Through five points in generic position passes a single conic curve.

which variables changed here?

The red points are moved and the (a,b,c,d,e) coefficients are computed to pass through the points.

@ylecun Unfix the real coordinate “1” on the real number line: riemann surface spinning in 4-space from the POV of an arbitrary location d. This is just the slice it makes through 4-space “coordinated” at point d and normalized to only the real component of 3d slice through 2d. (Donut)

Honest question, could any combination of "linearly independent" xy-nomials achieve the same as long as there are still 5 free coefficients or is this particular for conic curves? if the latter is true, is that someway related to quadric matrices or so?

Yes exactly

And a circle with 3 non-collinear points (or infinite conic sections).