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Hoping someone may be able to point me in the right direction so I can research this topic further.

Scenario: You have a vector field (either 2D or 3D) and you wish to find the shortest path between two points located within the vector field.

For example imagine an object floating on the ocean surface. The object has a simple rudder that allows it to steer. How can we find a path that will get the object from its current location (point on vector field) to (or as close to as possible) the destination point.

In this scenario the vector field would represent ocean currents in a 2D space and the desired path should make optimal use of these currents to reach it's final destination (or a point as close to the final destination as possible).

Can anyone offer any suggestions/hints around how this could be calculated?

Note: Post has been edited to improve clarity of question.

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  • $\begingroup$ What is the relation between the path and the vector field? $\endgroup$
    – Alexandre Eremenko
    Commented Jun 5, 2020 at 11:11
  • $\begingroup$ Yes, what do you mean by "within the vector field"? do you mean that one can only move following the vector field? If so the problem is hopeless because two arbitrary points are extremely unlikely to be connected by an integral curve. If not I guess your problem is meaningless and you should reformulate your question. $\endgroup$
    – leo monsaingeon
    Commented Jun 5, 2020 at 11:14
  • $\begingroup$ Thanks to both of you, Leo the first portion of your response is what I was referring to, that you can only move following the vector field. I was interested in finding paths that are as close to, if not all the way to the 2nd point in the vector field. $\endgroup$
    – dza
    Commented Jun 5, 2020 at 11:34
  • $\begingroup$ Can you only move along the vector field (like a train along a track), or do you have a rudder to steer with (unlike a train, more like a boat)? $\endgroup$
    – Ben McKay
    Commented Jun 5, 2020 at 12:15
  • $\begingroup$ @BenMcKay Thanks for the question, it's made me rethink what I originally stated in the question. It would be more like an unpowered boat, with no sail, just a rudder so it would moved strictly by ocean currents. $\endgroup$
    – dza
    Commented Jun 5, 2020 at 12:20

1 Answer 1

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The problem you are struggling to express here is known as Zermelo's navigation problem, and you are not correctly writing it down. You can read about it in Vel Jurdjevic's excellent book Geometric Control Theory.

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  • $\begingroup$ Thank you, Zermelo's navigation problem certainly seems to describe what I was trying to express. $\endgroup$
    – dza
    Commented Jun 5, 2020 at 12:39

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