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It is well known that the Cartesian equation for a general parabola in 2D space is $$(Ax+Cy)^2+Dx+Ey+F=0$$ or in parametric form, $$(at^2+bt+c, pt^2+qt+r)$$

  • What is the Cartesian system of equations for a general parabola in 3D space?
  • Is there a matrix representation?
  • What is the parametric form?
  • What is the aperture?
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  • $\begingroup$ Curves in 3D space can’t in be represented by a single implicit Cartesian equation. You need a system of equations or a parameterization similar to the one you’re already got. $\endgroup$ – amd May 7 '18 at 17:01
  • $\begingroup$ @amd The question has been reworded to reflect this more accurately. Thanks. $\endgroup$ – hypergeometric May 7 '18 at 17:05
  • $\begingroup$ There’s not really a matrix representation for the same reason that you need a system of equations: each defines a surface. $\endgroup$ – amd May 7 '18 at 17:09
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    $\begingroup$ This seems relevant: math.stackexchange.com/a/1957430 $\endgroup$ – David K May 8 '18 at 13:23
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There is no "canonic" way to give a Cartesian system of equations for a parabola in 3D space. The simplest and oldest way is that of giving a parabola as intersection between a plane and a cone, see here for an example.

On the other hand, the locus of points whose distance from a given line (directrix) is the same as their distance from a given point (focus) is a parabolic cylinder, so you may find more natural to give the parabola as the intersection between this cylinder and the plane of focus and directrix.

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The projections of the parabola on the coordinate planes are also parabolas. Hence you can expect the parametric equations

$$(at^2+bt+c,dt^2+et+f,gt^2+ht+i).$$

For an implicit equation, you can use a linear change of coordinates

$$\begin{pmatrix}u\\v\\w\end{pmatrix}=T\begin{pmatrix}x\\y\\z\end{pmatrix}$$ and the system

$$\begin{cases}(Au+Cv)^2+Du+Ev+f=0,\\w=0.\end{cases}$$

This is the intersection of a parabolic cylindre and a plane.

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