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 +
-
; the red-blue angle is
+
/2.
So the sum of the angles in the triangle is 2-
+2
.
Gauss'Theorem tells us that the total curvature enclosed in the triangle is -
+2
. We can see that the
curvature is coming from the zip-locus. As we take smaller and smaller such triangles about the cone point, the
contribution will go to zero, leaving us with a point-concentration of curvature at the cone-point equal to
-
.
Back to Sphericon page 4.