The Differential Geometry of the Sphericon.
The area of the graph of a differentiable function f(x,y)
is given by
For the positive curvature example
the circle x2+y2=1 fits in the
surface exactly at height -1. The area enclosed is
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
In the graph this circle becomes
length of a parametrized curve
is given by the integral
The area computation goes the same way as for positive curvature, since the quantity 1+fx2 + fy2 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 .