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Some may enjoy this, using PovRay CSG to illustrate answers
to geometric problems.
http://astronomy.swin.edu.au/~pbourke/geometry/toruscut/
Before you look ask yourself,
How many ways can a torus be cut (with a single plane)
so that the resulting cross sections are perfect circles?
--
Paul Bourke
pdb_NOSPAMswin.edu.au
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The third way is surprising....
Paul Bourke wrote:
> Some may enjoy this, using PovRay CSG to illustrate answers
> to geometric problems.
> http://astronomy.swin.edu.au/~pbourke/geometry/toruscut/
> Before you look ask yourself,
> How many ways can a torus be cut (with a single plane)
> so that the resulting cross sections are perfect circles?
>
--
Samuel Benge
stb### [at] hotmail com
See my website@: http://www.goldrush.com/~abenge/Top/index.html
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Paul Bourke wrote:
> Some may enjoy this, using PovRay CSG to illustrate answers
> to geometric problems.
> http://astronomy.swin.edu.au/~pbourke/geometry/toruscut/
> Before you look ask yourself,
> How many ways can a torus be cut (with a single plane)
> so that the resulting cross sections are perfect circles?
Nice, but the really interesting question is how you can proove that
these three ways are the only ones...
Christoph
--
POV-Ray tutorials, include files, Sim-POV,
HCR-Edit and more: http://www.tu-bs.de/~y0013390/
Last updated 25 Oct. 2003 _____./\/^>_*_<^\/\.______
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Samuel Benge <sbe### [at] hotmailcom> wrote in
news:3FC### [at] hotmailcom:
> The third way is surprising....
Here's an image that shows this:
http://home.no/t-o-k/povray/Isosurfaces-Rotated_Tori.jpg
Also have a look at this page:
http://mathworld.wolfram.com/Torus.html
(See the illustration halfway down the page.)
Tor Olav
> Paul Bourke wrote:
>
>> Some may enjoy this, using PovRay CSG to illustrate answers
>> to geometric problems.
>> http://astronomy.swin.edu.au/~pbourke/geometry/toruscut/
>> Before you look ask yourself,
>> How many ways can a torus be cut (with a single plane)
>> so that the resulting cross sections are perfect circles?
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Christoph Hormann wrote:
>Nice, but the really interesting question is how you can proove that
>these three ways are the only ones...
>
>Christoph
>
See Tor Olav's reference:
http://mathworld.wolfram.com/Torus.html
If I'm reading the proof at the bottom of the page correctly, (42)& (44)
establishes necessity as well as sufficiency. And I agree, that is the
interesting question.
Dave Matthews
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> > Some may enjoy this, using PovRay CSG to illustrate answers
> > to geometric problems.
> > http://astronomy.swin.edu.au/~pbourke/geometry/toruscut/
> > Before you look ask yourself,
> > How many ways can a torus be cut (with a single plane)
> > so that the resulting cross sections are perfect circles?
> Nice, but the really interesting question is how you can proove that
> these three ways are the only ones...
Fortunate for me I wasn't trying to prove it.
--
Paul Bourke
pdb_NOSPAMswin.edu.au
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> See Tor Olav's reference:
> http://mathworld.wolfram.com/Torus.html
I prefer PovRay to Mathematica :-)
--
Paul Bourke
pdb_NOSPAMswin.edu.au
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