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4 Nov 2024 17:37:11 EST (-0500)
  slicing a torus (Message 1 to 7 of 7)  
From: Paul Bourke
Subject: slicing a torus
Date: 26 Nov 2003 02:23:43
Message: <pdb_NOSPAM-A0CF36.18234526112003@netplex.aussie.org>
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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From: Samuel Benge
Subject: Re: slicing a torus
Date: 26 Nov 2003 03:04:54
Message: <3FC45EA5.7050003@hotmail.com>
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] hotmailcom
See my website@: http://www.goldrush.com/~abenge/Top/index.html


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From: Christoph Hormann
Subject: Re: slicing a torus
Date: 26 Nov 2003 04:32:08
Message: <du6e91-0p2.ln1@triton.imagico.de>
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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From: Tor Olav Kristensen
Subject: Re: slicing a torus
Date: 26 Nov 2003 10:10:19
Message: <3fc4c25b@news.povray.org>
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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From: Dave Matthews
Subject: Re: slicing a torus
Date: 26 Nov 2003 11:15:00
Message: <web.3fc4d128240d9e448062416c0@news.povray.org>
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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From: Paul Bourke
Subject: Re: slicing a torus
Date: 27 Nov 2003 03:51:55
Message: <pdb_NOSPAM-A8A2C9.19515427112003@netplex.aussie.org>
> > 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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From: Paul Bourke
Subject: Re: slicing a torus
Date: 27 Nov 2003 03:52:34
Message: <pdb_NOSPAM-FBEC04.19523327112003@netplex.aussie.org>
> 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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