aren't, after all, infinitely many such circles that would enclose a given area Z of the square.
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turtleturtleturtle
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Re: Moving circle on a square with fixed overlap

Postby turtleturtleturtle » Fri May 18, 2012 12:05 am UTC

I believe there should only be one circle that meets all of the requirements for any given length of CE. If it helps, I was thinking chiefly of situations where the overlap was less than half the area of the square (in other words, where point F lies on the same side of line BD as point C).


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