Sounds like you should take a look into graph theory.
When the route is fixed, you can create a mathematical form (formula) for it or use vectors to describe the route.
Using a formula makes display easy as you can create a set of values and draw them directly on a (xy) graph. Using vectors makes creation of a set of values a bit more difficult. However, once done, you can also use an xy graph or an image display to draw lines:
Norbert
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CEO: What exactly is stopping us from doing this?
Expert: Geometry
Marketing Manager: Just ignore it.