How to make the total length of a trail as short as possible?

[moonman239]

I'm just curious. Let's say a trail designer who works for a Boy Scout camp wants to build two trails that connect two campsites to the OA ring. How would he go about doing that?

I know that he would not use reflections (he would if he were, say, locating a point across the river such that the two trails that lead from the campsites to that point are as short as possible) and I don't think he'd find the circumcenter, incenter, or any other -center in the triangle and then make both a long trail through the circumcenter to the ring and a shorter trail that connects to the long trail at the circumcenter.

 

[HallsofIvy]

If the only condition is to have two trails connecting the two campsites to the campfire circle, just draw a straight line from each campsite to the campfire circle.

 

[moonman239]

If the only condition is to have two trails connecting the two campsites to the campfire circle, just draw a straight line from each campsite to the campfire circle.

The total length of the two trails must also be as small as possible. I trust that's the way?

 

[genericusrnme]

The total length of the two trails must also be as small as possible. I trust that's the way?

A straight line is the shortest path from one point to another (if everything is flat and there are no other constraints).

If you have a situation where there is some extra 'cost' depending on what path you take, for example if there's a river in the way or if there's a hill or something, you'd use the calculus of variations to minimise the a path integral given some 'cost' function of the coordinates.

 

[theorem4.5.9]

If you have a situation where there is some extra 'cost' depending on what path you take, for example if there's a river in the way or if there's a hill or something, you'd use the calculus of variations to minimise the a path integral given some 'cost' function of the coordinates.

This may be true if you want to to model the situation, but the question posed seems to be more conceptual in nature. Here's what I suggest.

First put a pulley at the two endpoints that you want to find the shortest path from. Sloppily guess the shortest path by running a single rope through the two pulleys. Pull on both ends of the rope until it's taut.

If there's a lake in the way pretend it's fenced off, that way the rope conforms around the boundary (this should be true for any obstacle).

Then the rope has to be the shortest path between the two points.

 

[haruspex]

Let's say a trail designer who works for a Boy Scout camp wants to build two trails that connect two campsites to the OA ring. How would he go about doing that?

The question is not clear to me, but I think you might be asking how to make the total length of trail connecting 3 points as short as possible.
If any angle of the triangle is 120 degrees or more, just connect each of the others straight to it. Otherwise, find that point where each pair of vertices subtends 120 degrees and connect each vertex straight to that.

 

[genericusrnme]

This may be true if you want to to model the situation, but the question posed seems to be more conceptual in nature. Here's what I suggest.

First put a pulley at the two endpoints that you want to find the shortest path from. Sloppily guess the shortest path by running a single rope through the two pulleys. Pull on both ends of the rope until it's taut.

If there's a lake in the way pretend it's fenced off, that way the rope conforms around the boundary (this should be true for any obstacle).

Then the rope has to be the shortest path between the two points.

You'd have to try several different initial placments of the rope if there are obstacles though or you could end up with the situation where the rope is wrapped around some obstacle and doesn't actually take the shortest path (when you do this you are essentially doing what the calculus of variations does).

 

[HallsofIvy]

So that, rather than "make the total length of the trails as short as possible", the problem is really "make this rather trivial problem as complicated as possible"?

 

[moonman239]

The question is not clear to me, but I think you might be asking how to make the total length of trail connecting 3 points as short as possible.
If any angle of the triangle is 120 degrees or more, just connect each of the others straight to it. Otherwise, find that point where each pair of vertices subtends 120 degrees and connect each vertex straight to that.

That's it.

So, a straight line will do the trick, I guess. I thought of the possibility of connecting each campsite directly to the ring.

Now, let's say this trail designer has to not only connect the campsites to the OA ring (FYI, OA stands for Order of the Arrow, a program operated by the Boy Scouts of America. Order of the Arrow is for exemplary Scouts.) but he also has to connect them to each other.

 

[Vargo]

Δ. From another angle, ∇.