M. Schilling catalog

Cheese-shaped surface of constant positive Gaussian curvature

Geodesic lines–Surfaces of revolution of constant Gaussian curvature

#constant curvature #constant Gaussian curvature #elliptic integral #geodesic line #Brill

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Cheese-shaped surface of constant positive Gaussian curvature

Description

The model shows a rotationally symmetric surface of constant positive Gaussian curvature, so called cheese-shaped (the shape resembles a head of cheese, name due to W. Blaschke). 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 doesn't intersects the $z$-axis (axis of revolution) at all.

The lines on the model are geodesic lines.

In the given above parametrization, the tangents to the profile curve at the end-points $t=\pm\pi/2$ are horizontal. Thus after rotation the resulting two holes in the surface can be glued up smoothly ($C^{1,1}$-smoothly, in fact) with two discs. The resulting surface looks like a head of cheese (illustration from W. Blaschke, Kreis und Kugel, de Gruyter, 1956):