That's also how Eigen defines it: https://libeigen.gitlab.io/eigen/docs-nightly/classEigen_1_1...

Basically this is the ax^2 + bx + c + d = 0 in 3D space.

Of note is that once you've got planes, you can define points as intersections of n hyperplanes.

In 2D, 2 intersecting hyperplanes (=lines here) will define a point.

But what if these lines are parallel? Well you just got the "point at infinity" abstraction for free. And if you defined operators on points as intersections of lines they will also work with the points at infinity.

All this being nicely described under Projective Geometric Algebra: https://projectivegeometricalgebra.org/projgeomalg.pdf

Also: with a few modifications you get conformal geometry as well; with everything being defined as intersections of spheres. After all, what is a plane but a sphere that has its center at infinity?