Re: Hyperboloid Question

"Shytot" <Shytot@xxxxxxx> writes in article <GpJPe.6182$iM2.606202@xxxxxxxxxxxxxxx> dated Sat, 27 Aug 2005 06:43:51 +1200:
>
>"Keith A. Lewis" <klewis@xxxxxxxxxxxxxxxx> wrote in message
>news:denj1n$meo$1@xxxxxxxxxxxxxxxxxxxxxx
>> "Shytot" <Shytot@xxxxxxx> writes in article
><k5yPe.5997$iM2.588352@xxxxxxxxxxxxxxx> dated Fri, 26 Aug 2005 17:51:25
>+1200:
>> >I read that a two-*** Hyperboloid is f ormed by a rotation of volume
>about
>> >an axis adjoining the two focal points.
>> >I also read that a one-*** Hyperboloid is formed by a rotation about an
>> >axis at 90 degrees to the axis adjoining teh two focal points.
>> >
>> >Can somebody tell me therefore if we have a two *** Hyperboloid, is the
>> >'space' bewteen the two sheets the same as the one sheet case? ie if the
>> >space between the two *** volumes (which look a bit like two cones
>joined
>> >near their apexes) were solid it would be the same as a one ***
>> >Hyperboloid?
>>
>> They are related, but not equal. In the two-sheet case, the sheets are
>> inside the pair of cones formed by rotating the asymtotes around the axis;
>> and in the one-*** case the *** is outside the cones.
>>
>> If you start with the hyperbola xy=k and decrease k until it is a small
>> positive number, and you form a shape by rotating it around the line y=x,
>> the result will be a two-*** hyperboloid which is very close to the
>cones.
>>
>> If you go "too far" and k=0, the shape *is* the pair of cones.
>>
>> If you go even farther, and k<0, it's a one-*** hyperboloid and it's on
>> the other side of the cones. But... the foci have shifted and are no
>longer
>> on y=x but on y=-x, so you have in fact rotated the hyperbola around a
>> perpendicular line.
>
>I was just wondering if the *** that is outside of the two cones in the
>two *** case is the same as the one *** case? It seems to me that they
>have to be?

It depends what you mean by "the same", I guess. They are not congruent for
any two different values of k, but the limit as you approach k=0 from the +
side (two *** case) is the same as the limit as you approach from the -
side (one *** case).

--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
.

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