**The Differential Geometry of the Sphericon**

##
Area and arclength calculations for:

The Differential Geometry of the Sphericon.

The area of the graph of a differentiable function *f*(*x*,*y*)
is given by

where *f*_{x} and *f*_{y} are the partial derivatives of *f* with respect to *x*and to *y*.

For the positive curvature example
*f*(*x*,*y*)=-(*x*^{2}+*y*^{2}),
the circle *x*^{2}+*y*^{2}=1 fits in the
surface exactly at height -1. The area enclosed is

where *D* is the disc
of radius 1. Using polar coordinates
the integral becomes

which can be evaluated (use the substitution *u*=1+4*r*^{2}) as
.

The negative example is more complicated because first one must
find the circle in the (*x*,*y*)-plane which gives a circle in the
graph of circumference .
I had to do it by trial and
error. The circle of radius *r* in the (*x*,*y*)-plane can be
parametrized as
,
.
In the graph this circle becomes
,
.
The
length of a parametrized curve
is given by the integral

In this case the length is

using the identity
before
differentiating, and the identity
after. For *r*=.715 one gets length 6.28 by numerical integration
which is close enough.

The area computation goes the same way as for positive curvature,
since the quantity
1+*f*_{x}^{2} + *f*_{y}^{2} is the same for both functions.
The area enclosed by the circle of circumference 6.28 is calculated
as before but using .715 instead of 1. It comes out to be
.

Back to Sphericon page 2.

© copyright 1999, American Mathematical Society.