### Computing Sphere Eversions

Published in **Mathematical Visualization**,
(H.-C. Hege, K. Polthier, eds.), Springer, 1998, pp 237-255.
**Plain-English Abstract**
It was a surprising consequence of an abstract theorem of Smale that a
spherical surface can be turned inside out without tearing or creasing,
if we do allow the surface to pass through itself. Over the intervening
forty years, mathematicians have designed different ways to see explicitly
how such a ``sphere eversion'' can be achieved.

Here, we consider several tools for computing and
visualizing sphere eversions. First, we discuss a family
of optimal sphere eversions with rotational symmetry.
These are computed by minimizing an elastic bending energy
for surfaces in space called the Willmore energy. We start
with a complicated self-intersecting sphere, which has the
desired rotational symmetry, and is also halfway inside-out
in the sense of having its inside and outside equally exposed.
This halfway model is a saddle critical point for the Willmore energy.
When we push off the saddle in the two directions and then flow
downhill in energy to the ordinary round sphere, it is inside-out
one way, but not the other way.

In any sphere eversion, the "double locus" of self-intersections
is a key item to follow through time as the sphere turns inside out.
We describe real-time interactive computer animation programs
to compute and display the double locus of any immersed surface,
and to track this along a homotopy like a sphere eversion.
Finally, we consider ways to implement computationally
the various eversions originally drawn by hand;
this requires interpolation of splined curves in time and space.