If you want to describe (for example) a loop or a part thereof in numbers, you will find it a very complex excersise. The examples you show in your link are snapshots. The reality is a continuously changing set of variables as, most significantly, speed changes and gravity acts at at a different angle to the z axis. At least with a loop you don't (or shouldn't) need to worry about all that cosine bank angle stuff. But you also need to consider the change in control effectiveness and control authority with changing airspeed - very relevant for the F-16 as an example.
In practice, the equations give way to known loop dimensions for given speeds, hence the use of gates - essentially known parameters at various, critical points in the display. Pilots flying the displays are well versed in the theory of flight, but it's not a very practical way of designing or flying a display sequence.