Brain hurts - Working out course to steer

[El Pinguino]

Continuing with the GC thread drift...
I mentioned above that some charting programs display GC by default... these pics show what the old original CMap program does.... two waypoints over 4000 miles apart in 36*S... track shown as E/W.... route shows course to steer is 122* as is the bearing of waypoint 2 from waypoint 1... ie rhumb line displayed ...GC data ( course/bearing and distance ) also displayed.

I wonder how much other consumer grade charting software is out there with this problem?

I know i-sailor struggles with the dateline ( 180* E/W ).

[Q Xopa]

Originally Posted by El Pinguino

Continuing with the GC thread drift...
I mentioned above that some charting programs display GC by default... these pics show what the old original CMap program does.... two waypoints over 4000 miles apart in 36*S... track shown as E/W.... route shows course to steer is 122* as is the bearing of waypoint 2 from waypoint 1... ie rhumb line displayed ...GC data ( course/bearing and distance ) also displayed.

I wonder how much other consumer grade charting software is out there with this problem?

I know i-sailor struggles with the dateline ( 180* E/W ).

I assume what you are saying about Rumb lines and Great Circle tracks is all correct.

However I this isnt what the OP, and others, are talking about.

GC and Rumb lines are, for want of a better way of describing it, lines on the earths surface. So in this case it would be relevent if we were traveling on the sea bed.

However as sailors we are not travelling on the earths surface. We are travelling in the body of water on top of the earths surface/ sea bed.

So to complicate this, for us sailors, the water we are travelling in moving, ie Currents and Tidal flow etc.

When these Tides and Currents are moving travelling from our sides particularly, and then reversing as the tide reverses can cause us to travel further than we need to if we try stay on a fixed track over the ground (be it GC or Rumb line).

So if we travel further through the water at a given sailing speed, say hull speed on a good day, it will take us longer to get us to our destination than if we can travel a shorter distance at hull speed through the water.

So saying this another way the water we are travelling in the above example will go in a 'S' pattern in relation to a GPS track on the Sea bed.

Imagine putting a GPS on a float. It will of course plot a track where the tides and currents take it from where you dropped it in the water. If we dropped it at a point on our GC or Rumb line GPS track you can imagine all the extra distance we travel in the water. Of course this will take us longer.

[boatman61]

Originally Posted by Q Xopa

I assume what you are saying about Rumb lines and Great Circle tracks is all correct.

However I this isnt what the OP, and others, are talking about.

GC and Rumb lines are, for want of a better way of describing it, lines on the earths surface. So in this case it would be relevent if we were traveling on the sea bed.

However as sailors we are not travelling on the earths surface. We are travelling in the body of water on top of the earths surface/ sea bed.

So to complicate this, for us sailors, the water we are travelling in moving, ie Currents and Tidal flow etc.

When these Tides and Currents are moving travelling from our sides particularly, and then reversing as the tide reverses can cause us to travel further than we need to if we try stay on a fixed track over the ground (be it GC or Rumb line).

So if we travel further through the water at a given sailing speed, say hull speed on a good day, it will take us longer to get us to our destination than if we can travel a shorter distance at hull speed through the water.

So saying this another way the water we are travelling in the above example will go in a 'S' pattern in relation to a GPS track on the Sea bed.

Imagine putting a GPS on a float. It will of course plot a track where the tides and currents take it from where you dropped it in the water. If we dropped it at a point on our GC or Rumb line GPS track you can imagine all the extra distance we travel in the water. Of course this will take us longer.

Actually.. to my mind one travels further over the sea bed than you do through the water.
If for example your heading is say N to S or viki verki across the English Channel UK-France-UK, Poole to Cherbourg.. CTS 180*
Although the tide from the East is pushing you West for 6hrs, then East for the next 6hrs you are still technically travelling in the same body of water.. its the seabed beneath that changes so that whereas you travel 60nm through the water you actually travel 72nm over the seabed.. assuming an average of 1kt each way.
But hey.. I aint 'Intelligentsia'..

[Q Xopa]

Originally Posted by Dockhead

These problems are also fun because the answers are often non-obvious, or counter-intuitive, and sometimes require a lot of thought to figure out.


You can work on this one yourself -- plot out some passages where you steer to max VMC and see where you end up.


The answer is not immediately obvious, but I think this one can be solved by logic.



One thing to define is what you mean by this -- set a constant heading at that heading which, at the start of the passage, gives you max VMC? Or keep steering to keep that VMC at max?


The first will obviously not work -- because you'll just sail past the destination and never arrive.



The second is more subtle, but I think you will see that it won't work, if you think about it. Here is the point -- steering this way maximizes the MOMENTARY VMC, without regard to the future. That's by definition -- the future is not considered at all, when you steer this way. So this course will give up FUTURE VMC, and a disproportionate amount of it, by putting you downtide from your destination, in order to get higher VMC in the moment. A bad bargain.


That's the abstract logic; but I think you will see that it actually works this way, if you'll plot out a few passages.


A good disproof of this would be to consider the case of absolutely constant current. We know that if the current is constant, then the straight water track coincides with the straight ground track, and you sail along the the rhumb line*. We all know that this is fastest. But you will obviously not be sailing max VMG to the destination, sailing like that. The max VMC heading at the beginning of the passage would have you heading off downtide from the destination at a high SOG. But you will be losing ground against the current which you have to make up later, so you will end up with drastically reduced VMC, later in the passage, which more than wipes out the gain in VMC at the beginning.


Logical? As I said, you can plot out a few example passages to see for yourself, whether it's really like this, or not.






* To the great circlists: I think I found a much better way to explain why all navigators ignore great circle on all passages of less than many hundreds of miles, and many passages of even thousands of miles. What is a "line"? Mathematically, a line has no width. But in the real life of real navigation, a line, like a course line, like a rhumb line, might be considered to be as wide as what we can practically steer. So if the apex of a great circle course is say 20 meters from the idealized rhumb line, or even a cable or two, then the rhumb line and the great circle course is the same -- because you can't steer the difference between them, even if you wanted to bother with it. The great circle path is contained within the rhumb line, if you consider the rhumb line to be as wide as what you can realistically perceive and steer.


It is true that a constant heading is not an exactly straight path through water, except in a few special cases (along the equator; along a meridian; etc.). But when steering a great circle would require you to hold a course to tenths or hundreds of a degree, it IS, for all practical purposes, a straight line, and the navigator correctlytreats it as a straight line, unless he is just trying to show off.


It's a bit like Newtonian physics. We know now that Newtonian physics isn't exactly right. Newtonian physics aren't right at all at the atomic level, and they are very slightly not right at a macro level. But we don't use quantum mechanics at all when we engineer a bridge -- all the assumptions are based on classical Newtonian physics, even though we know they aren't precisely correct. It would be bad engineering to attempt to introduce quantum effects, into engineering a bridge, as a useless complication greatly increasing design costs and the risk of an error, and for no benefit at all, since the differences between quantum physics and classical physics for these problems are practically meaningless, even if they are not non-existent.

Ok, drew some pictures with triangle example vectors and tried to understand this. I havent managed to see the link Im obviously missing. Im obviously slow.

Any more clues greatfully received.

As far as I can still see, I still think maximising our rate of closure, by definition, must mean that we will travel the shortest, ie quickest, distance through the water.

I cant understand what you are saying about it not working as we get close to our destination. From what I can see, even if you are at your destination any movement away from our destination will give us a decreased, actually it will go Negative from zero, rate of closure. Ie we are increasing our rate of moving away. So opposite to what we want.

Even as you say allowing for future by setting ourselves up in advance.

Im also not convinced about this. As Yogi Bear says "predictions are difficult, especially about the future". I think this is our reality, atleast often.

But even if we set ourselves up for an accurate (skeptical) predicted cross current, in the mean time we missed out on optimising our current condition.

Autopilots are not operating in a 'pedictive' way. They are reactive, as in the sense an error first then and make a correction. So if we can define our reference/null point we can display errors and get our autopilots to drive to a minimised error.

[parkstone bay]

Sorry to disagree QXopa, but shortest ISN'T always the quickest. To take a simple example, I live in port A 5 miles from port B; my motorboat does 5 knots. On the day I want to go to port B, there's a 2.5 knot counter current. Straight-line it, it will take 2 hours to get there. If I rock-hop via the shallows, it's a lot further, the charted distance is 7.5 miles, but there's no current. Taking the nominally longer route, it's 1 hour 30 mins to get there. That is passage planning.

[Q Xopa]

Originally Posted by parkstone bay

Sorry to disagree QXopa, but shortest ISN'T always the quickest. To take a simple example, I live in port A 5 miles from port B; my motorboat does 5 knots. On the day I want to go to port B, there's a 2.5 knot counter current. Straight-line it, it will take 2 hours to get there. If I rock-hop via the shallows, it's a lot further, the charted distance is 7.5 miles, but there's no current. Taking the nominally longer route, it's 1 hour 30 mins to get there. That is passage planning.

Agree entirely. In fact we are saying the same thing.

I was, as you are talking about distance through the water, not geographical distance. Port A to Port B is a geographical distance.

So in your case your motor boat at 5 kts has travelled through less water in its 7.5 Nm geographic rock hop trip so gets there sooner.

If we motor into a current we are still travelling 5 kts, ie 5 Nm distance per hour but dont travel 5 Nm geographical distance.

Yes and also agree about passage planning. However the big trick comes with trying to plan for conditions like currents and tides that arent always known. More specifically its easy to plan and use currents to your advantage by planning to go when its heading in the direction we want to go and hitching a ride.

However if currents are effecting us on our beam its harder to calculate the quickest route.

This is the OPs original question.

[Q Xopa]

Originally Posted by parkstone bay

Sorry to disagree QXopa, but shortest ISN'T always the quickest. To take a simple example, I live in port A 5 miles from port B; my motorboat does 5 knots. On the day I want to go to port B, there's a 2.5 knot counter current. Straight-line it, it will take 2 hours to get there. If I rock-hop via the shallows, it's a lot further, the charted distance is 7.5 miles, but there's no current. Taking the nominally longer route, it's 1 hour 30 mins to get there. That is passage planning.

In fact nice example.

Doing the maths-

A to B direct @ 5kts in 2 hours means you travel through 10 Nm of water.
Ie 5 Nm/h x 2 hrs = 10 Nm.
2.5 kts average opposing current. Ie 5 kts B Spd - 2.5kt current = 2.5 kt Net Boat spd. 2.5 kts x 5 Nm = 2 hrs.

A to B rock hop in 1.5 hrs means you travelled through only 7.5 Nms of water.
Ie 5 Nm / hr x 1.5 hrs = 7.5 Nm. Even though its 7.5 Nm geographical on your chart plotter.

This is a good result having Nil net opposing current. Even better if you can get some current going your way to assist.

So in your example you travelled 2.5 Nm less (through the water) than A to B direct.

As you say, Passage planning.

[El Pinguino]

Originally Posted by Q Xopa

I assume what you are saying about R(h)umb lines and Great Circle tracks is all correct.

However I this isnt what the OP, and others, are talking about.

GC and R(h)umb lines are, for want of a better way of describing it, lines on the earths surface. So in this case it would be relevent if we were traveling on the sea bed.
.

You obviously missed the bifurcation buoy at #108..... some took the lesser channel because it looked more interesting.......

[Q Xopa]

Originally Posted by El Pinguino

You obviously missed the bifurcation buoy at #108..... some took the lesser channel because it looked more interesting.......

Ooops, my bad. Got me not reading every post.

[Dockhead]

Originally Posted by Q Xopa

Ok, drew some pictures with triangle example vectors and tried to understand this. I havent managed to see the link Im obviously missing. Im obviously slow.

Any more clues greatfully received.

As far as I can still see, I still think maximising our rate of closure, by definition, must mean that we will travel the shortest, ie quickest, distance through the water.

Not indeed -- even in a constant current, if you bear off downtide a bit, you can increase your VMC compared to sailing the optimum course, which is equal to the rhumb line in a constant current.


Calculating the OPTIMUM VMC at any one moment is complicated and makes my head hurt. Here's a handy guide: https://soltools.wordpress.com/2011/03/30/manual-vmc/


But we can disprove this idea in a simpler way:


1. We know that if the water is moving constantly, the optimum path will be to steer uptide until COG = BTW. Everyone agrees on this, right? That is the constant heading which gets us to the waypoint without changing course and therefore losing distance through water. It's the shortest path through water, which in this particular case happens to coincide with the shortest path over ground.


2. If we show even one example of GREATER VMC by steering other than the above, then we see that maximixing VMC does NOT give us the best course, even if the current is constant. Everyone agree so far?


3. So here's a problem:


You are going from some place to some other place 60 miles away, and your waypoint bears 180. You are making 6 knots through the water and there is a constant current of 4 knots running due East West.


If you steer 180, then your COG will be 213.7 and your SOG will be 5.99 knots.


Now let's compare to steering up tide to make COG = to BTW, what we all know is the optimum course.


To make COG = BTW, we have to steer 146.3. Like that, our COG will be 180, but our SOG will be only 4.99 knots.


4.99 is less than 5.99.


So here is good mathematical proof, that optimizing VMC will not give you the optimum course, EVEN if the current is not changing.

Originally Posted by Q Xopa

I cant understand what you are saying about it not working as we get close to our destination. From what I can see, even if you are at your destination any movement away from our destination will give us a decreased, actually it will go Negative from zero, rate of closure. Ie we are increasing our rate of moving away. So opposite to what we want.

What I mean is that if we steer a constant heading based on max VMC at the beginning of the passage, then we will sail right by the destination and never arrive. I don't think that's what you meant -- I think you meant that you keep adjusting your heading to maintain max VMC, correct?


But in a constant current, if you sail max VMC, you will be carried downtide from your destination. As you do so, the bearing of the destination will change, and you will have to constantly change your heading to maintain max VMC. As you do so, not only will your heading change, but your VMC will fall rapidly as your are sailing more and more into the tide. Pretty soon it will be less than the VMC you would get from the COG = BTW straight line path, and will get less and less, the further you go. The AVERAGE will be less than the COG= BTW path.


I don't have time to do this plotting -- I'm at work -- but if you will just take some passage with a constant cross current, and plot the heading for max VMC, then do just one vector triangle to get COG, and then plot your position in one hour or half an hour or whatever. Then repeat, and keep repeating until you get the ship to the waypoint.



You will get a series of positions which will represent the path sailing your maximizing VMC course. Then just add it all up to get the time required to complete the passage, and compare it to the straight rhumb line passage. You will see it takes you longer.


The closer the current speed is to your STW, the more vividly you will see this effect.

[Q Xopa]

2. If we show even one example of GREATER VMC by steering other than the above, then we see that maximixing VMC does NOT give us the best course, even if the current is constant. Everyone agree so far?

I think this is all, as you have recently said, most a question of Definitions and descriptions.

So I’ve gone back to VMC school. I think ‘Velocity Made Good to Course’ is referring to a ‘Course’ over ground (still trying to understand VMC). Please correct me if I’m wrong.

I had also forgotten the definition of Rhumb line (had to re look it up)- constant bearing to True or Mag north. Again this wont work because this is talking about lines on the earth, ie Sea bed. We are traveling in a body of water moving different directions above the Sea bed. This is how we mostly used to navigate with our compasses and paper charts.

But this may actually work reasonably well. Because it we get swept left by the tide then equally right on the opposite tide. The problem is non equal tidal movements. So coincidentally, although it’s probably better than most of the other current methods in many circumstances.
But as we know it is very rare to have equal tidal deviations during a trip.


GC (Great Circle) Shortest distance of 2 points on the Earth’s curved surface. So as apposed to Rhumb lines the bearing will change. This is what most Chart Plotters and Aircraft FMSs will direct to steer. However with no, or negligible tidal or current influences this will be the shortest over ground and therefore the quickest route to sail.


What I mean is that if we steer a constant heading based on max VMC at the beginning of the passage, then we will sail right by the destination and never arrive. I don't think that's what you meant -- I think you meant that you keep adjusting your heading to maintain max VMC, correct?

Yes correct continual adjusting of heading.

But getting back to definitions again, I’m not sure if VMC is the exact definition of what I was thinking. As far as I can understand about VMC, the ‘C’ part is referring to a ‘Course’. Which again is (according to Wiki) ‘direction of Travel’. Travel is a reference to over the ground. So no, not exactly what I am thinking of. VMG might be closer, but I’m still trying to get my head around that definition.

So getting back to what I was trying to describe before, (but not very successfully), ‘Rate of Closure’ to our destination. This is the best way I can describe it. So actually nothing to do with any ‘Course’ over the earth like VMC is referring to. I’m more talking only about a rate/ speed to our destination. Not talking about a direction/ course/ track/ bearing etc, or any compass pointing or line on the surface on the earth.

Simply if we are getting closer to our destination at the fastest rate we can in the conditions we are sailing in then our course and heading etc will happen.

[Dockhead]

Originally Posted by boatman61

Actually.. to my mind one travels further over the sea bed than you do through the water.
If for example your heading is say N to S or viki verki across the English Channel UK-France-UK, Poole to Cherbourg.. CTS 180*
Although the tide from the East is pushing you West for 6hrs, then East for the next 6hrs you are still technically travelling in the same body of water.. its the seabed beneath that changes so that whereas you travel 60nm through the water you actually travel 72nm over the seabed.. assuming an average of 1kt each way.
But hey.. I aint 'Intelligentsia'..

You may not be "Intelligentsia", but your understanding of this is absolutely and precisely correct -- and it's something that a lot of sailors find hard to understand. So kudos

[Dockhead]

Originally Posted by Q Xopa

2. If we show even one example of GREATER VMC by steering other than the above, then we see that maximixing VMC does NOT give us the best course, even if the current is constant. Everyone agree so far?

I think this is all, as you have recently said, most a question of Definitions and descriptions.

So I’ve gone back to VMC school. I think ‘Velocity Made Good to Course’ is referring to a ‘Course’ over ground (still trying to understand VMC). Please correct me if I’m wrong.

I had also forgotten the definition of Rhumb line (had to re look it up)- constant bearing to True or Mag north. Again this wont work because this is talking about lines on the earth, ie Sea bed. We are traveling in a body of water moving different directions above the Sea bed. This is how we mostly used to navigate with our compasses and paper charts.

But this may actually work reasonably well. Because it we get swept left by the tide then equally right on the opposite tide. The problem is non equal tidal movements. So coincidentally, although it’s probably better than most of the other current methods in many circumstances.
But as we know it is very rare to have equal tidal deviations during a trip.


GC (Great Circle) Shortest distance of 2 points on the Earth’s curved surface. So as apposed to Rhumb lines the bearing will change. This is what most Chart Plotters and Aircraft FMSs will direct to steer. However with no, or negligible tidal or current influences this will be the shortest over ground and therefore the quickest route to sail.


What I mean is that if we steer a constant heading based on max VMC at the beginning of the passage, then we will sail right by the destination and never arrive. I don't think that's what you meant -- I think you meant that you keep adjusting your heading to maintain max VMC, correct?

Yes correct continual adjusting of heading.

But getting back to definitions again, I’m not sure if VMC is the exact definition of what I was thinking. As far as I can understand about VMC, the ‘C’ part is referring to a ‘Course’. Which again is (according to Wiki) ‘direction of Travel’. Travel is a reference to over the ground. So no, not exactly what I am thinking of. VMG might be closer, but I’m still trying to get my head around that definition.

So getting back to what I was trying to describe before, (but not very successfully), ‘Rate of Closure’ to our destination. This is the best way I can describe it. So actually nothing to do with any ‘Course’ over the earth like VMC is referring to. I’m more talking only about a rate/ speed to our destination. Not talking about a direction/ course/ track/ bearing etc, or any compass pointing or line on the surface on the earth.

Simply if we are getting closer to our destination at the fastest rate we can in the conditions we are sailing in then our course and heading etc will happen.

OK, but I've demonstrated exactly that steering to be "simply getting closer to our destination faster", at the beginning of a passage, will mean that we get there later, if there is a cross current, even an unchanging one.


Let's clear up confusion over definitions:


1. VMC -- I don't like this term, but it's what modern navigators use to mean actually velocity made good to a waypoint. So that's all that means -- "the fastest rate of getting closer to our destination" = "max VMC", OK? I understood your idea; I'm describing it as "steering to maintain max VMC", meaning, steering to always close your destination at the fastest rate, as of that moment.


2. "Rhumb line" -- that's the constant course "relative to True North", not indeed "towards True North". For us it means the course which will take us from A to B with a constant COG. It looks like a straight line on a flat map, but on a globe it's not in reality straight in most cases, hence the whole great circle discussion.



Forget great circle! As I said, it's a confusing and useless distraction.



Assume that the earth is flat between Needles and Cherbourg -- if you can maintain a constant COG of 181, you will travel in a straight line over ground. That's the rhumb line.


Now the problem should make more sense.

[cyan]

I think that the whole non-intuitive issue for students of CTS versus Rhumb Line course plotting is often NOT addressed well enough. In my experience, instructors will use vector math and trigonometry to show that CTS is best... but it often still "doesn't make sense" intuitively. At least Dockhead explained the shortest-course-through-moving-water concept, to which I say well done.

My favorite example is the Zig-Zag Paradox. (that's not really a thing, just my name for it) For me, this shows BOTH the non-intuitive bit, and also the "ah-ha" moment. (for MY brain, anyway) The beauty here is that there is NO vector addition and NO trigonometry. (threw in a couple Greek letters for angle to look official, though) I included diagrams, page 1 and page 2 of this "Zig-Zag Paradox", which I sketched one day to graphically deal with math that didn't seem right to me.

This example is a simplified form of crossing the Gulf Stream. The boat must cross West to East, from point A to point B in the shortest amount of time. There are 3 equal-width "zones" or bands to cross in which the first one is still water ("-0-"), the middle one has a 90 degree current (Vc), and the third one is still water. ("-0-")

The two boat captains have chosen different strategies. Mr. Rhumb Line (aka Mr. GPS) will take a straight line over ground, then change heading twice in order to crab along only for the middle zone where there is actually current. Mr. CTS has chosen a constant heading for all 3 zones, current or no-current. Since they are both going to the Bahamas from Florida, the last one to point B buys the rum.

On Page 1, at the top we see that Mr. Rhumb Line has a direct course over ground, and he needs to change his heading for the (middle) current zone to some big upstream angle that we'll call gamma. Mr. Rhumb Line takes his pain all at once, but only inside the (middle) current zone. He has to make a course correction entering and leaving this "crabbing" zone. The distance, "d" is the total amount of downstream drift that both either need to overcome due to the current zone.

At the bottom of Page 1, we see that Mr. CTS has what looks like a funky zig-zag over ground, with a constant (but smaller) upstream heading angle that we'll call theta. Mr. CTS is a steady dude, and spreads his pain across all 3 zones, current or no current. In each of the three zones he is making up 1/3 of the total downstream displacement, d. His smaller crabbing angle in the current zone has him drifting downstream, but he makes up for that 'drift" in both zero-current zones.

Mr. CTS seems to be doing the wrong thing, intuitively. This zig-zag boy is initially going out of his way (upstream) when he doesn't have to- the water is still. In the current zone he drifts so far that he then has to make up for it in the final still water zone. (Weirdo.) Meanwhile, Mr. Rhumb Line seems to be taking the direct path, just like his GPS and autopilot decided to do.

However, on Page 2 we play a little mind game to "straighten-out" the crooked water. (stay with me here) We can agree that any boat needs to overcome the fact that this current will push a crossing boat sideways by a distance of "d". If the water was actually STILL in all 3 zones (but with same boat behavior), then the same problem could be effectively shown by displacing the destination (Point B) by the distance "d" to the South, up stream. It's a funny diagram now, but still shows the same problem: arriving to the South by an offset distance "d" through still water is the same thing as arriving straight across the channel while fighting current that wanted to offset you by a distance "d" to the North. We have simply straightened out the water in this diagram, shifting the land instead.

We now see on page 2 that the zig-zag path THROUGH WATER was actually taken by Mr. Rhumb Line. (who's the zig-zag boy, now??!) Mr. CTS never made a heading change, thus he made a straight line THROUGH WATER to the destination, point B. It is clear here that the most direct route to B is the one with NO HEADING CHANGE. If Mr. CTS wins on page 2, then Mr. CTS wins on page 1. Bob's your Uncle. (should be a light-bulb-moment here about now)

With a bit of math, for Mr. CTS we could show the angle theta = arcsin(Vc/3Vb) without even using the dreaded Law of Cosines, due to the symmetry of the problem. (Vc=scalar current speed, Vb=scalar boat speed) For Mr. Rhumb Line, we could show that his angle gamma = arcsin(Vc/Vb). Adding up the 3 segments for each boat, we could compare real numbers. That wasn't the point here, though. The goal was to pick away at that nagging non-intuitive feeling about the superiority of CTS with varying cross currents.

Moral of the story:
The rhumb line pays the rum fine!

-Cyan

[Q Xopa]

Well done, Great diagrams! A true Picture tells a thousand words example.


What causes most if this issue is the terms used. As unfortunately these terms get so missed used and mixed up its nigh on impossible to know what some people are meaning.

In fact your 1st guy Mr Rumb, aka Mr GPS, is what I would call Mr Great Circle, aka Mr GPS.

Your 2nd 'weirdo' guy Mr CTS, (like the name), He is steering a contant heading, but sailing a zig zag course over the ground.

But as you so eloquently illustrated he sailed through the least amount of water.

Ok agree with Dockhead that this is a constant heading.

Also agree with him that VMC is a very sloppy term.

However Im still thinking maximising VMG would take you on this line.