After finding the NASA publication I mentioned in my previous post, I poked around a while longer in the NASA Technical Reports Server and found a few curiosities. One was that at one time NASA was interested in shapes that similar to the superellipses and squircles I’ve written about before.
A report  that I stumbled on was concerned with shapes with boundary described by
The superellipse corresponds to α = β = 2.5, and the squircle corresponds to α = β = 4 (or so), but the report was interested in the more general case in which α and β could be different.
By changing α and β separately we can let the curvature of the sides vary separately. Here are a couple examples. Both use A = 0.5, B = 0.8, and β = 1.8. The first uses α = 3.5
and the second creates a straighter line on the vertical sides by using α = 6.
So why was NASA interested in these shapes? According to , “The primary objective of the current research has been the optimun [sic] design of structural shapes” subject to the equation above and its three dimensional analog.
In order to provide material useful to the space program, it was decided to initiate the research with a determination of the geometrical and inertial properties of the above classes of shells. This was followed with a study of shells of revolution which were optimized with respect to maximum enclosed volume and minimum weight. A study on the vibration of beams was also reported in which the beam cross-section was defined by (1). Since bulkheads for bodies of type (2) require plate shapes of type (1), investigation was continued on clamped plates defined by (1).
Here (1) refers to the equation above and (2) refers to its 3-D version. The goal was to optimize various objectives over a family of shapes that was flexible but still easy enough to work with mathematically. The report  is concerned with computing conformal maps of the disk into these shapes in order to make it easier to solve equations defined over regions of that shape.
 The conformal mapping of the interior of the unit circle onto the interior of a class of smooth curves. Thomas F. Moriarty and Will J. Worley. NASA Contractor Report CR-1357. May 1969.