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Superhyperbola

Posted on 27 March 2025 by John

An ellipse has equation

\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1

and a hyperbola has equation

\left(\frac{x}{a}\right)^2 - \left(\frac{y}{b}\right)^2 = 1

Similarly the superellipse has equation

\left|\frac{x}{a}\right|^p + \left|\frac{y}{b}\right|^p = 1

and the superhyperbola

\left|\frac{x}{a}\right|^p - \left|\frac{y}{b}\right|^p = 1
When p = 2, the absolute value signs are unnecessary and the
superellipse and superhyperbola reduce to the ellipse and hyperbola
respectively.

Increasing p makes the superellipse more like a rectangle. But unlike
a rectangle with rounded corners, the change in curvature is
continuous.

[superhyperbola2]

Increasing p makes the superhyperbola more blunt at the vertices.

[superhyperbola1]

Marketing

The superellipse is a fairly well known variation on an ellipse. Even
if you're not familiar the term, you've probably seen the shape. I
give a couple examples here. The superhyperbola is the obvious analog
of a superellipse, but the term is far less common. I'd never hear
the term until yesterday.

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.

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.

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John D. Cook

John D. Cook, PhD

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