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Let $x\in R^3$. What is the name of the curve that satisfies $x_1\cdot x_2\cdot x_3 = c$ and $x_1 + x_2 + x_3 = d$ for appropriately chosen $c$ and $d$ so that the curves intersect? Note that the plane is perpendicular to the hyperbola's major axis.

Note that what I am referring to as a 3-dimensional hyperbola ($x_1 * x_2 * x_3 = c$) is not a hyperboloid in its canonical form but rather an extension of the rectangular hyperbola ($x_1 \cdot x_2 = c$) into $R^3$ (and $R^n$); so it too may have a different proper name in geometry.

Intersection of plane with 3D-hyperbola

Contour of intersection on the plane

Sven
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  • Do you have a parametric equation for the intersection (if image no. 2 was in the $xy$-plane)? It is beautiful. – Bobson Dugnutt Aug 19 '16 at 08:32
  • Your curve is just some 3rd degree curve (or rather, part of it). I don't think it has a proper name; "oval" might be the right word, but it is much more broad. – Ivan Neretin Aug 19 '16 at 08:33
  • Using an orthogonal change of coordinates, here is one of the curves: $8 - 2 a^2 - 2 a^2 b - 6 b^2 + 2 b^3 = 1$. Don't forget the intersection occurs in the other octants as well. By the way, quadric surfaces are the usual generalization of a hyperbola to 3D, for instance hyperboloids of one or two sheets. I can see where $xyz=1$ comes from, though. – Kyle Miller Aug 19 '16 at 09:18
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    It is a cubic plane curve with three linear asymptotes. That curve I gave factors as $0=1+(1 + a - b) (-1 + a + b) (1 + 2 b)$, and you can see the three lines. It looks like "Asymptotes, Cubic Curves, and the Projective Plane" by Nunemacher describes a theorem that asymptotes appear as factors in this way. – Kyle Miller Aug 19 '16 at 09:30
  • Thanks for the comments so far! @Kyle, thanks for pointing out the other intersections. In my problem I am only interested in $R^3_+$, so the other orthants aren't that crucial (to me). The curve you provide seems to be the parametric equation for the open-ended branch in one of the negative actants. Do you have the one for the closed curve? – Sven Aug 19 '16 at 14:06
  • @Kyle, I am familiar with the hyperboloids. As you can see, I am generalising the product to $R^n$. Is "$n$-dimensional hyperbola" be too much of a misnomer you think? – Sven Aug 19 '16 at 14:11
  • @Kyle, thanks for the Nunemacher paper on cubic curves, of which this seems indeed to be an example (thks @Ivan). I need to do some work, obviously. – Sven Aug 19 '16 at 14:15
  • @Sven Make sure you do an implicit plot of that curve with a domain centered at the origin: it is of the entire intersection. About your question about an equation for the closed curve alone, unfortunately it cannot be done as a polynomial because the polynomial has no factors. (The zero set is an "irreducible variety.") – Kyle Miller Aug 19 '16 at 18:10

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