M. Schilling catalog

Surface of revolution of constant negative Gaussian curvature, hyperboloid type

Geodesic lines–Surfaces of revolution of constant Gaussian curvature

#surface of revolution #constant curvature #constant Gaussian curvature #negative curvature #hyperboloid type #geodesic line #geodesic circle #Dyck #Brill

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Surface of revolution of constant negative Gaussian curvature, hyperboloid type

Description

Periodic surface with rotational symmetry and constant nagative Gaussian curvature. The surface is obtained by revolution of the profile curve $(x(t),y(t))$ around the $y$-axes with

$x(t) = \sqrt{1 - a^2} \sinh t$,

$y(t) = \int_0^t \sqrt{1 - a^2 \sin^2 \tau} d\tau$,

and $a > 1$.

The model shows behaviour of geodesics on such surfaces, and behaviour of geodesic circles, that is points at a given distance (along the surface) from a given point.