Planning for a LONG walk

February 6, 2008 8:34 AM

Plan your trip to a far away spot on the globe. You might wish to walk in a straight line or maybe just take the shortest route (other than, perhaps, digging). Take your camera in case you pass one of these.

Its all to do with great circles and rhumb lines I think. We have talked about confluences before but the project still has about 5,000 on land, to be visited so I thought they could do with another mention.
I was delighted to notice that a journey due south from Edinburgh would take me to the west of Liverpool and onwards to Timbuktu.

posted by rongorongo (28 comments total)

The straight line thing is interesting. I find it difficult to grasp why I can't leave my home and walk in a straight line that leads me in a spiral around the world, why every line actually brings me back home directly. I guess I don't understand the topology of a sphere as well as I intrinsically feel I do, or I don't quite understand what a "straight line" is...
posted by benzo8 at 9:05 AM on February 6, 2008

benzo8: Straight lines are typically straight. A spiral would require an ever-so-slight veer one way or the other.
posted by Sys Rq at 9:18 AM on February 6, 2008

Either this thing is messed up or I don't understand something. My origin was Texas and I chose 90 degrees as my heading, which I believe to be due east. My path was not a spiral, but a sine wave going throught south Florida, down to the Botswana region of south Africa, over through Australia, then returning home through Baja California. Feel free to educate me!
posted by Daddy-O at 9:47 AM on February 6, 2008

* through south Florida *

I thought due east would keep me in the northern hemisphere.
posted by Daddy-O at 9:49 AM on February 6, 2008

"Due east" would keep you in the northern hemisphere, but it wouldn't be a geometric straight line. Unless you were on the equator, but then you aren't strictly in the northern hemisphere.
posted by DU at 10:02 AM on February 6, 2008

Yeah, I'm with Daddy-O. In what sense of "straight line" does "straight line heading east" make me cross the equator (twice)?
posted by Flunkie at 10:06 AM on February 6, 2008

Daddy-O, recall the sine wave paths of space shuttles and satelites when drawn on a Mercator projection world map. Draw that same path on a globe, and it's a perfect circle equal to the circumference of the earth. If you went due east from your point of origin, in order to stay on that same latitude in the northern hemisphere, you would have to keep turning ever so slightly left -- it's not a straight line on the globe. (You could not put a satellite in orbit on that path, unless it has navigational power to keep turning slightly north.) The sine wave path is actually due east instantaneously at your starting/ending point. A satellite flying overhead in an identical earth orbit would also be going exactly due east at that point. But if you shine a laser beam straight ahead and keep following it exactly, you'll be on that sine wave path, but your direction relative to the latitude/longitude grid will continually change.
posted by beagle at 10:07 AM on February 6, 2008

It's a perfect circle equal to the circumference of the earth

Well then they should've said that, not "straight line", and especially not "straight line east".

A geometric straight line is the shortest distance between two points. The only way to get what they did as a "straight line" is "straight line through the point you pick and its antipode, starting in the direction that you choose".

What they wrote, and the directions they give, though, strongly imply "see what happens if you walk east" instead.
posted by Flunkie at 10:14 AM on February 6, 2008

A straightline path around a sphere* describes a circle.

Imagine yourself standing on some point on the surface of a sphere. Now, because a sphere is such a nice uniformly shaped object, it has infinite axes of rotational symmetry—you can spin it however you like and, ignoring surfaced decorations, it's still precisely the same shape.

So you pick a random point on the sphere. That point is equivilant to any other point on the sphere. Okay! Now pick a random direction and walk straight forward that way for a long time (25,000 miles or so on Earth). What you're doing is walking your own circumfrence, your own equator as it were, around the sphere. Because every starting point is functionally identical, and every direction of travel is too, you can consider whatever point and direction you chose to be equivilant to starting at the actual equator and walking along it.

No spirals or other strange shapes. The "sine wave" Daddy-O sees is the result of projecting a circular path around a sphere onto a flat projection of that sphere onto a 2D map.

If you want to walk due East and get a straight line? Start on the equator. Anywhere else on the globe, walking due East means necessarily means constantly slowly turning to one direction: you're not walking in a straight line, you're walking along a curve that gets projected onto a 2D map as a straight line.

If that isn't clicking, take it to the extreme. Go up to the North Pole, the very exact-to-the-inch point of extreme northerliness on the globe. Now, stand five feet south of it. Walk due east. You'll be back where you started in a few seconds, having walked a tight circle five feet in diameter.

^*Okay, the Earth isn't quite a sphere—it's actually an avoid, bulging slightly along its equatorial circumfrence thanks to the power of SCIENCE!—but close enough for funk.
posted by cortex at 10:16 AM on February 6, 2008

Yeah, in the mathematical sense of "straight line", a straight line around a sphere is a circle. But, first, not every such straight line goes through both a point and its antipode, and second, more importantly, the page to me reads as if it's using "straight line" in a non-mathematical sense - e.g. "east".

For example, they say "choose the place that you will start from and what direction you are going to walk". Not "the direction you are going to start walking in".
posted by Flunkie at 10:21 AM on February 6, 2008

A geometric straight line is the shortest distance between two points. The only way to get what they did as a "straight line" is "straight line through the point you pick and its antipode, starting in the direction that you choose".

These are the same thing. A great circle is the shortest distance between two points on a sphere.
posted by DU at 10:21 AM on February 6, 2008

Yes, I know that a great circle is the shortest distance between two points on a sphere.

I also know that the page doesn't say "great circle" anywhere on it, and also that it told me to choose a direction to walk, not a direction to start walking in.
posted by Flunkie at 10:25 AM on February 6, 2008

Well, the talleye site also doesn't say "choose a heading and follow it all the way around the globe", either. The sidebar of the site does disclaim the geometric gotcha, though:

If you don't understand why the path has this shape, Google Earth Export feature could make the things more clearly for you, in 3D perspective.

So I'm not sure there's any attempt whatsoever on the site to sell the idea that a straight line means "a straight line as seen on a projection map" or that picking a direction to walk in means walking along a fixed heading for the whole path.

Beyond which, the English on the site isn't exactly superb, so it's bordering on silly to object on points of fine parsing.
posted by cortex at 10:31 AM on February 6, 2008

I'm not saying the site "attempts to sell the idea".

Whatever. This is silly.
posted by Flunkie at 10:33 AM on February 6, 2008

Well then they should've said that

Yeah, the page's explanations leave something wanting, given the apparent paradox of the sine curve resulting from a 90 degrees East choice, IF you're not starting from the equator.

Cortex's example of starting 5 feet south of the North Pole is a good illustration of my point above that if you maintain a due East path, in the northern hemisphere, you'll have to be turning slightly left the whole way, relative to the straight-ahead laser beam. This becomes very apparent within shouting distance of the Pole.
posted by beagle at 10:33 AM on February 6, 2008

Meet in the middle
posted by hortense at 10:35 AM on February 6, 2008

What's really important is that the plane takes off.
posted by Dave Faris at 11:09 AM on February 6, 2008

I have to say that if you do open up the .kml file in Google Earth, you are definitely going in a straight line all the way around the globe until you get back where you started, as you might expect. Actually it's pretty cool, and worth downloading Google Earth just to convince yourself that this is so.

How this ties into the weird sine-curve shape on the 2D projection is something I am still trying to wrap my head around, but for sure this is no tricksy definition of "straight line."
posted by so_necessary at 12:18 PM on February 6, 2008

A man leaves his house, walks South for 3 miles, then walks East for 3 miles, then walks North for 3 miles, where he arrives back at his house and sees a bear. What color is the bear?
posted by noble_rot at 12:22 PM on February 6, 2008

There are no bears a little over 3 miles north of the south pole. So he must have brought it himself. So it must be a trained bear. Thus its a brown bear.
posted by vacapinta at 12:38 PM on February 6, 2008

I can throw in one piece of knowledge: on a mercator map, any straight line is a line of constant bearing (NOT a great circle path), i.e. due east, which, if followed, will eventually spiral toward one pole or the other. So if you really want to see that kind of "straight line" you can just draw a line on a mercator map. This is interesting because we're drawing great-circle paths on maps which don't show great-circle paths as straight lines, so they look funny! Ha ha!
There are map projections in which some or all of the straight lines you can draw are great-circle paths. There are map projections to do pretty much whatever you want, because making a new projection is a good geography doctoral thesis.
posted by agentofselection at 12:58 PM on February 6, 2008

A straightline path around a sphere* describes a circle.

Good explanation, but maybe I can add to the confusion by mentioning that walking due East from my position in the northern hemisphere around the top half of the sphere is, though not a Great Circle, still a circle, and thus would be a great circle on some other sphere. So although it's not a straight line on this hypothetical spherical Earth, the exact same path through space *would* be a straight line on some other sphere we could imagine.
posted by sfenders at 1:02 PM on February 6, 2008

How this ties into the weird sine-curve shape on the 2D projection is something I am still trying to wrap my head around, but for sure this is no tricksy definition of "straight line."

If you really want to bruteforce yourself into seeing it, get yourself a globe and a spare world map and a rubber band and a pen.

1. Wrap the rubberband around the globe along whatever axis you like, so long as you make sure it's a Great Circle—that it cleanly divides the globe in two.
2. Spread out your map, grab your pen, and start marking points on the map where the rubberband crosses the same points on the globe. Cities, border crossings, whatever landmarks are handy, so long as you mark at least a dozen or so.
3. Connect the dots.

Map projection is a fun topic but not one that necessarily gets more than a passing mention in grade school.

So although it's not a straight line on this hypothetical spherical Earth, the exact same path through space *would* be a straight line on some other sphere we could imagine.

Suggestion: we crash the moon into the Earth and let the survivors walk around the border of the collision to settle the spatial relations question first-hand.
posted by cortex at 1:33 PM on February 6, 2008

How this ties into the weird sine-curve shape on the 2D projection is something I am still trying to wrap my head around,

Surely, that figure looks familiar to people who have seen a night-day map of the Earth?
posted by vacapinta at 1:58 PM on February 6, 2008

metafilter: close enough for funk
posted by CitizenD at 2:58 PM on February 6, 2008

Just after posting this I found the site for Expedition 360 - which involved circumnavigating the globe using only human-powered transport. It took them from 1994 until last year. Their site has a page explaining how the idea of what constitutes a true circumnavigation has evolved since Magellan had a go in 1522.
posted by rongorongo at 2:17 AM on February 7, 2008

Map projection can cause the gears of one's brain to slowly grind to a sad, clunking halt, but this seems like a cruel way to try and get people to pick up on it without prior introduction.

To try and help people with the whole sine wave thing: it looks like that because the circle you're travelling on (the line) doesn't go all the way to the south or north. Mercator originally designed his map for ships, so if you want to go from A to B and you draw a line, and the line is (say) pointing 270 degrees (west) from A, you point yourself west, and start going. You constantly check your compass on the way, always adjusting so that it says 270º, and bang, you get to B.

The key here is that you are constantly adjusting.

If you start at A, turn to 270º, and start walking absolutely razor straight, your compass will eventually say 271º, 272º, etc, after hundreds of miles, because (and this is the key craziness) "lines of constant bearing" — what you're interested in if you are going somewhere — are actually very slightly curved the entire way, because you're walking in a big circle, around the north pole. If you take one step east of where you are right now, you've just increased the distance slightly between you and the north pole.

Think of a very tall, thin triangle, where the bottom is composed of a 90º angle on the right, and a (say) 89º on the left, and the sides continue up from the bottom until they meet. If you think of the apex of that triangle as the north pole, the bottom-right (square) angle as where you start walking from, and the bottom-left (slightly unsquare) angle as where you step to, you can see that your distance from the north pole has increase, and it's at a different, new angle from where it was before you took the step. Your choices are now: turn slightly, reorienting yourself so that the angle is the same as pre-step, or keep on walking in the same direction, continuing to change that angle.

The confusion comes in because everyone pretty much knows that if you'd like to head east, you keep checking your compass to ensure that you really are going east the whole time, and then you don't end up sine waving for 40000 km, but instead, at point B (which is getting very crowded right now, I wish Point A people would quit moving.)

Other projections that are interesting: Goode, Winkel Tripel, and the only map where straight lines on the map are straight lines (i.e. great circles) in reality, Buckminster Fuller's Dymaxion^*.

* - Until you cross the boundary between one of the faces of the dymaxion, in which case a great circle route would bend slightly, but hey, it's still awesome.
posted by blacklite at 2:31 AM on February 7, 2008 [1 favorite]

blacklite: Thanks - not just for a great explanation but for an introduction to the Dymaxion map. I see the Buckminster Fuller institute offers these for sale in various formats.
posted by rongorongo at 4:54 AM on February 7, 2008