Shame this doesn't include my favorite description, a 4d vector in projective affine coordinates.
Instead of a normal and point or constant you get (x,y,z,1) . P = 0. The translation between the two is trivial. If you want a plane spanned by 3 points you just can use the generalized cross product to find P.
One advantage is that you can avoid all the special cases with 3 intersecting planes. There exists exactly 1 point that is on all 3 planes, but as this is in projective coordinates it might lie at infinity.
I honestly don't know, I encountered the concept during my study, but it took lots of practice to get comfortable with it. Most articles I encounter are too practical to really foster understanding, or too technical to work as an introduction.
You can start with a description of how projective matrices work (and how translation and rotation are related to it). After that, best tips I can give are start with 2D until you can't bear to see another cross product. Then get familiar with Cramer's rule and higher dimensions. You'll need sone fluency in linear algebra.
My first practical use of the concept was to rectify photographs where e.g. a building was not quite upright. That might be a good starting point.
Instead of a normal and point or constant you get (x,y,z,1) . P = 0. The translation between the two is trivial. If you want a plane spanned by 3 points you just can use the generalized cross product to find P.
One advantage is that you can avoid all the special cases with 3 intersecting planes. There exists exactly 1 point that is on all 3 planes, but as this is in projective coordinates it might lie at infinity.