The Differential Geometry of the Sphericon


5. Curvature on the Sphericon: the cone-points

The cone-points on the Sphericon are of a special type, because they are on the ends of the zip-locus. The Sphericon geographers can still analyze them using Gauss' Theorem.

In this figure they have drawn a right triangle around one of the cone-points.

What are the other angles in this triangle? Here a little plane geometry leads to the answer. The black-blue angle is gamma" + pi - beta; the red-blue angle is gamma" + pi/2.

So the sum of the angles in the triangle is 2pi-beta+2gamma".

Gauss'Theorem tells us that the total curvature enclosed in the triangle is pi-beta+2gamma". We can see that the gamma" curvature is coming from the zip-locus. As we take smaller and smaller such triangles about the cone point, the gamma" contribution will go to zero, leaving us with a point-concentration of curvature at the cone-point equal to pi-beta.

On to Sphericon page 6.

Back to Sphericon page 4.

© copyright 1999, American Mathematical Society.