M. Schilling catalog

Spindle-shaped surface of constant positive Gaussian curvature

Geodesic lines–Surfaces of revolution of constant Gaussian curvature

#constant curvature #spindle-shaped #constant Gaussian curvature #elliptic integral #geodesic line #Brill #brass

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Spindle-shaped surface of constant positive Gaussian curvature

Description

The model shows a rotationally symmetric surface of constant positive Gaussian curvature, so called spindle. For a given curvature $K > 0$, and the radius $R$ of the longest parallel with $R < 1/\sqrt{K}$, the meridian (profile curve) of a spindle-shaped surface is uniquely defined by

$x(t) = R \cos t$, $y = 0$,

$z(t) = \int \limits_{0}^{t} \sqrt{1/K - R^2 \sin^2 \theta} \,\, d\theta$

for $|t| \leqslant \pi/2$. The meridian intersects the $z$-axis (axis of revolution) for $t=\pm\pi/2$.

The lines on the model are geodesic lines.

The fact that the curvature is constant can be observed by sliding the brass plate presented on the model without its deformations and without losing tight contact with the surface. This brass plate can be continuously (isometrically) deformed into the respectful plate for the sphere. This fact illustrates that the Gaussian curvature is a intrinsic quantity.