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?
On
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.
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.