Geodesic on the pseudosphere

Arya23 · 2021-09-18 01:02:53

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 ?

Jai Ganesh · 2021-09-18 01:25:42

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.

Arya23 · 2021-09-18 01:34:40

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.

Jai Ganesh · 2021-09-18 14:23:46

Hi,

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

The link.

Math Is Fun Forum

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

Geodesic on the pseudosphere

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

Re: Geodesic on the pseudosphere

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

Re: Geodesic on the pseudosphere

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

Re: Geodesic on the pseudosphere

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