General Equation for Parabola in 3D space

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?
• 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. – amd May 7 '18 at 17:01
• @amd The question has been reworded to reflect this more accurately. Thanks. – hypergeometric May 7 '18 at 17:05
• There’s not really a matrix representation for the same reason that you need a system of equations: each defines a surface. – amd May 7 '18 at 17:09
• This seems relevant: math.stackexchange.com/a/1957430 – David K May 8 '18 at 13:23

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.

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.