Surface of revolution of constant negative Gaussian curvature, conic type

Categories

• Curves
  • Types of curves on surfaces
    • Geodesic lines

• Curves
  • Types of curves on surfaces
    • Asymptotic lines

• Surfaces
  • Surfaces of constant Gaussian curvature
    • Surfaces of revolution of constant Gaussian curvature

Description

Surface with rotational symmetry, constant nagative Gaussian curvature with conical type singularity. The surface is obtained by revolution of a 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 $0 < a < 1$.

The model shows behaviour of geodesic and asymptotic lines on such surfaces.

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