What curve does a kayak make...

[djlewis]

What curve does a kayak make when crossing in a crosswind or crosscurrent, keeping the bow aimed at the target feature?

It's common knowledge that the kayak does not go in a straight line, but as it drifts downwind, it has to aim farther and farther upwind to keep the target off the bow. So the course ends up kind of "hooking" back toward the target near the end. The solution usually recommended is to start with a ferry angle, calculated from the drift rate and distance to travel, and then keep reducing the angle as you approach the target. Done properly, that will give you a straight line course. Whether that course is actually the most efficient one is a different question to be dealt with later.

Anyway, it's often asked what is the shape of the hooking-back track if you do it "wrong", that is, don't use a ferry angle. The answer is, it's a pursuit curve, which is a family of mathematical curves that arise when a predator chases a moving prey. Obviously the kayak in this case is not chasing a moving prey. But the effect is the same, since the crosscurrent is constantly changing the frame of reference for the fixed target, so the target is, in effect, moving with respect to the frame. Here is a mathematical analysis of this exact curve for the kayaking problem. This has not, to my knowledge, been published anywhere to date, but I haven't done a thorough search.

Any questions?!

Yes -- what about the relative efficiency of the pursuit curve and the straight line? I've seen it suggested that the pursuit curve may actually be more efficient even though it is longer, because you are going with the drift as much as possible; when you use a ferry angle, you are constantly working against the current. On general thermodynamic grounds (a steady, smooth expenditure of energy is generally more efficient than a varying one), I suspect the ferry-angle-straight-line course is still the best. But I will ask the mathematicians after the buzz from this question has died down.

--David (substituting thinking about paddling for actual paddling on this picture-perfect day)

[leong]

What curve does a kayak make when crossing in a crosswind or crosscurrent, keeping the bow aimed at the target feature?

It's common knowledge that the kayak does not go in a straight line, but as it drifts downwind, it has to aim farther and farther upwind to keep the target off the bow. So the course ends up kind of "hooking" back toward the target near the end. The solution usually recommended is to start with a ferry angle, calculated from the drift rate and distance to travel, and then keep reducing the angle as you approach the target. Done properly, that will give you a straight line course. Whether that course is actually the most efficient one is a different question to be dealt with later.

Anyway, it's often asked what is the shape of the hooking-back track if you do it "wrong", that is, don't use a ferry angle. The answer is, it's a pursuit curve, which is a family of mathematical curves that arise when a predator chases a moving prey. Obviously the kayak in this case is not chasing a moving prey. But the effect is the same, since the crosscurrent is constantly changing the frame of reference for the fixed target, so the target is, in effect, moving with respect to the frame. Here is a mathematical analysis of this exact curve for the kayaking problem. This has not, to my knowledge, been published anywhere to date, but I haven't done a thorough search.

Any questions?!

Yes -- what about the relative efficiency of the pursuit curve and the straight line? I've seen it suggested that the pursuit curve may actually be more efficient even though it is longer, because you are going with the drift as much as possible; when you use a ferry angle, you are constantly working against the current. On general thermodynamic grounds (a steady, smooth expenditure of energy is generally more efficient than a varying one), I suspect the ferry-angle-straight-line course is still the best. But I will ask the mathematicians after the buzz from this question has died down.

--David (substituting thinking about paddling for actual paddling on this picture-perfect day)

I don’t consider the “Pursuit Curve” to be a simple named curve. It’s just a name given to the path taken by a fox as he chases a rabbit. The shape of a pursuit curves depends on the path the rabbit will take as she’s chased by the fox. If the rabbit stands still then the pursuit curves becomes a simple straight line.

If you want a name for your curve I’ll give you one. Let’s call it the “Kayak Waypoint Curve”. I’m not sure, but I don’t believe it falls into the family of pursuit curves. Of course, in your case the waypoint is standing still. But the Kayak Waypoint Curve is different than the fox chasing a dead rabbit. Which one of the family of pursuit curves do you think is equivalent to the Kayak Waypoint Curve?

[djlewis]

I don’t consider the “Pursuit Curve” to be a simple named curve. It’s just a name given to the path taken by a fox as he chases a rabbit. The shape of a pursuit curves depends on the path the rabbit will take as she’s chased by the fox. If the rabbit stands still then the pursuit curves becomes a simple straight line.

If you want a name for your curve I’ll give you one. Let’s call it the “Kayak Waypoint Curve”. I’m not sure, but I don’t believe it falls into the family of pursuit curves. Of course, in your case the waypoint is standing still. But the Kayak Waypoint Curve is different than the fox chasing a dead rabbit. Which one of the family of pursuit curves do you think is equivalent to the Kayak Waypoint Curve?

Did you follow the "Here" link to the mathematical account? It's an area I'm not particularly experienced in, so I'd appreciate your opinion about the math.

These mathematicians do claim it's a classical pursuit curve, and while that's not as celebrated a curve as: Parabola, Circle, Witch of Agnesi (sic!), Catenary(alysoid, funicular), Brachistochrone, Cycloid, Epicycloid, Rose (rhodonea) and the like, I think it's a legit family of curves and a meaningful, widely-recognized name.

--David (wondering why, if he has time to look all this up, he's not out paddling)

[leong]

Did you follow the "Here" link to the mathematical account? It's an area I'm not particularly experienced in, so I'd appreciate your opinion about the math.

These mathematicians do claim it's a classical pursuit curve, and while that's not as celebrated a curve as: Parabola, Circle, Witch of Agnesi (sic!), Catenary(alysoid, funicular), Brachistochrone, Cycloid, Epicycloid, Rose (rhodonea) and the like, I think it's a legit family of curves and a meaningful, widely-recognized name.

--David (wondering why, if he has time to look all this up, he's not out paddling)

David, you and I are probably the only NSPN-ers with both the mathematical background and nutty desire to even think about all this.

Anyway, before I looked at the solution you referenced, I took a quick dive into the mathematics and I couldn’t solve the resulting differential equation that I came up with. The DE that I got looked a little different than the referenced solution but it might be equivalent. I was going to solve it numerically (with the help of Mathcad) but then realized there was an easier approach for dummies like me.

If you wanted an approximate plot of the kayak’s position using a paper and pencil, you could move the kayak towards the target one unit and then slip it side ways rho units, where rho is the ratio of the crossdrift speed (due to current and/or wind) to the paddler’s speed. So I did it this way, but with an iteration performed in Mathcad. I did it fast so there may be some trivially mistakes, but I think that the results look like they should look. See attached pdf of the Mathcad sheet.

With regard to naming the curves, I’m not sure that “pursuit curve” is really right, although the math is similar. If it is a pursuit curve, then I would think that there must be some simple rabbit motion that results in a pursuit curve that is equivalent to the kayak waypoint curve. Perhaps you ought to pose this to the authors. I think that they’re much more knowledgeable in DE’s than I am.

Anyway, this problem is interesting but I’m now spending this kind of my intellectual energy on regression models for implied volatility of stock options, which is more useful for me in a $ sense.

With regard to the relative efficiency of the kayak waypoint curve (I don’t want to call it a pursuit curve until I know that it really is a pursuit curve) I’m almost certain that it takes much more energy than the straight-line path resulting from the correct ferry angle. But, I could be wrong. Perhaps when I have more time I’ll compute the total work done for each strategy (ferry angle vs. always point to the target).

Gotta go now.

Leon

Mathcad - Bow aimed at target with crossdrift.pdf

[JohnHuth]

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.

[djlewis]

David, you and I are probably the only NSPN-ers with both the mathematical background and nutty desire to even think about all this.

Well, there is John Huth -- a top-drawer physicist and NSPN denizen.

I have the desire to think about this, but not the specific math background -- never did much analysis before heading off into theoretical CS. So I'm at the mercy of you and the guys on math.stackexchange that did this work. Your plot looks reasonable.

With regard to naming the curves, I’m not sure that “pursuit curve” is really right, although the math is similar. If it is a pursuit curve, then I would think that there must be some simple rabbit motion that results in a pursuit curve that is equivalent to the kayak waypoint curve. Perhaps you ought to pose this to the authors. I think that they’re much more knowledgeable in DE’s than I am.

I think "pursuit curve" is being used here as a general term for a certain kind of curve defined mathematically rather than the result of a specific pursuit scenario. But still, doesn't thinking of the side current as a moving frame of reference make the fixed target waypoint into a moving one, in effect a moving prey? That's what the guys who solved it on math.stackexchgnage seem to be saying. But as I said, I have to take everyone's word for it.

As for energy, the guy did point out that his model assumed constant kayak speed over water, not over ground. I do know enough to know that means a shorter path is perforce a more efficient one, energy-wise.

[djlewis]

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.

Interesting -- thanks, John. It does seem to say that a "pursuit curve" is a rather general family. I wonder if the one in question for kayakers -- with a fixed prey and moving frame of reference -- has been analyzed and published before?

[leong]

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.

Thanks, John and David.

From the reference John linked, a pursuit curve is “One particle travels along a specified curve, while a second pursues it, with a motion always directed toward the first. The velocities of the two particles are always in the same ratio.”

Yes, that’s clear. But let’s define a kayak waypoint curve as the track over ground of a kayak paddled at a constant speed always heading towards some fixed waypoint and being dragged “sideways” by a constant current and/or wind. By sideways I mean perpendicular to the line from the kayak’s starting position to the waypoint. So my question is this:

If a kayak waypoint curve is just a pursuit curve, then what is the specified curve of the rabbit (the first particle) that results in a pursuit curve that is equivalent to a particular kayak waypoint curve? The waypoint doesn’t move; but, obviously, a stationary rabbit is not the answer since it results in a pursuit curve that is a trivial straight line.

Later, David said,

“But still, doesn't thinking of the side current as a moving frame of reference make the fixed target waypoint into a moving one, in effect a moving prey?”

Aha, that that might be the key. Perhaps that implies that the rabbit moves along a straight line, perpendicular to the original line from the kayak to waypoint at the side-drift rate (or something like that).

Leon

[JohnHuth]

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.

[JohnHuth]

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.

[jason]

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.

.xls were allowed as attachments. My guess is you were attempting to upload a .xlsx, I have updated the forum to allow an upload of .xlsx files too.

-Jason

[JohnHuth]

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

[leong]

What curve does a kayak make when crossing in a crosswind or crosscurrent, keeping the bow aimed at the target feature?

It's common knowledge that the kayak does not go in a straight line, but as it drifts downwind, it has to aim farther and farther upwind to keep the target off the bow. So the course ends up kind of "hooking" back toward the target near the end. The solution usually recommended is to start with a ferry angle, calculated from the drift rate and distance to travel, and then keep reducing the angle as you approach the target. Done properly, that will give you a straight line course. Whether that course is actually the most efficient one is a different question to be dealt with later.

Anyway, it's often asked what is the shape of the hooking-back track if you do it "wrong", that is, don't use a ferry angle. The answer is, it's a pursuit curve, which is a family of mathematical curves that arise when a predator chases a moving prey. Obviously the kayak in this case is not chasing a moving prey. But the effect is the same, since the crosscurrent is constantly changing the frame of reference for the fixed target, so the target is, in effect, moving with respect to the frame. Here is a mathematical analysis of this exact curve for the kayaking problem. This has not, to my knowledge, been published anywhere to date, but I haven't done a thorough search.

Any questions?!

Yes -- what about the relative efficiency of the pursuit curve and the straight line? I've seen it suggested that the pursuit curve may actually be more efficient even though it is longer, because you are going with the drift as much as possible; when you use a ferry angle, you are constantly working against the current. On general thermodynamic grounds (a steady, smooth expenditure of energy is generally more efficient than a varying one), I suspect the ferry-angle-straight-line course is still the best. But I will ask the mathematicians after the buzz from this question has died down.

--David (substituting thinking about paddling for actual paddling on this picture-perfect day)

David, no matter what path you paddle, there’s a smooth expenditure of energy by definition (at least if there’s no wind) because you paddle at a constant power output? Of course the ferry-angle-straight-line path is the shortest of all possible paths to the waypoint. But the question becomes is it the fastest path too (the one that minimizes the time to get to the waypoint)? Because the power is constant, the fastest path is the one that minimizes the total work done. I haven’t proved it in general, but I’m fairly certain that the ferry-angle-straight-line path not only is the shortest path but it’s the fastest path too. I think this holds true in the presence of wind as well. A nice problem to solve sometime.

Leon

[JohnHuth]

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.

[leong]

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.

That's the beauty of using a GPS. No need to do any trig and calculate continuously changing ferry angles. Just follow the GPS's arrow and it will get you to the waypoint on the shortest straight line.

[leong]

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.

Here's the beauty of using a GPS: No need to do any trig to calculate continuously changing ferry angles (based on drift speed and direction of wind and current, paddling speed and direction, etc.). Just follow the GPS's arrow to the coordinates of the waypoint and it will get you there on the shortest straight line.

[ThomasL]

Leong,

All the math and informed speculation are certainly interesting, but your GPS "solution" makes the best sense to any 21st century layman. Less talk....MORE PADDLING!! LOL

Tom

[Suz]

A straight line on the GPS is not always the best solution. Things to consider that the GPS can't, wind, swell direction, CURRENT.

[lhunt]

Things to consider that the GPS can't, wind, swell direction, CURRENT.

Because the GPS computes its pointer based on course, not heading, it is a very efficient way to get a straight line across all these things if you have a waypoint on your destination. The GPS doesn't know it's correcting for anything other than your direction of travel - if you move too far left it will simply keep telling you to bear right.

Of course, if you have a situation where a straight line is not the fastest route based on current or wind forecast or any other factors (like an island being in the way :-) ), you would have to program multiple waypoints, or follow an existing track. A GPS set to find a destination "off road" isn't smart enough to avoid obstacles or to determine the fastest route given prevailing paddling conditions.

But it's definitely good at straight lines.

-Lisa

[jason]

Because the GPS computes its pointer based on course, not heading, it is a very efficient way to get a straight line across all these things if you have a waypoint on your destination. The GPS doesn't know it's correcting for anything other than your direction of travel - if you move too far left it will simply keep telling you to bear right.

Of course, if you have a situation where a straight line is not the fastest route based on current or wind forecast or any other factors (like an island being in the way :-) ), you would have to program multiple waypoints, or follow an existing track. A GPS set to find a destination "off road" isn't smart enough to avoid obstacles or to determine the fastest route given prevailing paddling conditions.

But it's definitely good at straight lines.

-Lisa

I think that Suz's point is blindly following an arrow is a bad idea. Knowing the currents can make ones life much better., for example in long crossing you can be pushed two different directions and doing the math up front will let you paddle on a much shorter single course and not have to continually correct (incorrectly).

Admiralty charts will include tidal diamonds that will show you what the currents are doing based on the tidal offset. Most of the US charts don't have tidal diamonds but we can research with NOAA before a trip:

-Jason

[leong]

I think that Suz's point is blindly following an arrow is a bad idea. Knowing the currents can make ones life much better., for example in long crossing you can be pushed two different directions and doing the math up front will let you paddle on a much shorter single course and not have to continually correct (incorrectly).

Admiralty charts will include tidal diamonds that will show you what the currents are doing based on the tidal offset. Most of the US charts don't have tidal diamonds but we can research with NOAA before a trip:

-Jason

The beauty of using a GPS is that you get this minimum distance straight line even if the wind and current are changing with time and position. That is, following the GPS, your heading is always consistent with the correct instantaneous ferry angle. With a GPS you don’t need to know the currents and/or wind, and you never can know the values of these parameters no matter what kind of chart that you use. No chart can give you the side-drift rate for a kayak of unknown wind-cross-section. For that matter, the currents and winds are changing and no chart can possibly know what they are at any time and place (speed or direction). You intimate that with a GPS you “continually correct (incorrectly)” That’s just not true. With a GPS you are continually correcting correctly. There is no shorter course that you can paddle than the one by following the GPS’s arrow … I’m sure that you know that a straight line is always the shortest distance between two points on the Euclidean plane. For the distances we paddle, the surface of the earth is almost exactly a Euclidean plane.

Respectfully,

Leon

[ThomasL]

I am certainly unsophisticated when it comes to "calculating" an efficient course to a goal. However, a few painful efforts in varying conditions of wind and current surely teaches even the most unsophisticated the best approach.

Tom

[jason]

The beauty of using a GPS is that you get this minimum distance straight line even if the wind and current are changing with time and position. That is, following the GPS, your heading is always consistent with the correct instantaneous ferry angle. With a GPS you don’t need to know the currents and/or wind, and you never can know the values of these parameters no matter what kind of chart that you use. No chart can give you the side-drift rate for a kayak of unknown wind-cross-section. For that matter, the currents and winds are changing and no chart can possibly know what they are at any time and place (speed or direction). You intimate that with a GPS you “continually correct (incorrectly)” That’s just not true. With a GPS you are continually correcting correctly. There is no shorter course that you can paddle than the one by following the GPS’s arrow … I’m sure that you know that a straight line is always the shortest distance between two points on the Euclidean plane. For the distances we paddle, the surface of the earth is almost exactly a Euclidean plane.

Respectfully,

Leon

Leon,

Let's take an exaggerated example where the GPS arrow isn't' the correct direction.

You have a crossing that involves going over a large tidal stream that's 24 nautical miles across.

Your destination is at a compass bearing of 0 from the launch.

You and your group are paddling at 4 knots.

The tidal stream that's flowing at 12 knots at a heading 090 for the first 3 hours, then swapping over to 270 for the next 6 hours again at 12 knots.

[leong]

Leon,

Let's take an exaggerated example where the GPS arrow isn't' the correct direction.

You have a crossing that involves going over a large tidal stream that's 24 nautical miles across.

Your destination is at a compass bearing of 0 from the launch.

You and your group are paddling at 4 knots.

The tidal stream that's flowing at 12 knots at a heading 090 for the first 3 hours, then swapping over to 270 for the next 6 hours again at 12 knots.

Jason,

You are, of course (pun not intended), correct for the example that you gave where you know something that the GPS can’t possibly know. Lisa and I had discussed situations like this many times while paddling together. So I thought the caveat that Lisa stated in an earlier post covered the situation of your example. For the sake of convenience, here’s what she said:

“Of course, if you have a situation where a straight line is not the fastest route based on current or wind forecast or any other factors (like an island being in the way :-) ), you would have to program multiple waypoints, or follow an existing track. A GPS set to find a destination "off road" isn't smart enough to avoid obstacles or to determine the fastest route given prevailing paddling conditions.”

Because she posted that caveat, I incorrectly assumed that you were questioning whether a GPS would in fact keep you on the minimum “distance” path to some waypoint. Of course, when there is a discontinuity in wind or current or an island in the way, the minimum distance path (a straight line) doesn’t necessarily imply that it gets you to the destination in the shortest time. I know this well in a practical sense because I have raced sailboats.

Anyway, you’re right for what you were thinking, so sorry for the confusion. But, alternatively, when you know nothing about the future or the conditions in another place I think the GPS straight-line path is the optimal path to follow. I think you agree with the last sentence.

Respectfully,

Leon

[leong]

PS

Funny how Dr. Lewis get's these off-topic discussions (arguments?) going. Sometimes he throws a grenade and then runs away!