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I'm trying to derive an equation in $x, y, z$ of such a 3D surface, that:

  • Its cross-section in x-y plane is an ellipse
  • Its cross-section in y-z plane is a parabola
  • Its cross-section in x-z plane is a hyperbola
  • The cross-sections are not allowed to be trivial everywhere - e.g., a point as a cross-section doesn't count.
  • If it's impossible to have a singular required cross-section curve, it's fine to have multiple - for instance, if the y-z cross-section has more than one parabola, it's fine, but all the curves in that cross-section must only be of a parabolic shape - though, they don't have to be identical. Still, a single curve is preferred.
  • Final equation in $x,y,z$ doesn't have to be degree 2 even though all cross-sections represent degree 2 curves in their respective planes.

I usually have some effort to show, but here I'm not sure where to even start or how to prove that, for instance, such a shape cannot exist in 3D. I "imagine" that it involves some "rotation-like" transformation of a parabola into hyperbola along the geodesics of elliptical shape, but I can't figure out how to do it.

Alma Do
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1 Answers1

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  • Analogous to dual conics, we can construct one example below:

$$ \det \begin{pmatrix} 0 & x & y & z \\ x & \color{red}{0} & 1 & 1 \\ y & 1 & \color{lime}{-1} & 1 \\ z & 1 & 1 & \color{blue}{+1} \end{pmatrix}= x+y+z+1$$

  • See more on dual conics in my older post here.
Ng Chung Tak
  • 19,693