Superhyperbola
22 comments
·March 28, 2025zahlman
JKCalhoun
I was thinking the same. The hyperbola (and super hyperbola) are more about negative space.
munchler
> It’s not clear why the superellipse would be common and the superhyperbola obscure
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.
Animats
Alex Pentland was once into geometric modeling of soft objects using superquadrics, which are the rotational solids of superellipses.[1] This was a reasonable idea around the time the Symbolics 3600 was a thing and GPUs barely existed.
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.
BrenBarn
> The name is also off-putting: juxtaposing super and hyper sounds silly. The etymology makes sense, even if it sounds funny. Piet Hein used the prefix super– to refer to increasing the exponent from the usual value of 2. Its unfortunate that hyperbola begins with a root that is similar to super.
How about just "superbola"? :-)
adrian_b
> Its unfortunate that hyperbola begins with a root that is similar to super.
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" :-)
1970-01-01
Super-hyper-bola: the unstoppable bola. It's ARCH nemesis? Mathman.
https://en.wikipedia.org/wiki/Square_One_Television#"Mathman...
https://www.youtube.com/playlist?list=PLQYdOIKzgOwDk-QXhRVDz...
falcor84
Increasing the p doesn't seem appealing to me, but playing with this, I found that I'm actually quite fond of decreasing p to a value in between (1, 2) to be quite nice for having a curve make a sharper turn.
I couldn't find a name for this curve, but I propose "hypohyperbola".
lexlambda
I was confused by the legends between the first and second image? Both have p=2 and p=4 yet look different, I believe that this is wrong, as the sentence before the second image states "increasing p". Hope I'm not missing something
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TheRealPomax
Also known as a "squircle", https://en.wikipedia.org/wiki/Squircle, which is a way better name.
jihadjihad
The author of TFA has a couple neat posts about squircles that might be of interest, too:
0: https://www.johndcook.com/blog/2018/02/13/squircle-curvature...
1: https://www.johndcook.com/blog/2019/04/02/history-of-the-ter...
alex-robbins
A superellipse is only a squircle if a and b are 1. As with squares and rectangles, all squircles are superellipses but not all superellipses are squircles.
adharmad
Unrelated but on the topic of squircles, an interesting factoid:
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...
kretaceous
> if a and b are 1
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.
russellbeattie
Oh, I disagree entirely! This name makes you think of alternate dimensions and other cool stuff.
Sadly, maxisuperhyperbola and ultramaxisuperhyperbola don't seem to be things.
mitthrowaway2
Since a hypercube is a higher-dimensional cube, I think a higher-dimensional hyperbola would be a "hyperhyperbola".
itishappy
hyperboloid, my day is ruined
hnuser123456
I think the novelty of the article was in the 2nd image, and why it's so much less well known. But my theory is there are not as many applications for symmetrical curvy lines that diverge than symmetrical curvy lines that converge.
nlawalker
Yeah that bit was funny to me. "I'm not sure why closed shapes are so ubiquitous and pairs of diverging lines aren't."
>It’s not clear why the superellipse would be common and the superhyperbola obscure, but here’s some speculation. First of all, the superellipse had an advocate, Piet Hein. If the superhyperbola has an advocate, he’s not a very effective advocate.
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.