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The curve $1 = xy$ is hyperbola so I was expecting the surface $1 = x y z$ to be called a hyperboloid, but I don't think this is true.

In general I'd like to know the name of the class of $n$-dimensional objects with formulas $$ \prod_{i=1}^n x_i= 1 $$

Lucas
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  • For what it's worth, the way we generalize a hyperbola to "hyperboloids" relies on the alternative parameterization $X^2-Y^2=1$ viz. $x=X+Y,,y=x-y$, leading to a natural alternative to Euclidean geometry. There isn't a similar geometric interest in $\prod_ix_i=1$ with more than $2$ indices. – J.G. Aug 27 '20 at 13:28
  • @J.G. yeah, they're all quadratic. Would have thought such shapes with such a simple formula would have a name though... maybe not! – Lucas Aug 27 '20 at 13:41

1 Answers1

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Hyperboloids are quadric surfaces. There are two types of them:

  • hyperboloid of one sheet, like $x^2-y^2-z^2-1=0$,

  • hyperboloid of two sheets, like $x^2-y^2-z^2+1=0$.

https://en.wikipedia.org/wiki/Hyperboloid

There are paraboloids, but no hypoboloids (don't ask me why :).

The site Mathcurve does not give $xyz=a$ a specific name. It is a cubic surface.

https://mathcurve.com/surfaces.gb/cubic/cubic.shtml