What curve does a kayak make...

[PeterB]

Lisa,

aaaaahh, I think I see what you mean. The diagram means that while ferrying properly the arrow is pointing to the waypoint on the GPS screen, not the real waypoint. (in a large enough, cool-enough diagram, the waypoint/ tree would be on the screen, with the arrow pointed at it.)

So, if you're crossing with a good ferry angle," GPS is happy". meaning it shows you're on the right course by pointing straight to it on the screen .

If you're crossing sans ferry angle ( Case 2: just pointing your bow at your target) you're swinging/pendulum style to the right, so GPS is "not happy", arrow pointing to the left, meaning you're not on a straight line, but to the right of your target. (that pursuit curve thing.)

[EEL]

Lisa,

You diagram looks ... not right... to me. .. but I believe you set a waypoint and the arrow points to it. not in the direction your bow is pointed. While ferrying your boat is essential crabbing sideways, so arrow pointed in direction that bow is pointed would not the same as the waypoint.

Peter:

You are correct or at least I agree with your view of the illustrations. One problem with this thread is the failure to describe how the GPS is being used in technically appropriate language which leads to confusion and incorrect interpretations. Every decent book on kayak navigation that I have read in the past few years contains a section on how to use a GPS to follow a "perfect" ferry angle course over ground (COG) by creating a waypoint for the destination. To do that, at the starting point plug in a waypoint for the desired destination and the GPS creates a bearing to the destination which is typically shown on the display of simple units I have used as a big arrow. Think of that bearing as creating a line of position which is the shortest line to the desination from the starting point. It is also the "perfect" ferry angle track over ground. Typically the top center point or line on the display indicates the direction of travel. As Lisa says, ignore the issue of which way the GPS is physically pointed and its orientation to the kayak. If the big arrow and the top center point are aligned, your COG is on the bearing which means the COG is identical to the line of position and you are moving along the "perfect" ferry angle track over ground to the destination Technically this means you are maintaining a cross track error (XTE) of zero which is the ideal COG.

This may or may not be the best course of action in the real world and there may or may not be practical problems in keeping the arrow lined up with the COG when paddling and there may or may not be issues with how frequently the GPS calculates its direction and there may or may not be problems with outlier data being displayed. However the basic technique is sound in theory.

Ed Lawson

Further the affiant sayeth not

[leong]

Peter:

It is also the "perfect" ferry angle track over ground. Typically the top center point or line on the display indicates the direction of travel. As Lisa says, ignore the issue of which way the GPS is physically pointed and its orientation to the kayak. If the big arrow and the top center point are aligned, your COG is on the bearing which means the COG is identical to the line of position and you are moving along the "perfect" ferry angle track over ground to the destination Technically this means you are maintaining a cross track error (XTE) of zero which is the ideal COG.

Ed Lawson

Further the affiant sayeth not

Ed,

Wonderful! I wish that you had joined the team of us believers earlier in the thread.

-- Leon

[lhunt]

Lisa,

aaaaahh, I think I see what you mean. The diagram means that while ferrying properly the arrow is pointing to the waypoint on the GPS screen, not the real waypoint. (in a large enough, cool-enough diagram, the waypoint/ tree would be on the screen, with the arrow pointed at it.)

So, if you're crossing with a good ferry angle," GPS is happy". meaning it shows you're on the right course by pointing straight to it on the screen .

If you're crossing sans ferry angle ( Case 2: just pointing your bow at your target) you're swinging/pendulum style to the right, so GPS is "not happy", arrow pointing to the left, meaning you're not on a straight line, but to the right of your target. (that pursuit curve thing.)

Yes! Got it in one. I'm sure that Ed has it right - I didn't explain it well in the beginning. I was using the terms "heading" and "course" and being too wordy. Oh, well, practice makes perfect.

To answer Ed, the track I've been using to show that it does work in at least some "real world" conditions is the one from a race, here. The straight piece from Land's End to Brace Rock was done purely with the GPS. There was a stiff offshore crosswind and the rest of the racers were way out to sea. I have also used it very successfully in very rough water. There are plenty of other examples.

My GPS (which is a pretty standard model) calculates the direction every second or so, and the line is easy to see. It doesn't swing around any more than a compass. I'm not sure what you mean by "outlier data"? If you mean momentary swings because of satellite errors or being sideswiped by a wave - it happens but goes away so quickly it isn't an issue. During acceleration the line swings a bit, I've noticed.

-Lisa

[leong]

I couldn’t explain how it really works without offending everyone. My training partner, Lisa, is not only strong and fast like Nike (the goddess of strength and speed), but she's smart too.

You go girl.

Sorry racing partner, sometimes I just gotta brag.

Leon

[leong]

Just to let everybody know, John Huth, David, Lisa and I have been discussing this topic amicably offline. Hopefully, a resolution of the issues will come later.

Leon

[lhunt]

We seem to have reached agreement on the basic core of discussion which was in contention before. That is, that a GPS is capable of directing a paddler into the correct ferry angle to make a straight line in a cross current or wind. Some reservations (about particular models of GPSs) might remain open until people get a chance to actually try it.

Here is one snippet from my side of the conversation (such as it was), in case it is of general interest. Note that some GPSs have an internal magnetic compass that allows them to find magnetic North (or true North, depending on settings). This snippet has some references back to my picture from earlier, and also refers to that internal compass.

"Forget the waypoint and the tree for a moment. The GPS has an internal map which it can display on its screen. It needs to know how to orient the map on its screen. If you choose "North up", no problem, it always puts North at the top of the screen. If you choose "Track up", it orients the map so that the interesting part, which shows the places you haven't been to yet, is at the top of the screen. That's how most of us use it, so I'm talking about "Track up".

Scenario #1: If your GPS has a magnetic compass, and you hold it level and point it Northwest, it displays the Northwest part of the map at the top of the screen. Scenario #2: If your GPS's compass is disabled, and you move towards the North, it displays the North part of the map at the top of the screen. It figures that out by getting the point you were at a second ago and comparing it with where you are now.

If you are in a kayak, pointing Northwest, but moving North because of a current, the GPS has to decide whether to use the compass and put Northwest at the top of the screen (#1), or ignore the compass and put North at the top of the screen (#2).

In other words, the GPS can orient its map based on either the compass or the direction of travel, but not both.

Now, having understood that, mentally put the tree back on the shore directly North of your kayak, and put a virtual tree into the GPS's map as a waypoint directly North of your little virtual kayak dot. Current is coming from the West. You have a good ferry angle - your real kayak is pointing Northwest but moving North. The pointer from your virtual kayak dot to the waypoint (virtual tree) [on the map] is definitely going North. But in Scenario #2, North is at the top of the screen, so the pointer is straight up, encouraging you to keep the ferry angle. In Scenario #1 Northwest is at the top of the screen. North, and the pointer, are to the right. If you follow the pointer, it takes you out of your ferry angle and has you point directly at the real tree, giving you a pursuit curve.

-----------------

If you are standing still, obviously it's better to determine direction based on the compass than the direction of travel. My GPS (and also the eTrex Vista) allows you to tell it to use the compass below x mph. Setting it to 0 (or maybe 1) mph disables the compass under most conditions. Looks like factory default is 10 mph. This is done under settings->bearing for both types of GPSs.

The compass issue is further complicated by a couple of things which are off topic here: You have to hold the GPS flat to get the compass to work, so if it is enabled it will sometimes stick or swing around unhelpfully in a kayak in rough water. Also, even GPSs with no magnetic compass have a "compass" screen where it displays a round thing that looks like a compass and gives you your heading based on direction of travel rather than the magnetic or "true" north."

And another snippet, about GPS usage in general.:

"I use the map page but don't pay much attention to the map. There is a pointer to the waypoint and that's all I have to see. I zoom in on the map so that the pointer will be long even when I get close, and I just keep it at the top of the screen. Of course, I seldom use the GPS to find my way or to know where I am. I'm always doing day trips in familiar places. The crossings aren't that long (Isles of Shoals being the longest), and I never go alone. If these things weren't true, I would not rely solely on the GPS.

I use the GPS primarily for speed and distance. I keep a GPS log of everywhere I've been and can display all the tracks on a map. I use it for getting a straight line in a cross wind or current. Once or twice I was glad to have it in a fog, but at no time was it a matter of life or death. Once I used it to backtrack and find a lost paddle.

I think absolutely the biggest reason not to use a GPS is to avoid dumbing yourself down. You can lose your sense of connection with North, South, East and West with a GPS. The "old fashioned" way of navigation increases the challenge and, for many people, the fun. It's like deliberately going out in rough weather, or surfing, or rock play - we all seek something that makes us work harder for exercise (mental and physical) and to keep ourselves young. The other reasons most people give for not using a GPS (batteries running out, it might break, etc.) are legitimate but more manageable. You wouldn't leave on a trip without your chart, and I wouldn't leave on a trip without topping off the juice in my rechargeable batteries. I keep the gadget in an electronics drybag and check the drybag every so often for leaks. If I think there is any chance of my needing the GPS for "real" navigation, I bring a spare for backup."

I think on difficult expeditions or dangerous crossings even diehard chart/compass users might want a GPS for backup, and vice versa.

I also agree with Jason that sometimes the shortest trip (in time) between two points isn't a straight line. In that case, the "old fashioned" way might be better. You might like having a GPS around to help you zero in on the endpoint during the last leg.

Hope this summary helps. I really learned a lot from this discussion.

Lisa

[leong]

We seem to have reached agreement on the basic core of discussion which was in contention before. That is, that a GPS is capable of directing a paddler into the correct ferry angle to make a straight line in a cross current or wind. Some reservations (about particular models of GPSs) might remain open until people get a chance to actually try it.

Here is one snippet from my side of the conversation (such as it was), in case it is of general interest. Note that some GPSs have an internal magnetic compass that allows them to find magnetic North (or true North, depending on settings). This snippet has some references back to my picture from earlier, and also refers to that internal compass.

Lisa

I think that Lisa’s picture shows what happens. Here’s why it has to happen that way:

"The GPS knows the kayak’s location and the waypoint’s location in some fixed earth coordinate system. It doesn’t know it’s own orientation (what direction it’s facing or even if it’s pointing backwards or straight down). It also doesn’t know the heading angle of the kayak. But it does know the Course-Over-Ground (COG) that the kayak is moving along as you paddle. If the COG coincides with the rhumb line from your present location to the waypoint, the GPS’s arrow (at least it’s an arrow with my GPS) points to the top of the screen. If the kayak’s COG for this latest update is to the left of the present rhumb line, then the GPS’s arrow points to the right (indicating that you should change your heading to the right), and vice versa. Of course, your COG is the result of the vector sum of the kayak’s forward speed and the cross-velocity vector due to current, wind and wave action. So, the GPS doesn’t really have any idea of current velocity (I mean water speed and direction) or wind or anything else that’s pushing the kayak. All it really knows is the COG and the latest rhumb line to the waypoint. If you try to keep the arrow always pointing to the top of the screen (via heading changes as necessary) you will be following a straight line to the waypoint. And, as Lisa stated, this will only work if the GPS’s internal compass is not used."

Respectfully,

Leon

[djlewis]

I think that Lisa’s picture shows what happens. Here’s why it has to happen that way:

"The GPS knows the kayak’s location and the waypoint’s location in some fixed earth coordinate system. It doesn’t know it’s own orientation (what direction it’s facing or even if it’s pointing backwards or straight down). It also doesn’t know the heading angle of the kayak. But it does know the Course-Over-Ground (COG) that the kayak is moving along as you paddle. If the COG coincides with the rhumb line from your present location to the waypoint, the GPS’s arrow (at least it’s an arrow with my GPS) points to the top of the screen. If the kayak’s COG for this latest update is to the left of the present rhumb line, then the GPS’s arrow points to the right (indicating that you should change your heading to the right), and vice versa. Of course, your COG is the result of the vector sum of the kayak’s forward speed and the cross-velocity vector due to current, wind and wave action. So, the GPS doesn’t really have any idea of current velocity (I mean water speed and direction) or wind or anything else that’s pushing the kayak. All it really knows is the COG and the latest rhumb line to the waypoint. If you try to keep the arrow always pointing to the top of the screen (via heading changes as necessary) you will be following a straight line to the waypoint. And, as Lisa stated, this will only work if the GPS’s internal compass is not used."

Respectfully,

Leon

Hey, Leon... I'll bet you $100 that... oh, wait... not supposed to do that... unghhh... can't resist... compulsive gambler...$100 that... nono... resist...yow...that...that...that...that.............

(nice music from a compulsive gambler involving boats and divine consequences)

[djlewis]

I also agree with Jason that sometimes the shortest trip (in time) between two points isn't a straight line. In that case, the "old fashioned" way might be better. You might like having a GPS around to help you zero in on the endpoint during the last leg.

Hope this summary helps. I really learned a lot from this discussion.

Lisa

Me too.

A mapping GPS is also particularly useful in a marsh or other complex place where there are lots of twisty little passages, all alike (an energy bar to the first to identify the reference, without googling, binging or the like).

Also, I think there's a theorem in there somewhere that amounts to -- a GPS is useful (as outlined so nicely by Lisa) only when there is no significant reversal of current over the course. I think that's the same as saying that the GPS is useful on the last leg, that is, after paddling past all the significant changes in current direction.

[JohnHuth]

Hi, all -

Yes, to repeat, Leon, Lisa and I have been working on this offline, quite amicably. I still have some tests I want to play with on my GPS when I get back from Switzerland, but right now I'm up to my neck in particle physics stuff. I'm envious of you folks who can get out on the water!

Best,

John

[Gcosloy]

Now if we could only get the bunch of you to collaborate on fixing the economy? Or climate change, or health care. I'm convinced now more than ever that science more than any other subject or endevour trains the mind how to think! Thanks for all the fun and fireworks guys and gal.

[leong]

Now if we could only get the bunch of you to collaborate on fixing the economy? Or climate change, or health care. I'm convinced now more than ever that science more than any other subject or endevour trains the mind how to think! Thanks for all the fun and fireworks guys and gal.

I hear you buddy. But unfortunately, unlike most science and mathematics, those other topics are governed by politics more than by logic.

[leong]

Now if we could only get the bunch of you to collaborate on fixing the economy? Or climate change, or health care. I'm convinced now more than ever that science more than any other subject or endevour trains the mind how to think! Thanks for all the fun and fireworks guys and gal.

Speaking of science and mathematics, Gene, earlier in this thread I demonstrated how the natural path that the a GPS leads you along is just the straight rhumb line to the target. If it can also lead you along a pursuit curve, then it should be possible to demonstrate that with a similar argument. I can’t, perhaps someone else can. But if no one can do it then the statement “The GPS leads you along a “pursuit curve” is probably false. Sort of a “reductio ad impossibilem” proof that the statement is false. Any takers?

For your convenience here’s a repeat of my earlier demonstration that it’s a straight line:

“"The GPS knows the kayak’s location and the waypoint’s location in some fixed earth coordinate system. It doesn’t know it’s own orientation (what direction it’s facing or even if it’s pointing backwards or straight down). It also doesn’t know the heading angle of the kayak. But it does know the Course-Over-Ground (COG) that the kayak is moving along as you paddle. If the COG coincides with the rhumb line from your present location to the waypoint, the GPS’s arrow (at least it’s an arrow with my GPS) points to the top of the screen. If the kayak’s COG for this latest update is to the left of the present rhumb line, then the GPS’s arrow points to the right (indicating that you should change your heading to the right), and vice versa. Of course, your COG is the result of the vector sum of the kayak’s forward speed and the cross-velocity vector due to current, wind and wave action. So, the GPS doesn’t really have any idea of current velocity (I mean water speed and direction) or wind or anything else that’s pushing the kayak. All it really knows is the COG and the latest rhumb line to the waypoint. If you try to keep the arrow always pointing to the top of the screen (via heading changes as necessary) you will be following a straight line to the waypoint.” Note: Assume that the GPS is not using an internal compass.

Respectfully,

CAM-able Leon

[JohnHuth]

I had a question along these lines - although people probably never try this, if you ask for a long distance path with a GPS, does it return a rhumb line or a great circle route? It's probably not a practical issue, unless you're crossing the Atlantic, but I wonder what the default is.

I just looked up my own question - evidently it's a great circle route, but I wonder if you can switch over to a rhumb line, if you want, as an option.

[leong]

I had a question along these lines - although people probably never try this, if you ask for a long distance path with a GPS, does it return a rhumb line or a great circle route? It's probably not a practical issue, unless you're crossing the Atlantic, but I wonder what the default is.

I just looked up my own question - evidently it's a great circle route, but I wonder if you can switch over to a rhumb line, if you want, as an option.

John,

I’m not sure what you mean by a rhumb line in the context of the surface of the earth. For instance, to take an extreme example, suppose one point is the North Pole and the other point is the South Pole. Then by a rhumb line, do you mean any of the infinite number of meridians going through these two points or do you mean the single straight line going through the two poles as well as the center of the earth? I think you mean the former. Given that, I believe the answer for the GPS is it gives you the great circle that is equivalent to the rhumb line. Or do you have a different definition for a rhumb line?

Leon

[JohnHuth]

A rhumb line, as I understand it, is a path or heading that maintains a constant azimuth of travel throughout. When you begin to travel long distances on the globe, this isn't the shortest path, but a great circle route is.

A Mercator projection will preserve rhumb lines as straight lines.

The rhumb line is an example of a kind of a curve called a loxodrome. A rhumb line has the curious feature that if you continue to extend it, it spirals into either the North or South Pole.

Now there are some degenerate cases where a great circle route will coincide with a rhumb line. If you travel from the South to the North Pole, one of the meridians will be a great circle route, and in this case it should also be a rhumb line, as your heading will maintain a due north.

In general, if you travel to a point directly opposite your point on the Earth, I believe that there are an infinite number of great circle routes, but if you are traveling between any two random points, a rhumb line does not generally coincide with a great circle route.

As a curious side note, when Muslims pray to Mecca, they face in the direction of a great circle route, although there was a time when there was some confusion as to whether the direction should be a rhumb line or great circle route. In this case Mecca acts like a kind of religious 'north pole' and the prayer lines are like lines of meridians. From Boston, a rhumb line to Mecca would be to the SE, but a great circle route would be to the NE.

Figure 209.pdf

[leong]

A rhumb line, as I understand it, is a path or heading that maintains a constant azimuth of travel throughout. When you begin to travel long distances on the globe, this isn't the shortest path, but a great circle route is.

A Mercator projection will preserve rhumb lines as straight lines.

The rhumb line is an example of a kind of a curve called a loxodrome. A rhumb line has the curious feature that if you continue to extend it, it spirals into either the North or South Pole.

Now there are some degenerate cases where a great circle route will coincide with a rhumb line. If you travel from the South to the North Pole, one of the meridians will be a great circle route, and in this case it should also be a rhumb line, as your heading will maintain a due north.

In general, if you travel to a point directly opposite your point on the Earth, I believe that there are an infinite number of great circle routes, but if you are traveling between any two random points, a rhumb line does not generally coincide with a great circle route.

As a curious side note, when Muslims pray to Mecca, they face in the direction of a great circle route, although there was a time when there was some confusion as to whether the direction should be a rhumb line or great circle route. In this case Mecca acts like a kind of religious 'north pole' and the prayer lines are like lines of meridians. From Boston, a rhumb line to Mecca would be to the SE, but a great circle route would be to the NE.

Okay John, fair enough. I guess you just taught me the correct definition of a rhumb line on the earth (and we both realize that it’s almost equivalent to a great circle for short distances). I know all about loxodromes (was a sailor before I became a kayaker). So my answer is that the GPS directs you to follow a great circle route, not a rhumb line.

Here’s what I think about loxodromes: There’s nothing natural about a loxodrome. A loxodrome comes to exist because of the arbitrary (but very useful) lat/long coordinate system used to pin point a location and direction on the earth. One could have defined a coordinate system where the North Pole was moved to the Greenwich point and the South Pole to its antipode and defined the circles of latitude appropriately for this different coordinate system. Then telling someone to sail west forever would create a different loxodrome.

Hmm, I’m not good at explaining what’s in my head. Let me put it another way. Suppose the earth was one big parking lot and you were told to start driving straight in some direction (say southwest), but never turning the steering wheel. Then you’d be driving along a unique great circle around the earth, right? If you didn’t have a compass you wouldn’t even know how to follow the southwest loxodrome, right? It’s the artificial construct of the lat/long coordinate system that’s the problem. But that problem will (I think) always exist whenever you define some coordinate system for a sphere and define a compass direction (okay, it’s not exactly a sphere, but you know what I mean).

Make any sense?

Leon

[JohnHuth]

The rhumb line was the product of the invention of the compass. If you look at charts that arose at the time of the use of the compass in western Europe, you'd see them criss-crossed with rhumb lines. The use of latitude and longitude grew starting around 1480 and began to be used more extensively for navigation as celestial navigation took over. With Mercator's projection, you could then connect dead reckoning and rhumb lines with latitude and longitude measurements from celestial observations (well, latitude at first, and then longitude with clocks or lunars).

So, yes, it's somewhat historical, but it makes sense in the context of a compass as being the main technology of way finding from about 1300 onward.

Rhumb lines are never extended in practice - I just mentioned the loxodrome as a curiosity.

[leong]

The rhumb line was the product of the invention of the compass. If you look at charts that arose at the time of the use of the compass in western Europe, you'd see them criss-crossed with rhumb lines. The use of latitude and longitude grew starting around 1480 and began to be used more extensively for navigation as celestial navigation took over. With Mercator's projection, you could then connect dead reckoning and rhumb lines with latitude and longitude measurements from celestial observations (well, latitude at first, and then longitude with clocks or lunars).

So, yes, it's somewhat historical, but it makes sense in the context of a compass as being the main technology of way finding from about 1300 onward.

Rhumb lines are never extended in practice - I just mentioned the loxodrome as a curiosity.

Understood. Wow, John, you know a lot more than elementary particles!

I’m not really familiar with long-distance navigation with a compass, but I can guess. So please let me know if the following example is at least plausible:

Suppose a sailor has good charts of the world, a way to accurately determine his position (lat/long) and an accurate compass. He decides to sail to a far-away waypoint. From the chart he sees that a great circle route to the waypoint crosses the local meridian at 45 degrees (he doesn’t figure out the rhumb line direction). So he starts sailing NE. Since the waypoint is very far away, holding a NE course won’t ever get him to the waypoint because: 1) He’ll be sailing along the wrong loxodrome and 2) He’ll be slowly moving off of the great circle from his original position to the waypoint. In fact, if he keeps going NE he’ll spiral towards the North Pole. But, instead, every once in a while he could look at his current position on the chart to figure out a new compass heading for the great circle route to the waypoint. If he does this often enough, I think his track should approximate the great circle to the waypoint, right? And his total distance traveled will be less than it would have been if had he chosen to sail along the proper loxodrome.

--- Leon

[JohnHuth]

Yes, actually in practice a great circle route is often accomplished by periodic shifts in compass heading, so you sail a series for rhumb lines, which approximate a great circle route, but it's close enough.

[leong]

A Mercator projection will preserve rhumb lines as straight lines.

I couldn't remember the name “gnomonic” when I entered my last post, but that’s the map projection that displays all great circles as straight lines (I just looked it up). It says that it’s actually the oldest of all map projections (Thales, 6’th century BC). Obviously, there are distortions with this projection as there are for every projection from a sphere to a plane.

BTW, in an earlier life, I studied vectors on curved spaces (parallel transporting of a vector along a geodesic curve and all of that stuff). But don’t get me started; besides, I’m sure you could cream me on any of that given your background.

Anyway, back to the GPS topic. I’ve always wondered why people think it’s odd that great circle arcs are the shortest distance between two points. I guess it’s because great circles look like curved paths on most common map projections. Yet, anyone walking from A to B on the surface of the earth will naturally walk along the arc of a great circle, not a loxodrome or any other curve. So, to me, it’s natural that a GPS will lead you along a great circle and not a loxodrome.

[lhunt]

Found these links that may help, though I'm definitely unqualified for this conversation otherwise. There were other references but these seemed to be the most relevant. One of the people in the first discussion apparently asked Garmin, and got the answer "Great Circle" (different model, no doubt, though, from what we use).

http://www.ybw.com/forums/showthread.php?t=248240&highlight=great+circle+gps

http://www.ybw.com/forums/showthread.php?t=266942

[JohnHuth]

I think the perception of great circle routes as longer than rhumb-line courses has to do with the projection onto Mercators charts, which is what are most prevalent. If you look at world-wide ocean shipping routes (google it up), you can find how coastlines, great circle routes and major canals (Panama and Suez) and straits (Malacca) tend to shape the course of shipping routes. Time is money!

[leong]

Re: Higgs boson.

Congratulations John!