Superhyperbola
johndcook.com
johndcook.com
Piet Hein was a designer. The superellipse, being closed, is a much more useful shape for designing physical objects, and fitting a roundish shape into a rectangular space is a useful property in our world of architectural square corners.
I think the explanation is pretty obvious: The hyperbola itself is way more obscure than the ellipse to begin with, so it’s not surprising that hyperbola variations are also obscure.
Once weighted skinning of skeletons was invented, that idea went away.
I once wasted a few months trying to do analytical collision detection for superquadrics using an early symbolic mathematics system. Dead end.
How about just "superbola"? :-)
The choice of the word "similar" by the author is not really appropriate, because this is not a random similarity.
The words "super" and "hyper" mean exactly the same thing and they descend from a single Indo-European word, the former being the Latin variant and the latter being the Ancient Greek variant, while the corresponding English variant is "over".
"Superbola" would be ambiguous, because that could be a "superparabola".
I am among those who dislike the mixing of distinct languages inside a compound word when there is no need for that.
Therefore, while "superquadrics" is a correct term, instead of "superellipse" and "superhyperbola" I would prefer "hyperellipse" and "hyperhyperbola". And instead of "Superman", "Overman" :-)
https://en.wikipedia.org/wiki/Square_One_Television#"Mathman...
https://www.youtube.com/playlist?list=PLQYdOIKzgOwDk-QXhRVDz...
I couldn't find a name for this curve, but I propose "hypohyperbola".
Sadly, maxisuperhyperbola and ultramaxisuperhyperbola don't seem to be things.
Since Archimedes until and including Euler, paraboloids and hyperboloids of 2 sheets were named "conoids" (parabolic conoids and hyperbolic conoids), which makes more sense than "hyperboloid", because they look like rounded cones (Archimedes analyzed only their variants that are surfaces of revolution).
The hyperboloid of 1 sheet has been named by its discoverer (Christopher Wren) as "hyperbolical cylindroid", which is also more suggestive of the shape of this surface.
The change in terminology to paraboloids and hyperboloids was justified by the fact that in the older literature "conoids" and "cylindroids" had been used only for surfaces of revolution, because those with elliptical sections were not discussed before Euler, but this justification fails, because we now also talk about elliptical cylinders and cones, so there the names "cylinder" and "cone" have been retained, even if they also referred strictly to surfaces of revolution in the older literature.
A more consistent terminology would have been to retain the names "spheroid", "cylindroid" and "conoid" that were used in the old literature and add "elliptical" whenever they are not surfaces of revolution, like it has been done for "cylinder" and "cone".
0: https://www.johndcook.com/blog/2018/02/13/squircle-curvature...
1: https://www.johndcook.com/blog/2019/04/02/history-of-the-ter...
Among all squircles having arbitrary exponents (|x|^p + |y|^p = 1), PI (3.14159...) is the smallest value of ratio of circumference to diameter.
There is a paper on this with the pithy title "π is the Minimum Value for Pi": https://www.tandfonline.com/doi/abs/10.1080/07468342.2000.11...
If a and b are equal* (not just 1). A circle is a special case of ellipse where a and b are equal and the eccentricity is 0. This is the same principle.