Path Homotopy in the Context of Fuel Optimal Trajectories

It's not uncommon to see the term "homotopy" used in papers when solving for minimum fuel spacecraft trajectories. While I've used the process before, I've never really understood the theory from which it originates nor what the concept broadly is in any thorough manner. I thus decided to teach myself rudimentary algebraic topology and, in doing so, have notably expanded my understanding of not…