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Find surface of constant negative Gauss curvature
On Aug 11, 10:02 am, 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? 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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Find surface of constant negative Gauss curvature
On Aug 11, 12:41 pm, Hannu Poropudas wrote:
On Aug 11, 10:02 am, 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? 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 |
#3
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Find surface of constant negative Gauss curvature
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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Find surface of constant negative Gauss curvature
On Aug 11, 3:31 pm, Hero 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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Find surface of constant negative Gauss curvature
"Hero" wrote in message
ups.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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