How many different parametric cubic curves are there? 2. The “same” game

I´m interested in both the answer to the question in the title and in finding the clearest possible derivation of that answer. My first efforts were very algebra-intensive, but had an aura of concreteness about them. As I played with the question, I came up with more and more elegant ways to ward off the brute-force algebra and make the answer more intuitive without reams of calculation. For these elegant techniques to work, however, I needed to build up a collection of tools (lemmas) that themselves take a little time to explain. One tool is the use of a particular matrix identity, which I review in this article, and which I use to answer the title question: there are three different parametric cubic curves. However, deriving that answer in a clear and obvious manner is still a little way away, but that´s OK: the reason I´m doing this is really not so much to answer the question as to build intuition about the algebraic formulas and the geometry that they represent. The journey is the reward