# Calculate the aircraft's movement path in which its side speed is required

I am working on an implementation problem in which given an aircraft and the objective of the system is to find the quickest route (optimal in terms of time) from point A (contains the information of the aircraft's 2D position and heading which is looked from top view) to point B (contains the same information as point A). There are several conditions:

1. The aircraft must use the full speed. By full speed, I mean the engine must use its maximum speed as much as it can provide in the current condition (e.g., if some of the engines are destroyed, the maximum speed decreases). There are three types of the speed:

• Forward speed
• Side speed
• Turning speed
2. The aircraft doesn't have to use all three types of the speed simultaneously.

The answer of the following post here has answered some of my problems, but there still some of the issues I want to solve, which is if I want to include the side speed of the aircraft to the movement, how will the calculation of the movement path (especially when the aircraft turns) look like?

Please kindly recommend and I would appreciate any ideas for this problem.

• Also, is the desired route the fastest in terms of distance (the shortest) or time (the quickest)? The shortest path may involve a high number of manoeuvres, and this may take time; the quickest may take just few seconds, but it's not necessarily the shortest one, I guess. Aug 6, 2020 at 9:20
• It's the quickest one. Sorry for the ambiguity. Aug 6, 2020 at 9:41
• If your aircraft can move at its full side speed and forward speed simultaneously, then the fastest way to get anywhere is to fly at it diagonally (though that might look odd). You'd use the same method from the linked answer, but just apply a twist to your start and end positions until their new "forward" direction lines up with this fastest diagonal. If your aircraft has to reduce forward speed to go sideways or vice versa, then whichever speed is greatest is the preferred direction of travel, and again you can use the linked answer as-is, or with a pre-processing twist. Aug 6, 2020 at 11:51
• Can you please explain more about the forward direction lining up with the fastest diagonal? Because I don't think the forward direction will align with the fastest diagonal direction if it flies diagonally, somewhat like this image. Aug 6, 2020 at 18:45