JohnHuth

Interesting photo, but this one is from Cape Cod http://www.latimes.com/news/nationworld/nation/la-na-cape-cod-shark-20120710,0,7741361.story that time of year. Remind me to not purchase a kayak colored like a seal.

Thanks! If you want to read my attempt at humor on the Higgs discovery, if you dare: http://www.quantumdiaries.org/2012/07/05/the-celebrated-god-particle-by-mark-twain/ Sorry, off topic. I can't figure out a kayak angle on this one.

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!

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.

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.

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

That's a pretty interesting article. Not that this is completely aligned, but there was a period called the "Medieval Warming Period," where ocean temperatures in the North Atlantic were supposedly about 1 degree C higher than they are today. This is correlated with the Viking exploration of North America. It's a bit of a controversial subject, however, as people try to use this to refute climate change. Still, it makes me curious about the coastlines about 1000 AD.

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.

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

Leon - Here's where I think the nut of the problem is:. My old Garmin Etrex Vista *only* pointed to a waypoint with no calculation whatsoever that factors in additional motion. So, the idea that my GPS device *only* points to a waypoint is what obscured the conversation. Now, I'm perfectly willing to be taught that a new GPS device factors in additional things that can be calculated, like current. But, the way it was phrased was to say "you just don't understand how GPS works", as opposed to "you may not realize that current GPS devices have the following features..." At least the way I was approaching the problem was based on the experience with my old device, which was probably less sophisticated, being older. I'm assuming that everyone, however, agrees with the assertion that the 'continually heading along the bearing that points directly across' is longer than factoring the ferry angle. Right? I certainly apologize for not appreciating the capabilities of your GPS device. I for one will not be contributing to NSPN website for some time to come, however, as the tone of the dialog on here is just too difficult for me. I apologize and sign off. No bets, not boasts about paddling skills, no nothing. I'm not into winning at the cost of offending and I apologize if I crossed that line. Good bye John

Just to help the discussion, let me suggest something along these lines 1.) Do we now agree on the construction that the pursuit curve, which is derived from a continual set of headings equal to the bearing to an object on the opposite shore in the presence of current is longer than a heading that takes into account current? No words about GPS devices. If so....then item 2.) We can compare the features on different GPS devices. I'm sure the newer ones have many nifty features. In my old Garmin etrex vista, it could only point toward a waypoint and couldn't compensate for current. It could also display a compass simultaneously. I'm sure newer GPS units can do many things. item 1.) is a pretty clear statement of math. item 2.) has more to do with features on a device, and they will certainly vary with time. So, it would be OK to say "my GPS device has these features..." but it does not follow that all GPS devices have (or had) identical features. People relate their experience, so it would be reasonable to say "you may not be familiar with GPS device X which has the following features.... "

David - Exactly! The point is not whether there are more sophisticated functions in a current day GPS. It is a "manner of saying" that if you paddle in the direction of the bearing to the object on the far side, you'll end up with a pursuit curve, which is longer than the ferry. Again, it's simply a statement of math. It would be the first case in David's manual, if you want to reference a GPS-like analogy. As a matter of usage one should really say "GPS device" rather than GPS as GPS is the system, not the device. My old Garmin etrex vista only could display a 'bearing' as the direction of travel. I'd happily admit that there are other features on newer GPS devices. But, again, if you maintain a heading equal to the bearing to an object on a crossing in the presence of current you will get a pursuit curve, and if you compensate for the current you'll get the straight line. The pursuit curve is longer.

Here's a figure, attached that illustrates what I'm saying and what I believe David's saying. The "GPS" is just a way of saying that you are always following a heading toward an waypoint on the far side of a crossing. It uses the waypoint feature - you punch in the lat/long of a location you want to head to, or choose it from a chart. The GPS then computes the heading to that point, assuming a fixed reference frame (ie. no current). But, if you are a die hard fan of GPS, just leave that concept aside and look at the concept of having a heading that always points to the waypoint on the opposite side. It's just a mental construct to aid visualization. On the top illustration, I've broken the crossing into 5 discrete steps, where the arrow points to the landmark on the opposite side of the crossing at each stage. You physically change direction at each of those steps in order to compensate for the fact that the current has swept you downstream. The path will include paddling that will have both downstream and upstream components. In the lower case, one takes a ferry angle, which implies that there is no component of the path that is either upstream or downstream. For a fixed paddling speed, the ferry angle crossing will be faster. This has nothing do with paddler A is an adonis and paddler B is a wimp, it's just a mathematical statement. Now, to go to the other question of using the GPS as an aid to determine current and even ferry angle, I'll try to come up with some illustrations for that. Waypointing.pdf

No need for a bet, a straight line path is faster than the pursuit curve as stated, (using the GPS waypoint following as an example). It just is. No arguing or equivocation. It's like making a bet whether or not the tangent of 45 degrees is 1, you don't need to touch water to settle that (I would've thought).

Let me try again. OK, a GPS has many features. You will get your present position, either displayed on a map, or as a latitude or longitude. I will also give you your 'true' heading - that is to say, the direction in which you're moving and also your speed, as measured against a "fixed" Earth. It will also point you in the direction of a pre-determined waypoint. It will also give you a heading to that waypoint, assuming that fixed reference frame. It will also give you a compass showing true north and your 'true' heading with respect to true north. It will also keep a record of your travels and you can upload them to a computer etc. Now, what a GPS typically doesn't have is the ability to download apps like you might with an iPhone. I was only suggesting that the ability to download certain aids to navigation might be helpful. Yes, I've used a GPS - it was a Garmin and I've burned through 3 of these as the salt water tends to chew up the receivers rapidly and I've just given up on them. They also have a tendency to fail in heavy rain and fog - just an observation. Now, using a GPS to assess current set and drift is doable. You paddle out into your stream, and just set there. Leave the GPS on, and it will give you, effectively the current set and drift, as this is just the direction you're moving and speed. OR....you can use a heading that's based on your GPS' compass referenced to true north as you cross, but then you'll find your 'true' heading (i.e. wrt a fixed reference) deviates from that due to the effect of current. If you adjust your heading (direction in which you're paddling) so that the 'true' heading is pointing at the waypoint you've programmed in, you'll have just found a ferry angle using the GPS. Again, to get back to the pursuit curve point of the thread.... the issue as I see it is that if you program in a waypoint that's at the other end of the crossing, the GPS will keep you pointed in that direction. If you head in that direction as the GPS "thinks" it should if you were not in a current stream, then you will find yourself continually adjusting the heading from the GPS, as you will get swept downstream and the GPS will want to compensate for that. All David is saying is that if you, instead, calculate your compass heading, assuming that there is a uniform current, you'll take a straight line path, rather than the pursuit curve that results from following the multiple headings the GPS will give you in trying to direct you toward the waypoint on the far side. This is faster. What I think Leong is saying is that a) there are other factors, like non-uniform current and waves and wind (true) and there are ways of using the GPS to determine current and even take it into account (true), although those are not "smarts" built into the GPS - it is using the GPS as an instrument to determine current and even help you figure out an optimal ferry angle. However, the issue is not the inherent precision of the GPS, it is simply the difference of a pursuit curve, using the GPS path as an EXAMPLE of what happens if you continually try to point toward a landmark on the far side of the crossing. There's a difference between what we (or I) am saying about a highly reductionistic statement of a mathematical problem using the GPS feature of pointing toward a waypoint as a way of putting this idea across as a concept and the real world, where the GPS can help you determine current and adjust for it (again using some native smarts). Whew.... clearer?

I have to say, this is quite an entertaining thread. I find that in message boards things frequently devolve this way. I realized how you can use a GPS to advantage to find a heading in the presence of currents (partly through that URL Leong posted). This, in part uses the GPS as a compass, but also as an aid to measuring currents. If you are able to track your path toward a destination, you can try an initial heading directly toward the destination. The current will make that heading shift, and, in fact, if you have a display of your actual track, you can estimate the angle off, and then adjust the heading until you're heading in a straight line toward the destination. OR, you can go into the current, let yourself drift and measure that drift and adjust accordingly - then switch on the compass feature and it's pretty much the same as an ordinary compass, but you've also measured the local current. So "GPS as a current measuring device" seems possible in principle. I don't use one, so I don't know if it's practical, but it strikes me as possible. As far as the $100 bet is concerned, I think it misses the point David was trying to make. Let's go back to the more garden-variety use of a GPS, where it is more of a hiker's device and it gives you a 'straight line' to your destination, and is not employed in a slightly more sophisticated way to measure current, either through drift measurement or adjustment of heading. In that 'garden variety' case, you will be changing your heading constantly and taking a slower path than if you had figured current initially and figured a heading based on a compensation for that. A 'bet' along these lines would be something more like a crossing to an island in the presence of a current where person A uses a compass and figures in current and person B rigidly adheres to a projected GPS line that tries to always move toward a target, assuming no current. Then, the bet is whether person A or person B reaches the target island first. The target island should be big enough that it presents a minimum 10 degree spread from the starting point. This takes out the meter-ish precision of the GPS as a factor in actually locating a very small target. The lobster-buoy finding exercise only uses the enhanced position accuracy of a GPS and isn't really pertinent to the question of the original 'pursuit curve' idea of the thread, namely, is the straight line path that factors in current faster than one where the heading is constantly pointing toward the target. Having said that, I embarked on a rather fun exercise with Mark Schoon. We were crossing from Swan Island to Placentia Island in a thick fog with the current draining out of Blue Hill Bay. The 'game' was to compensate for the current as precisely as possible an hit a buoy halfway across (I think the full crossing was two miles and the buoy was halfway). It was all map-and-compass. We figured a 10 degree ferry angle. After paddling maybe 15-20 minutes, damn if we didn't hit right on that buoy! I was a pretty happy camper that we were able to do that. I wouldn't have expected that precision, but the angular precision was pretty darn high. It might have been luck that we did that well, but it made me convinced you can do quite well with a standard map-and-compass and seat-of-the-pants estimate of current.

Yes, that will do it. There are many ways to skin a cat, no doubt. Yes, you'll need to know the kayak's speed, clearly. As I said, I personally don't use GPS'es, but to the extent that it acts like a compass *and* gives you progress to a waypoint that strategy in the link is certainly viable. This makes me think that perhaps GPS'es should have a little mini-calculation thrown in to give a ferry angle on-the-go, with some inputs from the user. It seems like a useful feature. I would imagine that some enterprising GPS programmer must have done this by now. If not, a great opportunity! I should note that Pacific Islanders had some ingenious ways of dealing with current. In some cases, they would paddle or sail away from an island some distance, then stop and monitor their drift to get some idea of the current and adjust the heading appropriately. In other cases, they would line up two landmarks on a near shore, and then adjust the heading so that they were moving in a direction that kept these in line. The anthropologist Raymond Firth noted that some islanders in the Santa Cruz Reef Islands had specific names of landmarks on islands that were used to find the proper heading in the presence of currents. With a technique like this, no calculations are necessary.

I suspect, perhaps, we might actually all agree....although perhaps not? Let's see - 1.) A GPS gives you a highly accurate position fix 2.) A path given by your typical hand-held GPS from one point to another doesn't factor in conditions like current, etc. 3.) A straight line ferry angle in the presence only of a uniform current, factoring in the current is probably the most efficient 4.) You can calculate this ferry angle even with a piece of paper and a pencil 5.) Non-uniform currents, winds etc will create deviations from the idealized ferry angle that assumes a uniform current Places where perhaps we don't agree: 1.) In the 'real world' are GPS'es just as good or better than using a compass and calculating a ferry angle assuming a uniform current? I'd maintain that there's something of an 'art' that's lost in all this. I might know that there's a 2 knot current in the middle of a crossing and start out figuring out a ferry angle, but then realize that the current near shore is less than that, and back off the ferry angle by a bit. I'd then look at the wind conditions, and use some figure of merit about the sideways slip of the kayak due to windage. Some of the stiffest tests of this is are crossings in the fog, where you lose visuals of the far shore, and you only learn how well you did once you reach the shore - no room for fudging. In some cases, the more foolhardy of us will make a crossing to a relatively small off-shore island in the fog in downeast Maine (mea culpa) where the tidal currents are large. In cases like that, I often don't have a table of currents, but just know the phase of the tide and some notion of what the max current is based on the topography and bathymetry. I'll still do a correction rather than follow a GPS heading, and in my experience you can do rather well, but a person has to throw in all the knowledge they have about the conditions. The straight-line ferry correction is just one factor that informs your decision about heading.

Actually, I don't use GPS'es for cars or anything. I don't know if that's a bad idea, but it makes me feel disconnected from my surroundings.

Just to amplify Suz and Jason's comments: if you take a direct GPS line in the presence of current, the GPS is likely to continually tell you to alter your heading, as you will not be moving as the GPS "thinks" you should. This in itself probably generates an interesting 'pursuit curve', and I'd wager takes a lot more energy than a course that uses knowledge of current. That Excel spreadsheet can be downloaded to an iPhone or equivalent, and takes a lot of the pain out of adjustments for current, so it's not so much of a headache. I do know some airplane pilots who have apps like this to calculate headings in the presence of wind, so it's not too difficult to take the pain out of it.

Well....not to be too science geeky, but I think you need to state the "problem" concisely to get an answer. If the goal is to minimize energy expenditure on a crossing, you have to consider the possibility that you go under and exceed your 'steady state' power output for portions of the crossing. If this is the case, I suspect that the problem is unconstrained to the point where there might be multiple allowed solutions. If you limit yourself to a steady power output, I suspect the answer is one that has a constant heading (i.e. adjusting for current with a fixed ferry angle). The other problem that I find interesting is the curve that's generated if you always point toward an object that you initially identify as being directly across from you when you start the crossing. That's kind of a cool curve, and I'm curious about conditions where there is no valid solution. Presumably this is when either a) the current is too strong or the crossing is too wide. Having said all this, I think the hardest part of a *real* crossing (e.g. in a fog where you don't have visual references) is trying to estimate the effect of the change of current as you leave from a shore, get into the middle and then approach the far shore. Even then the bathymetry can play a role. It might be fun to try to put in a quadratic current variation in a crossing and see what path that implies to minimize energy expenditure with a constant power output - I reckon that just means multiple ferry angles to maintain a 'straight' crossing.

Jason - Yup, that was the case. OK, here's the Excel file attached. It should be self explanatory. You put in your speed (vessel's), you put in the speed of the current (kts) and the direction of the current (degrees wrt true north), and then your desired heading. The result is the heading you have to take to achieve the desired heading. It also gives you your speed with respect to a static (i.e. global) frame of reference. This is not a pursuit curve, just the vector work on velocities to get a straight line heading. If there is no solution, the spreadsheet tells you that. I'm working on a piece of mathematica code to do the pursuit problem where you keep your kayak pointing at a fixed landmark, which is different from the simple vector math of keeping a desired heading in the presence of current. Corrections to heading 3.xlsx

I have a spreadsheet to calculated a heading needed for a desired heading in the presence of current but it seems I cannot attach Excel spreadsheets. I'll see if I can forge a link in google docs.

Yes, pursuit curves are a general class in mathematics. There are many scenarios one can paint, and the differential equations can get hairy very quickly.

http://curvebank.calstatela.edu/pursuit/pursuit.htm First considered by Pierre Bouguer, the frenchman who also first described the 'metacenter' in terms of the stability of a vessel. It's an interesting general problem.