sphericon 8 ## 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 thesurface 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 mustfind the circle in the (*x*,*y*)-plane which gives a circle in thegraph of circumference .I had to do it by trial anderror. The circle of radius *r* in the (*x*,*y*)-plane can beparametrized as ,.In the graph this circle becomes ,.Thelength of a parametrized curve is given by the integral

In this case the length is

using the identity beforedifferentiating, and the identity after. For *r*=.715 one gets length 6.28 by numerical integrationwhich 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 calculatedas before but using .715 instead of 1. It comes out to be.