I am afraid you need to get a little education in mathematics and physics before you can follow the thought experiment I was proposing.
Alan: I propose this experiment: two balls moving in space collide into each other and move apart again.
Graph of experiment: I put "time" as vertical axis. I put "space" as horizontal axis.
So far, you are ok except for your use of the term "space". Under most graphical representations, the single horizontal axis you wish to use would be called "the x axis". The fact that you did not do so, tends to indicate an attempt to make it a relativistic "space-time" geometry. The fact that it is a relativistic "space-time" geometry resides in mathematics required to change from one coordinate system to another and not in the names of the axes. As I say, there is no error here; only a disturbing indicator.
Alan: I show the two balls at the base of the graph with a gap between them, both balls on the horizontal axis at zero time. Above here I show the balls are now closer together at one unit time. Looking further above I have the balls closer and closer for higher values of time till they are together, then further up they progressively move apart as climb higher up the time axis.
Again, this far everything is fine. You are clearly representing two balls approaching one another then bouncing apart. The fact that you chose such an experiment implies your understanding of relativity is limited as it isn't going to provide much information on relativistic phenomena. In particular, you give no information on the forces in your experiment; thus the problem, as stated, cannot be analytically solved. (We want to work with the numbers here, not just the pictures. We need to know x as a function of t for both balls: i.e., you must be able to solve your problem.)
Alan: I also draw another graph representing the view from on one of the balls. The left ball is seen on the graph to describe a straight vertical line parallel to the vertical time axis.
The right ball is seen to approach it from a bigger distance than in my first graph; to meet it part way up the time-line of the left-ball whose view I'm adopting here, then move away again. Note in the first graph I got equal curved paths for each ball, in the second graph I got a straight line for one ball and a deeper curve for the other.
Just two ways of looking at two balls approaching, colliding, moving apart; on a space-time graph.
Ok, you have now introduced a second frame. At this point, if the two graphs are representing exactly the same phenomena it is important that the transformation of your data is in exact accordance with the proper relativistic transformation required to go from your first frame of reference to this second frame of reference.
I doubt you took the care to make those calculations as your description implies quite a complex change. First, the left-ball is now following a straight vertical line parallel to the vertical time axis which implies your second frame is the rest frame of the left-ball. You go on to say that you "got .. a deeper curve for the other". The fact of that statement implies that the paths in the first drawing were curved. This means that the mathematics required to go from the first drawing to the second involves an extremely involved transformation which changes continually as time progresses. You are dealing with accelerating frames here and transformation between accelerating frames require an understanding of General Relativity.
If you cannot "solve the problem and analytically represent the relativistic correct solution" then you cannot perform the thought experiment I proposed. For you, there are strong indications that this is a major problem here.
Alan: I can take an arbitrary location of start and finish for the left ball and for the right ball I guess.
Again, I get a distinct impression that you do not understand what I was talking about. By establishing those lines which represent the trajectories of your balls on a piece of graph paper you have already elected a start and finish by not drawing infinitely long lines. Since I am talking about analytical representation, I need to specify a selection of end points.
Alan: "p" appears to designate a confining definition of the object. For example if the right ball had previously bumped into something else it would carry the effect of this with it so that pre-set defining characteristic of the right ball could be called "p".
No, that is not the definition of a parametric representation of a line. In a parametric representation of a line, a parameter is used to label a specific point in the line. "p" is a number who's value tells the observer which point in that line is being referred to. Fundamentally, it makes no difference how that parameter is laid out so long as a specific value of the parameter refers to a single point on that line. In a parametric representation of the line, the specific value of the coordinates of each and every point in the line can be represented by functions of that parameter.
The importance of parametric representation is that curved lines which intersect themselves can be easily represented analytically. A decent knowledge of analytical geometry is skill central to understanding physics.
At this point in your post, there is no possibility that you are following the thought experiment I proposed as that thought experiment requires a facility with parametric representation you do not seem to possess.
I will not go on any further because from this point on there exists no bridge between what I suggested you think about and what you are thinking about; none to my knowledge at least. I am really not trying to give you a hard time Alan; what I am trying to say is that you need a better understanding of mathematics and physics. Without that, it will never be possible for you to understand what I am talking about. It is a question of communication.
You said once that you knew a math professor who was willing to talk to you. Take the thought experiment to him and see if he can help you understand what I am proposing.
Have fun -- Dick