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#1 2021-09-18 01:02:53

Arya23
Member
Registered: 2021-09-18
Posts: 2

Geodesic on the pseudosphere

What is the geodesic on the pseudosphere, parametrically represented as r(u,v) = a Sin u Cos v i + a sin u Sin v j + a (Cos u + log (tan u/2)) k ?

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#2 2021-09-18 01:25:42

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 45,956

Re: Geodesic on the pseudosphere

Hi,

I can give you some idea on the subject.

The pseudosphere is the surface obtained by revolving a tractrix around its asymptote. In a suitable parametrization, the first fundamental form of the pseudosphere is the same as in Poincaré's half-plane model of hyperbolic geometry. Each pair of points in Poincare's half-plane is joined by a unique geodesic that is either a vertical line or a circular arc with center on the horizontal axis. Geodesics on the pseudosphere are then easily obtained by mapping the lines and circular arcs onto the surface. The fact that the pseudosphere can serve as a model for hyperbolic geometry was discovered by Eugenio Beltrami.

In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.

A pseudosphere of radius R is a surface in

having curvature −
in each point. Its name comes from the analogy with the sphere of radius R, which is a surface of curvature
. The term was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#3 2021-09-18 01:34:40

Arya23
Member
Registered: 2021-09-18
Posts: 2

Re: Geodesic on the pseudosphere

It's so good to get a reply from you, sir. But I actually need the solution with steps. I've been trying it for so many days with a lot of reference books. It would be quite helpful if you solve the question and share the solution, sir. Thank you.

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#4 2021-09-18 14:23:46

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 45,956

Re: Geodesic on the pseudosphere

Hi,

Here's a link on the subject. Hope you find it helpful.

The link.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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