Science > Physics > Re: Find surface of constant negative Gauss curvature
| Topic: |
Science > Physics |
| User: |
"Hannu Poropudas" |
| Date: |
11 Aug 2007 02:41:35 AM |
| Object: |
Re: Find surface of constant negative Gauss curvature |
On Aug 11, 10:02 am, Narasimham <mathm...@hotmail.com> wrote:
How to find the equation/parameterization of a surface of constant
negative Gauss curvature = -1 passing through four vertices of a
regular tetrahedron with its center to vertex distance = 1?
I'am not sure did I understood the problem right, but
I can think here approximately a pseudosphere surface
(see my PROFILE, there is a picture of pseudosphere which has
a constant negative Gaussian curvature)
but it does not have a center and it's length is infinite
(in it's mathematical equation) and it would not go exactly
through the forth corner of the regular tetrahedron you
mentioned.
If this interests you, then please take a look it's equation
more closely for example from the book:
Lipschutz,M.M., 1969.
Theory and Problems of Differential Geometry.
Schaum's Outline Series, McGraw-Hill Book Company,
New York, Printed in the United States of America.
269 pages, page 241, Example 11.8.
Hannu
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| User: "Narasimham" |
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| Title: Re: Find surface of constant negative Gauss curvature |
11 Aug 2007 04:04:43 AM |
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On Aug 11, 12:41 pm, Hannu Poropudas <hapor...@luukku.com> wrote:
On Aug 11, 10:02 am, Narasimham <mathm...@hotmail.com> wrote:
How to find the equation/parameterization of a surface of constant
negative Gauss curvature = -1 passing through four vertices of a
regular tetrahedron with its center to vertex distance = 1?
I'am not sure did I understood the problem right, but
I can think here approximately a pseudosphere surface
(see my PROFILE, there is a picture of pseudosphere which has
a constant negative Gaussian curvature)
but it does not have a center and it's length is infinite
(in it's mathematical equation) and it would not go exactly
through the forth corner of the regular tetrahedron you
mentioned.
So it is not an exact solution?
If this interests you, then please take a look it's equation
more closely for example from the book:
Lipschutz,M.M., 1969.
Theory and Problems of Differential Geometry.
Schaum's Outline Series, McGraw-Hill Book Company,
New York, Printed in the United States of America.
269 pages, page 241, Example 11.8.
Hannu
.
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| User: "Hero" |
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| Title: Re: Find surface of constant negative Gauss curvature |
11 Aug 2007 05:31:16 AM |
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Narasimham wrote:
Hannu wrote:
Narasimham wrote:
How to find the equation/parameterization of a surface of constant
negative Gauss curvature = -1 passing through four vertices of a
regular tetrahedron with its center to vertex distance = 1?
.....
So it is not an exact solution?
But of course there is an exact solution. Take a small tetra and place
it into the opening of the pseudosphere (trumpet). It touches three
points of the surface. Now slide it into the trumpet until it get
stuck.
With friendly greetings
Hero
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| User: "Greg Neill" |
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| Title: Re: Find surface of constant negative Gauss curvature |
11 Aug 2007 07:41:26 AM |
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"Hero" <Hero.van.Jindelt@gmx.de> wrote in message
news:1186828276.159465.97120@q75g2000hsh.googlegroups.com...
Narasimham wrote:
Hannu wrote:
Narasimham wrote:
[snip]
Guys, isn't it time to check the Newsgroups line?
The crossposting of this thread to sci.astro and
sci.paleontology seems to be silly. The physics
groups could probably also live without it.
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| User: "Narasimham" |
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| Title: Re: Find surface of constant negative Gauss curvature |
11 Aug 2007 07:05:24 AM |
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On Aug 11, 3:31 pm, Hero <Hero.van.Jind...@gmx.de> wrote:
Narasimham wrote:
Hannu wrote:
Narasimham wrote:
How to find the equation/parameterization of a surface of constant
negative Gauss curvature = -1 passing through four vertices of a
regular tetrahedron with its center to vertex distance = 1?
.....
So it is not an exact solution?
But of course there is an exact solution. Take a small tetra and place
it into the opening of the pseudosphere (trumpet). It touches three
points of the surface. Now slide it into the trumpet until it get
stuck.
With friendly greetings
Hero
OK fine,we get a symmetrical solution here,but my implicit question
was for the following discussion point ;)!
For the pseudosphere case we can rotate the tetra unsymmetrically
along two axes and and adjust it until we get a contact of the tetra's
corners to the inside of the pseudosphere..and still if we take the
hypo and hyper pseudospheres or Kuen's or Breather surfaces,there are
several more possibilities.
If one had asked the question about a tetra to be placed in a sphere,
the solution is uniquely one and quite easy to see by symmetry.
It becomes so complicated with the negative curvature pseudosphere and
we are not able to write out a parameterization in integrated or in
differential forms for the set of four lines traced by the four
vertices. Can we? I suspect at least one more arbitrary parameter is
involved when attempting to write for all possibilities.
Best Regards,
Narasimham
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