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Pseudosphere Geodesics

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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 pseudosphere does not correspond to the whole upper half-plane but only to the region
[1,∞)
. The fact that the pseudosphere can serve as a model for hyperbolic geometry was discovered by Eugenio Beltrami.
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