557-
A Google search for your puzzle shows that the question was first posed by Richard Rado, and the solution you described was given by Abram Besicovitch.
Google Books allowed me to see a sketch of the solution in action, so you can disregard my request for clarification.
I did notice that the solution has X getting closer to Y as well as Y getting closer to the circle’s edge with every move, so in the “real world”, even if X and Y were microscopic, X’s edge would eventually touch Y’s – it’s just that their exact centers wouldn’t ever be in exact alignment.
I’d still appreciate an answer to my “semicircle” and “diameter” questions.
Also, since maintaining distance isn’t what’s important, but rather just preventing X and Y from occupying the same spot, couldn’t Y use a similar or identical plan to evade X inside a square?
Just for your amusement, here are terms I don’t remember hearing:
radius vector
{t_1, t_2, t_3, …}
ith
half-plane
Furthermore, your application of the Pythagorean theorem and R_i’s being bounded was way over my head – oh, well.