quasi
Posts:
10,255
Registered:
7/15/05


Re: area of a chordal region
Posted:
Apr 18, 2014 3:31 AM


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

