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I would call it a saddle, but it's not the standard saddle. Is there a standard name for it, the way we have 'hyperboloid of one sheet' for example?

Henry
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  • What is the "standard saddle"? Are you sure this isn't the same thing, rotated 45° and possibly scaled? – hmakholm left over Monica Dec 09 '13 at 19:50
  • @Henning Sorry the "standard saddle" claim comes from wikipedia http://en.wikipedia.org/wiki/Saddle_surface – Henry Dec 09 '13 at 19:57
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    The right name, as said in some answers below (e.g., Paul's) is "hyperbolic paraboloid" ; it belongs to the category of ruled surfaces as is the case for the hyperboloid with one sheet you mention, but it is definitely a very different shape. The name "saddle" is too vague, like speaking of an oval when one deals with an ellipse. – Jean Marie Feb 04 '23 at 13:37

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It is simply a hyperbolic paraboloid, equivalents to the surface $z=x^2-y^2$. Its cross-sections are parabolas and hyperbolas.

Armando j18eos
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Paul
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It's called "the saddle" :). Substitute $x=(u+v)$ and $y=(u−v)$ to get $z=(u+v)(u−v)=u^2−v^2$, which is a more conventional parametrization of the surface, while the surface itself is unchanged.

John Hughes
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  • This is true, but then $z=(xy)^3$ is also a saddle, geometrically or analogously. Perhaps there is a more precise name? – abnry Dec 09 '13 at 19:54
  • @John oops thank you! – Henry Dec 09 '13 at 19:56
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    @nayrb: I see your point. My change of vars, however, is a linear map (scaling the two ordinate axes by $\sqrt{2}$ and the vertical axis by $1$); yours involves a cube (or cube root). Somehow that seems more extreme. I'd be happier if my substitution involved scaling all three axes by $1$, but that doesn't work. I grant you that such a nonuniform scaling can turn a sphere into an ellipsoid, and a right cylinder into a non-right one, so it's probably not the best notion of "sameness". But it does seem to satisfy the original asker of the question. :) – John Hughes Dec 09 '13 at 20:58
  • @John: However, if you scale the horizontal axis by a further $\sqrt 2$ and the vertical one by $2$ the net scaling will be isotropic, and the result is still similar to the standard saddle. – hmakholm left over Monica Dec 09 '13 at 21:40
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    There is no such thing as "the standard saddle" taken to mean a unique surface. – Narasimham May 02 '16 at 22:33
  • I disagree with that. There is no specific surface called "the saddle", and you missed the real name, given by @Paul. –  Sep 26 '16 at 13:17
  • OK. I should have said "it's called a saddle". Sometimes things have many names. I could have said "It's called 'the graph of $f(x, y) = xy$', " and that. too, would have been true. OP seems to have been happier with my generic name that with Paul's more specific one, but another reader may prefer Paul's; so be it. – John Hughes Sep 26 '16 at 13:22
  • How do we parametrize the surface? I really can not see the reason for doing it this way. – Ninja Oct 12 '17 at 12:10
  • The original surface is parameterized by the map $(s, t ) \mapsto (s, t, st)$; my suggested alternative is parameterized by the map $(s, t) \mapsto (s, t, s^2 - t^2)$. – John Hughes Oct 12 '17 at 13:00
  • I have to solve this question for a problem session and I still can not understand how can we parametrize it like this way? Why this way and not $s^2 + t^2 $ for example.. – Ninja Oct 12 '17 at 14:42
  • I'll be honest -- your question makes no sense to me. I've given a parameterization; you've given an expression, not a parameterization, that seems unrelated to the question. I have no idea how to answer this, so I'm going to stop trying. – John Hughes Oct 12 '17 at 15:47
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Saddle point is an attribute/ character of all surface points. If you hold a 3D model of it in your hands, its nomenclature is invariant by the direction of your view :) but lines of projection can have a separate name.

EDIT 1:

Sorry did not follow OP properly at that time, (now deleted phrase hyperboloid of 1 sheet). All points of negative Gauss curvature have Saddle points qualitatively, as against the ellipsoidal points. (synclastic/anticlastic etc.)

The real parts of $ ( x+ i y)^2 = ( x^2 -y^2+ i \,2 x y )\quad \, ( z= 2 xy ;\, z= x^2 -y^2 ) $ are intrinsically same, one can be obtained from the other by rotation through$ 45^0 $ about z-axis.Are ruled surfaces. The surface is called a hyperbolic paraboloid due to cross sections as parabolas/hyperbolas. Hypar is a term used in Civil engineering application.

AFIK there is no standard saddle implying a fixed geometrical parametrization and there perhaps need not be. It is just contrasted from convex bulbous surface geometries.

Narasimham
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  • Nope, this would be $z^2=x^2+y^2-1$. –  Sep 26 '16 at 13:15
  • Are you sure it is a hyperboloid and not a hyperbolic paraboloid? – edm Sep 26 '16 at 13:16
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    Thanks, added clarifications. "Saddle point" is generic for all anti-clastic surfaces. The Hyperbolic paraboloid is a special case but not rotationally symmetric. So $ z^2= x^2+y^2-1$ is one sheeted with saddle points, $ z^2= x^2+y^2+1$ is two sheeted synclastic, both rotationally symmetric . – Narasimham Sep 27 '16 at 08:54