Distinct Activities: Shackled by a Common Name?

[Seaworthy Lass]

Originally Posted by Andrew Troup

I think it would be a good idea to go back to your bed, Lass !

Sleep deprivation can trigger novel insights, but IME you can never be sure until they're reassessed in the cold clear light of dawn...

LOL, yes fatigue will produce odd effects .

OK, the disc idea was a little far fetched. I am now awake and trying hard to find a good analogy, as the treadmill idea only works for limited conditions. It is a plane free to move in all directions that the ant is walking in a straight line over (we can dismiss any considerations in vertical movement as the boat can arrive at any height and these measurements are small compared to the distance travelled).

The first of the two posts I made in my sleepy state is correct though I think, the treadmill concept is good if you just consider the component of current perpendicular to the bearing of the destination from the departure point (or at least if it doesn't skew at all during the journey).

[noelex 77]

I think the whole ant analogy is stupid.
Everyone knows their little legs are too small to push the buttons on the GPS

[Andrew Troup]

Originally Posted by noelex 77

I think the whole ant analogy is stupid.
Everyone knows their little legs are too small to push the buttons on the GPS

As they say in Latin "Pedes antibus non GePeSitudinem"

Personally I think that they were being a little PedAntic

[wolfenzee]

One interesting fact about boating is that basically you can cruise with what ever your economic position will allow. Which is why one sailor might have a $15,000 electronic navigation system and another has a $30 used handheld GPS. This applies to everything across the board.
Then you have the sailors that can afford the latest greatest fanciest stuff and realizes they can function quite comfortably on what they have and put the money to better uses. I have a couple of GPS systems, Radar, VHF, SSB, etc all bought really cheap because someone wanted the newest model, I have alot of nice "previously owned" stuff the only problem I can see is cosmetic.

[Andrew Troup]

The problem with applying a conveyor belt representation to multi-directional scenarios is that it needs to be slewed to represent current movement in a new direction.

This slewing makes it look as though the constant heading does not produce a straight track through the water, but I've persuaded myself the problem is entirely with the belt representation. In order to travel in a new direction, bodies of water do not slew, like some sort of mobile whirlpool.

Here's how I convinced myself that the constant heading solution DOES work even for currents travelling at multiple angles. I hope it might work for someone else.

The notional substitute for a belt was inspired by Seaworthy Lass' movable plane. (Thanks Lass, for the idea, and for your strenuous and thought-provoking brainstorming of alternative analogies, above and beyond the calls of duty and slumber)

Imagine a cheap plastic, semi-see-through outdoor picnic tablecloth from the fifties, with checkerboard squares (our one was red and white; other colours are available)

By hanging suitable weights on one of more edges of the tablecloth (or some other more realistic means), we can get it to move steadily in any direction we want. This is done in such a way that the lines of the pattern remain parallel with the edges of the table, meaning there is no slew.

We have trained an ant, and equipped it with crampons. These leave a trail of punctures in the tablecloth and dents in the table.

We pick from one hat a departure and destination point, expressed as coordinates on the table, the unit of measure being the same as one square on the cloth.

From another hat, we pick a series of tablecloth displacements, expressed as squares per minute.
A sample might read: (move cloth) 2 squares to the right for first minute, then 2 squares up for the next minute.

Say the departure we are given happens to be directly above the destination, at a distance of 5 squares.

To simplify things a bit, we'll assume the ant adjusts its speed to arrive over the destination after the predetermined period of time. The speed is assumed constant.

Here's how it plays out:

We prepare the tablecloth for the trial run as follows: Mark the position of the departure point. Move the cloth 2 squares right and 2 up. Mark the position of the arrival point (on the tablecloth, in both cases).
By simple subtraction and addition, we can mentally work out that the destination point on the tablecloth will be 3 squares up and 2 left of the departure point.

Draw a straight line on the cloth between these points.
Have the ant walk the line. The length will be the square root of: (3 squared) plus (2 squared), IOW sqrt 13, or 3.6 squares, and the ant has two minutes to do this, so it must walk at a rate of 1.8 squares per minute.

Now we can run the simulation.
The ant sets off along the line we drew, and we move the cloth according to the instructions. After two minutes, the ant is over the destination, having walked the shortest distance possible across the cloth.

At the end of the first minute, the ant will be half way along the line, 1.5 squares up and 1 square left on the tablecloth
The cloth will have moved 2 squares right.

We can either add these together to find out whereabouts on the table the ant has got to, or simply look at the crampon marks in the table.
Either method will show that during that first minute the ant travelled 1.5 square up and one square right relative to the table.

During the second minute, the ant moves a further 1.5 squares up and 1 square left on the tablecloth.
However the tablecloth is now moving upwards 2 squares, so the ant travels 3.5 squares up and 1 square left relative to the table.

By adding the two results, we see that the ant travels up five squares (1.5 + 3.5) and sideways no squares (1-1) as required.

The job of the ant is to get between two points on the tablecloth in a given time, at the lowest possible speed. The ant doesn't need to know anything about how the tablecloth is moving over the table, so it's a very simple abstraction from the point of view of the ant, and even an ant would probably recognise that a straight line is the best solution to their simplified problem.

Who am I to argue with an ant?

So this is one way to demonstrate not only that is a straight line through the water the optimum solution (which most people can be persuaded of), but that a constant compass heading achieves this result. Even when the current is from various random directions.

(Note that this current progression is conceptually identical to the one I raised for a boat in post #266, which lead to me expressing a bit of doubt in #269 about the constant-heading solution in such a case. I have now erased those doubts.

The "optimal solution" I proposed in that case, although I didn't realise it, was a straight rhumb line up the table. If the ant was required to do this, it would have had to walk two squares left during the first minute, and as many squares up as it could get.
During the remaining minute the ant would walk up the remaining squares.

Solving this requires striking two arcs of equal radius from departure and arrival points on cloth such that they intersect on the vertical line 2 squares left of the departure. (Because the ant travels the same distance across the cloth during the first minute as the second). The radius of those arcs is about 2.15 squares

This requires the ant to crawl at 2.15 squares per minute and is so far above the optimal 1.8 figure as above (it's about 20% more distance) that I'm embarrassed to have suggested it.

And I apologise to anyone who was persuaded by my previous line of reasoning, including one who posted to that effect based on my initial raising of some doubt in post #257.)

[rgscpat]

Button-pushing ants?
Everyone knows how hard it is for an ant,
to move a rubber tree plant.

[Astrid]

Everyone knows how hard it is for an ant,
to move a rubber tree plant.

Is that from a song called High Hopes?

[Jammer Six]

You know, sometimes the things I read on the internet concern me.

[Jammer Six]

Other times, it concerns me that I understand them.

[rgscpat]

Originally Posted by Astrid

Is that from a song called High Hopes?

Exactly so.

Sinatra's classic song High hopes - Video Dailymotion

and discussion

High Hopes (1959 song) - Wikipedia, the free encyclopedia

Versions of the song appeared in such places as the 1960 John F Kennedy campaign, the animated movie Antz, the movie Rocky, and the Philadelphia Phillies baseball stadium.

[Seaworthy Lass]

Originally Posted by Andrew Troup

............
Imagine a cheap plastic, semi-see-through outdoor picnic tablecloth from the fifties, with checkerboard squares (our one was red and white; other colours are available)
[/INDENT]By hanging suitable weights on one of more edges of the tablecloth (or some other more realistic means), we can get it to move steadily in any direction we want. This is done in such a way that the lines of the pattern remain parallel with the edges of the table, meaning there is no slew.

We have trained an ant, and equipped it with crampons. These leave a trail of punctures in the tablecloth and dents in the table.

We pick from one hat a departure and destination point, expressed as coordinates on the table, the unit of measure being the same as one square on the cloth.

From another hat, we pick a series of tablecloth displacements, expressed as squares per minute.
A sample might read: (move cloth) 2 squares to the right for first minute, then 2 squares up for the next minute.

Say the departure we are given happens to be directly above the destination, at a distance of 5 squares.
..........
So this is one way to demonstrate not only that is a straight line through the water the optimum solution (which most people can be persuaded of), but that a constant compass heading achieves this result. Even when the current is from various random directions.

Perfect model!
The constant heading regardless of any incremental changes in direction of current was not in doubt, but it is satisfying to have a model to fit it.

I was hung up on how to fix the departure and arrival points on the plane (equivalent to the edges of the treadmill), but of course they are not on the plane at all but on the "table" .

I would have the ant leave droppings that instantaneously seep through to the table though, rather than wearing crampons (unnecessary drag ). Maybe think of an ant who has had too much Mexican food the night before departing on his journey?

You are the ant's pants!

[Dockhead]

So everyone's got it now!!! Cool.

[nigel1]

Originally Posted by Dockhead

So everyone's got it now!!! Cool.

Which is great, the ant by now is knackered and giddy and any further analogies will doubtless prove worthless

[goboatingnow]

I just finished setting up Mathematica on my iMac. Ill do some error calculations between various CTS time quantums and the pure Track method. Hopefully ill get some useful answers.

Dave

[Hudson Force]

Originally Posted by Dockhead

So everyone's got it now!!! Cool.

No, 'sorry to say, putting mountain climbing gear on the ant does nothing for me. I've come some distance in accepting that the constant heading track and the GPS shortest distance track are in final outcome of idential time, with variables in distance and speed, but I still see no mathmatical advantage in time traveled with the constant heading plan. Both are vector summations.

If we look at another analogy, we can consider that employing the constant heading plan is slapping down a $100 bill for the task and the use of the GPS is paying a penny at a time to toal 10,000 payments.