The Differential Geometry of the Sphericon
4. Curvature on the Sphericon: the zip-locus
The Sphericon has more complicated curvature singularities
than cone points. Along the curves where the cones are zipped
together (the zip-locus) there is a 1-dimensional concentration of curvature.
The Sphericon World geographers can analyze this curvature
by another application of Gauss' Theorem.
Here they have drawn a quadrilateral that is bisected by the
zip-locus. (They can do this without actually crossing that
locus themselves!). Its red and blue edges are
drawn using straight line segments
from the cone-points to the zip-circles, so they meet those
circles at right angles.
In that way, when the zipping is done, each pair of edges
fits together without forming a corner as seen in the surface.
Since they are both straight line segments before the zipping, they will
form a single geodesic edge. The resulting
figure has four geodesic edges: one red, one blue and two black.
What are its angles?
By Gauss' Theorem, the total enclosed curvature
is equal to this sum minus 2 (here n=4), so the total
enclosed curvature is 2 = 2L/R.
- Suppose the pie-slices have radius R,
and that the length of the arc intercepted by the quadrilateral is
L. If the sides are extended to the cone-points, they will meet
at an angle = L/R, in radian measure.
- If the quadrilateral is drawn
symmetrically, its angles will all be equal, and equal to
( + )/2. The sum of the interior angles is therefore
2( + ).
This calculation does not depend on the height of the
quadrilateral away from the zip-locus. The only way to explain
the result is to say that the surface curvature is concentrated
along the zip-locus in such a way that any curve intersecting the
zip-locus in an arc of length L will enclose total curvature 2L/R.
On to Sphericon page 5.
Back to Sphericon page 3.
© copyright 1999, American Mathematical Society.