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Topic: area of a chordal region
Replies: 20   Last Post: Apr 18, 2014 5:19 PM

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quasi

Posts: 10,255
Registered: 7/15/05
Re: area of a chordal region
Posted: Apr 18, 2014 3:31 AM
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Richard Tobin wrote:
>quasi wrote:
>>
>>Given a square of side length 2, find the area of the region
>>consisting of all points in the interior of the square which
>>lie on some unit length chord having endpoints on adjacent
>>sides.
>>
>>3 pi / 8 ?
>>
>>Well, yes.
>>
>>Was it that easy?

>
>It's well known that the envelope of a ladder sliding down a
>wall is an astroid, and the area of an astroid is, if not well
>known, at least easily looked up.


Then consider that a warmup.

Now for the 3D version ...

Computational Challenge:

Given a cube of edge length 2, find the volume of the region
consisting of all points in the interior of the cube which lie
on some unit length chord having endpoints on the surface of
the cube.

Remarks:

I haven't tried it -- it looks formidable.

Moreover, although the 2D problem for the 2x2 square had a nice
answer, I don't know whether to expect the 3D problem for the
2x2x2 cube to also have a nice answer.

quasi



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