Curiosity: Sorry for not being so clear. Perhaps this will clarify…
Consider a closed unit circle centered at the origin (0,0) in the Cartesian coordinate system in the Euclidean plane (i.e. your standard unit circle). Now, consider two different points in the closed unit circle (these points can be inside the circle or on the boundary of the circle, but since the circle is closed, they can never be outside the circle). Let’s call these points A and B, respectively.
Now, assume A and B move freely in the closed circle with equal maximum speed. A tries to “catch” B by getting to the same location as B (i.e. A catches B if both A and B have the same exact coordinates at any given time). B tries to “evade” A.
Assuming A and B play optimal strategies (i.e. A and B make the best possible moves at all times in accordance with their goals), does A “catch” B? If so, how can A be guaranteed to “catch” B? If not, how can B be guaranteed to “evade” A?
To make it more interesting for non-math-oriented people, I had called A the rodef and B the nirdaf since in a literal sense this is the case (i.e. A is the chaser and B is the chased). However, you correctly point out that from a halachic standpoint, the din of rodef and nirdaf is more complex. I apologize for the confusion.
Aside (if it’s confusing, just ignore): This is an example of a pursuit-evasion game. There are several interesting properties of these games in general, but there is something particularly interesting about this game.
One of the basic properties of these types of games is to note the path that the chaser takes to catch the chased. This path is called a pursuit curve, which is aptly named because the path of pursuit need not be straight. For example, if a mother is chasing a little child running straight in some direction, chances are that the mother is faster, and she can get onto the child’s path of travel and then follow the child’s path until she catches him. However, if she runs at the same speed as the child, then she cannot just follow the child’s path because she will never catch up. Instead, the mother runs to where the child is heading and intercepts him at some point. If the child changes direction during the chase in response to the mother’s chase, then the mother’s path of pursuit will be curved. Hence the term pursuit curve.