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Finding the longest straight line you could sail without hitting land (technologyreview.com)
242 points by zeristor 79 days ago | hide | past | web | favorite | 145 comments

This reminds me of a discovery I made while vacationing in the Corn Islands, Nicaragua -- There was a giant cube buried in the middle of nowhere with one corner poking out of the earth. A plaque nearby said it was the 'soul of the world' and I looked it up to find this funny old organization (http://www.souloftheworld.com) had planted the cubes around the planet, saying that their positions were the vertexes of a cube that was positioned in such a way (and only possible in this way) that every corner touched land. Apparently it has a certain spiritual significance, which is funny to me, but I thought it was really interesting that someone went through the trouble to create them.

Looks like it's true - I threw this together over lunch and it looks pretty close:


This seems like a very interesting question.

After an exhaustive investigation, we found, surprisingly, that there was only one possible combination on the whole planet that would permit the eight vertices of the cube to emerge on solid land. (The results of the investigation made with the AMQ were later corroborated in a study carried out by highway engineer D. David Fernándes-Ordóñez.)[0]

I wonder how difficult it would be to replicate this finding (or a similar finding). I'm naturally skeptical given the extreme remoteness of the islands (Points include the Cocos Islands (Australia), the Corn Islands (Nicaragua), and the Hawaiian Islands (USA))[1] and the not-perfect-sphere shape of the earth, but I also know that I don't have a good natural intuition about the shape of the earth. This would be an interesting project!

[0] http://www.souloftheworld.com/genesis.html [1] http://www.souloftheworld.com/work.html

Well a good starting approximation might be to take a WGS ellipsoid model of the earth and see if their existing coordinates forms something close to a cube.


>This seems like a very interesting question.

>After an exhaustive investigation, we found, surprisingly, that there was only one possible combination on the whole planet that would permit the eight vertices of the cube to emerge on solid land.

Not really that interesting IMHO. After enough searching your're going to find a shape that meets the criteria. If a cube had more than one configuration, they likely would have just move up a notch, and if a cube had no solution they would have moved down a notch.

Someone just did that in this thread!

"I have cubed the earth, with 4 simultaneous _corner_ days in 1 rotation of Earth!"

-- Dr. Gene Ray, Cubic and Wisest Human.

How can you quote him and not link to Time Cube? http://timecube.2enp.com/

It was from memory. I thought his website had gone...

Gone it has, but the true disciples took care to maintain mirrors eternal.

It is. Gene died in 2015 and the original domain expired the same year. The link posted above is a mirror/tribute.

TIL. Couldn't remember the original domain.

Brings back many memories. Back in around 2000/01, the website had complete contact information for Gene Ray including phone number. So I called him up and he rambled at me for about 20 minutes before he had to go because his grandkids were coming over.

Wow, he's a real person? I just assumed the whole thing was a parody, but I was never quite sure...

Wow those are going to confuse the hell out of future beings (if there are any left). I wonder if they're all "built" already?

I don't believe so. I know there is one on the island of Molokai in Hawaii, which I always wanted to visit when I used to live in Honolulu, but otherwise its not a super tourist-friendly place to visit and I opted to visit other islands instead. The Corn Islands, however I highly recommend to anyone!

Good call. I've been there. Got attacked by fish. Would not recommend.

Given the tectonic nature of the planet's surface that has to be the worlds longest running piece of performance art.

On a similar 'frequency', this sort of 'spiritual significance' question has been around for quite a while. It appears that three Temples — of Parthenon (438BC), Poseidon (440BC), and Aphaea/Aphaia (500BC)  — are located at the vertices of a triangle. AKA 'Holy Triangle of antiquity'.

* http://www.goddess-athena.org/Museum/Temples/Aphaea/

* https://eyesofaroamer.com/2016/09/06/the-holy-triangle-the-p...

That's not unusual. Any three points can form the vertices of a triangle. :)

The first site claims that these three temples form an isoceles triangle, which really just means that the distances between two pairs of points are roughly equal. The second claims that they form an equilateral triangle, which would be more unusual -- but is clearly false.


Found a pic of the one in Nicaragua:


> "A really smart mathematician, Fra Luca Pacolini, demonstrated mathematically, that the four regular solid bodies: the Tetrahedron, the Cube, the Octahedron, and the Icosahedron, correspond respectively to the four elements: fire, earth, water, and air."

Does anyone have any idea what that even means?

I think it means that the writer doesn't really know what he is talking about.

And, by the way, there are five regular solids. The other is the dodecahedron. I imagine the writer would say that corresponds somehow to the "quintessence" some philosophers used to go on about. But that still doesn't mean this idea has anything to do with physical reality.

EDIT. By the way, in spite of all the mystical silliness that seems to surround it, personally, I think that the Soul of the World sculpture is a really cool idea.

the idea comes from ancient philosophy, when philosophers could make claims like "everything is made of fire" or "everything is made of water" or "everything is made of triangles" and argue it reasonably. the platonic solids, also, are five in number, where the fifth is a dodecahedron. i think it was plato, among others, who also believed in a fifth element, aether, but i cant remember the details.

how one would demonstrate any of these relationships mathematically is beyond me.

That was it! There was nothing around the area other than cows and some shanty houses. It was really cool to stumble upon.

Cool idea. Totally new to me.

They use height data rather than land/water data, and assume that there's just a cutoff at sea level. I think this gives them the correct answer for the water path, but it gets things wrong for the land path because there are a lot of lakes and rivers above sea level and also some dry land below sea level.

If you don't allow the path to cross any lakes or rivers at all then I think the land path has to be much shorter, since rivers makes it impossible to make any progress at all in most places. If you trace their path it definitely goes through several lakes. The optimal path is probably across a desert, my guess would be Antarctica.

If you do allow the path to cross lakes and rivers then I think there's a longer path than the one they give, starting in Liberia and ending near Fuzhou, China. They probably didn't spot this one because it passes too close to the Dead Sea, which is below sea level (and crosses the Suez canal, which is at sea level).

> They probably didn't spot this one because it passes too close to the Dead Sea, which is below sea level

This is adressed in the paper. The relevant part of the problem statement: "the longest distance one could drive for on the earth without encountering a major body of water"

and about the Dead Sea: "Guy Bruneau of IT/GIS Consulting services calculated [5] a path from Eastern China to Western Liberia as being the longest distance you can travel between two points in straight line without crossing any ocean or any major water bodies. However, the path crosses through the Dead Sea (which can be considered to be a major water body), and hence does not satisfy the constraints originally set out."

Oh, I didn't spot that they had already considered that path.

However I don't accept their defence.

1) Depending on environmental conditions, my path can cross the Lisan Peninsula. https://en.wikipedia.org/wiki/Lisan_Peninsula (EDIT: In fact there's enough clearance to just go completely south of the Dead Sea)

2) Their path crosses the Volga River, which is much larger in total surface area than the Dead Sea. And at the point at which they cross it, the Volga is just as wide as the Dead Sea.

But rivers don't fit in the conventional definition of "major body of water" even if their surface area is large.

It's odd to say you "don't accept their defense" when you're really just operating from a different set of assumptions in the first place.

>rivers don't fit in the conventional definition of "major body of water"

I suppose that's true. But it raises the question of what a river is. Lots of things that people would describe as lakes in fact have water flowing into them, through them, and out of them. So it's hard to distinguish them from a broad river.

Anyway, the distinction is moot since you can in fact squeeze just south of the Dead Sea and avoid hitting it at all.

Without looking up a formal definition, how I believe the common person would differentiate between a river and lake are the following:

    * Is the body of fluid, within a set of bounds, mostly still (and not flowing)?
    * Is it closer to a:
    * > bowl (hemisphere with cut side facing up)
    * > sliced cylinder (again flat slice facing up)
    * > sheet* (a large thin expanse)
    * Is the fluid at the bottom shifting location, how quickly?
Rivers tend to have high flow to volume ratios, faster moving flows at depth, are usually more shallow, and are defined more like a squashed tube in natural states.

Lakes tend to be deeper, more stable (slower flow, if discernible at all), and are generally placid. Lakes /usually/ have a large dimension in at least two surface directions while rivers usually have that in only one.

Exceptions to the above occur with canals/channels* (though that's an ocean thing) which might be closer to the fuzzy boundary.

River deltas also occur in high sediment deposit areas, such as the ends of rivers where they transition in to lakes / oceans; the extreme end of a river delta being bogs and other swamp like areas with shallow slow moving water. (I argue that such areas are neither river nor lake, but a third category.)

> It's odd to say you "don't accept their defense" when you're really just operating from a different set of assumptions in the first place.

Why is that odd? Their set of assumptions is exactly what he is criticizing.

I think of criticizing the defense as criticizing the argument or reasoning offered.

Put another way, if the problem is with the assumptions, the criticism goes far deeper than just that defense.

Is that a straight line? Maybe they allow crossing rivers but not seas/lakes.

In case people are wondering:

    A global map with resolution
    of 1.85 kilometers ...
1.85 km is roughly one Nautical Mile, which is roughly one arc-minute of a great circle. As such it's a natural unit of measure.

The original definition of the meter was 1 ten-millionth of the distance from the North Pole to the Equator via Paris, making the Earth's circumference 40 million meters. Divide by 360 degrees per circle, then by 60 arc-minutes per degree, and you get (40 x 10^6) / (360 x 60) which is about 1852 meters.

Just in case people were wondering ...

I was wondering how one would make a global map with a resolution of a nautical mile. Spacing dots a nautical mile apart on the equator is simple, but you can’t continue with a rectangular grid up north or south (by the time you’re at 60 degrees, the dots would be only half a mile apart)

A perfect solution doesn’t exist, and AFAIK no exact solution for the simpler “place N dots on a sphere in maximizing the minimum distance between dots” exists, but decent approaches exist. See http://web.archive.org/web/20120315152121/http://www.math.ni..., https://www.maths.unsw.edu.au/about/distributing-points-sphe...

See also gradian: https://en.wikipedia.org/wiki/Gradian

"one grad of arc along the Earth's surface corresponded to 100 kilometers of distance at the equator; 1 centigrad of arc equaled 1 kilometer."

i.e. both nautical miles and kilometres are derived from the size of the Earth. Plain old statute miles are just a mess, and best avoided.

Yup, so that will be the best thing once everyone switches to using grads for latitude and longitude.

No! Radians is the SI unit. I would possibly also accept fractions of a whole rotation.

The arbitrary factor of 400 (or 360), is simply not helpful for machine calculations.

400 is pretty great though. a straight angle is 100 gradians, and a bunch of things get simplified. 30 and 60 degrees get worse though

What can I do to convince you that including an additional (2 PI)/N or N/(2 PI) factor into every single angle calculation you will ever do is a complete waste of time?

At the very least, you could take out the arbitrary N.

100 gradians is 0.25 rotations, or PI/2 radians. 30 degrees is 1/12 rotations, or PI/6 radians. There is no advantage whatsoever in multiplying everything by an arbitrary scaling factor that is not based on PI (or TAU).

Polar circumference. Earth is an ellipsoid.

Equatorial circumference is 40075 km. One arc-minute of equatorial longitude is about 1.855 km.

People who do crazy things with measurements cause a significant fraction (at least 15/360) of the many pains in my ass. Degrees need to die a painful death, and so can grads. There is nothing at all "natural" about them as a unit of measure, as they are arbitrary divisions of one rotation, by 360 and 400 respectively.

But I admit, even if not natural they may be more convenient for some calculations than other numbers.

> As such it's a natural unit of measure

As natural as the number 360, and 60.

I would argue the meter seems more natural, since it is related to 10 fingers ...

Given that you are already using degrees, arc-minutes, and arc-seconds in your specifications of latitude and longitude, it makes sense to use a unit of distances that meshes with them. Using meters makes less sense because it doesn't work well with the size of the Earth.

Difficulties with measurements and conversions are like a dead cat under the carpet - no matter how you push it about there's always an inconvenient lump and a bad smell. The Earth is inconveniently shaped, and you simply have to deal with it. The existing system of measurements may offend your sense of taste, but it has evolved over time to be useful to those who have to use it. Attempts to devise systems a priori and without taking into account the extensive experience of those who actually use them have always failed.

There's probably a reason for that.

I think "natural" was meant in the sense "it naturally follows" or "it's natural to assume", where it's meant to imply a natural path given your preexisting knowledge and conditions, not natural as in "this comes about in nature".

With that in mind, 360 is a fairly natural way to segment a circle for modern Humans.

60 is a very natural number.

You have twelve joints on the fingers of one hand. (Use the thumb to count.) You have five fingers (including the thumb) on the other hand to track multiples of 12 with. Voila, 60!

360 and 60 are quite handy, by virtue of having a lot of integer divisors. For a civilization where long division is unknown or limited to a few scribes, being able to divide things by common fractions easily is an important criterion in a system of measure.

> since it is related to 10 fingers

If your parent comment is correct. 10 finger thing will merely be an afterthought.

If you like this sort of things, you absolutely should read "Which lines of longitude and latitude pass through the most countries?".


Through Greece and I didn't read the other one because the author is a terrible writer.

Central America (?)

I loved the final question in the article:

“The question now is: who will be the first to make these journeys, when, and how?”

These have the potential to become quite important journeys... with the potential for many “firsts”... and also much contention; how much did they deviate from the path, how much deviation from the path is acceptable?

So many interesting questions to ask.

That final question struck me as odd, the next question for my mind was could the technique be generalised to a map on a torus?

Same here, that’s what I thought as well :-) I’m sure that the big documentary channels will pick this up and we’ll see both journeys in a few years. Seems to me that the land journey will be harder to complete. When you’re sailing it’s easy to keep a “straight line” without having to consider mountains, rivers, permissions, politics and any other obstacles associated with crossing land.

It won't be easy to sail in anything resembling a straight line through the Drake Passage unless your ship is very big.

Speaking of big ships, you might also want to avoid icebergs.

For the land route a realistic approach would be to plot major checkpoints along the true great circle path and just navigate between those checkpoints in whatever way (preferably on land) is most feasible. That would still be interesting.

Yeah I was thinking about how hard it would be to stay “true” to the path as possible if a race or challenge was by wind power only. Also in competition, points could be given or taken based on the amount of deviation from the line etc.

“The question now is: who will be the first to make these journeys, when, and how?”

I wouldn't volunteer for the water journey any time soon considering you're basically launching into a pirate haven.

Non stop circumnavigations have already been done.

For example: https://en.m.wikipedia.org/wiki/Wilfried_Erdmann

Can you clarify this comment, how is circumnavigation relevant to this considerably different journey / path?

"A global map with resolution of 1.85 kilometers has over 230 billion great circles. Each of these consists of 21,600 individual points, making a total of over five trillion points to consider."

What? Is there something I'm missing here, or did they just decide to include "individual points" as an utterly useless way of inflating the difficulty of a brute-force approach?

The longest path needs to start and end somewhere along a great circle. So those individual points are needed.

The only relevant points are those that make up the coastlines - there is no need to test paths that start and/or end in the middle of sea or land.

Oh I see what you mean. Yes, I suppose it is worded that way to impress the reader with the big numbers involved, when many of them can be trivially rejected.

Coastlines are close to infinite in length (they grow inversely to the length of the measure used; idealised mathematical 'coastlines' are fractal and infinite), we can limit them by limiting the resolution.

at a 1.8 km resolution (without checking my maths) 21,000 points along a great circle is basically every 1.8km so i guess you are checking at each 1.8km if you just drove into water (or vice versa)

Your point that you only need to fail once is fair - there is a lot of boundary conditions one can apply quite quickly

Going from Pakistan to Russia seems like the wrong way around Cape Horn. Today you've got 30 kt winds gusting up to 50 kts. Who wants to beat into that?


I windsurf in 30 knot winds at sea. Yeah, it's rough, but it's not hurricane level madness that rips ships apart.

That sounds like a lot of fun :)

Wind surfing has a surprisingly high fun/expertise ratio. You can start surfing on your own in as little as 2 hours of training. I was chasing dolphins on my second day of surfing.

Of course, like any sport/hobby, it can take years to master but even for complete beginners it can a lot of fun.

Correct. You can make it easy or tough simply by choosing smaller/bigger sail and applying less/more force. So it can be very relaxing in 20 knots, especially on a lake.

For the water challenge couldn't you just sail the 60th parallel south forever? https://en.wikipedia.org/wiki/60th_parallel_south

Also could we challenge flat earthers to the same challenge and see who travels the furthest?

Straight line here is taken to mean geodesic, the shortest possible path on the surface between two points. Parallels on the sphere are not geodesics!

If we're going to split hairs, an actual straight line would be tangent to Earth at the point of the boat, and you could sail forever, so long as you do it at the speed of light and don't hit the land masses on some other celestial body.

There is some definition of "straight line" that includes a course of a constant bearing, or a rhumb line. It's straight when plotted on a Mercator projection.

By that definition, keeping to a true east or true west bearing would be a straight line.

Meh, this is the worst kind of pedantry imo. Like, first of all, doesn't matter. But if you must, you're going to have to go with 'as would be defined by mathematicians' - because they spend a lot of time thinking about these things, and thus are by far the most qualified to have an opinion....

And the mathematicians have thought long and hard about how the Euclidean concept of a straight line generalizes to other geometries... and came up with geodesics... aka great circles...

Why would mathematicians take precedence over people who navigate ships and airplanes for a living? Also, ellipsoidal geodesics aren't always circles.

"Thinking long and hard" isn't actually much of a qualification, if you think about it long enough and hard enough. Playing around with the definition of "straight line" is just an amusement, putting different theoretical constraints on the recreational problem. The whole thing is pedantry to begin with, so don't be surprised when someone pops their head in with something unexpected just to show off how clever they think they are.

Launching pointless academic arguments is almost the whole point. It shows the audience that everyone involved is very smart, and all possibilities have been duly considered, and therefore the agreed-upon answer must be very significant, reliable, and noteworthy.

If you navigate a ship while keeping the rudder straight, absent current or wind you will be following a geodesic. If you try to follow a parallel you will have to constantly be turning.

Planes follow geodesics too over oceans, not parallels.

Yes the geodesic on an ellipsoids aren't always great circle, but the earth's geodesics are commonly referred to as "great circles" because the earth is very nearly spherical.

Should we ignore wind and current? If you are in a medium that has winds and currents, and you do not touch your yaw controls, how will you know if your course remains on a straight line?

You either adjust your compass bearing (also correcting for magnetic variation), or you adjust course.


You’d have to constantly turn your boat to the right to stay on course.

That's not a straight line (it only looks like that on this map).

There are no actual straight lines.

And we'd need a frame of reference in any case.

The problem seems ill-defined.

I think the problem is fairly well defined, the headline just doesn’t define it fully (with good reason).

I love geographic puzzles such as this one. Here are links to some other interesting ones.

Longest straight lines of sight, modulo some atmospheric refraction (has been discussed on HN before): https://beyondhorizons.eu/lines-of-sight/

Furthest points from the sea/land: https://en.wikipedia.org/wiki/Pole_of_inaccessibility

Latitude and longitude integer degree intersections around the world: http://confluence.org/

The integer degrees is arbitrary, but it provides a random sampling of landscapes. I like the idea of going out to find some defined spot, like geocaching without any caches. It also provides a page of antipodes photos, places exactly on the other side of the earth from each other: http://confluence.org/antipodes.php

Must also mention earth sandwiches: http://www.zefrank.com/sandwich/

Fun fact: there are parts of the Pacific ocean that are opposite each other, therefore the Pacific ocean spans (not covers) half the globe.

I don't have a globe handy, but can't you sail indefinitely round and round Antarctica? Or doesn't following a line of lattitude count as a "straight line".

that would certainly not be a geodesic (the only sensible defintion of "straight line" on a sphere), more like a circle. It would be just like running circles around your house.

You would have to keep turning slightly South to maintain a bearing parallel to the lines of latitude - the only one that's "straight" (it curves downward, but not North or South) is the Equator.

Source? My gut tells me this is not true, but I'm willing to be convinced.

It’s confusing because the latitude line 66° south looks like a straight line on a globe. But to sail in a “straight line”, which means keep your steering centered forever, you will travel in a great circle.

All great circles lie on a plane that intersects the sphere (earth) through the center of the sphere. You can see that the only latitude line that’s on a plane intersecting the center of the earth is the equator, and that 66° south doesn’t. This also means that all straight line paths on earth touch the equator at 2 opposing points. Or said another way, you can start with the equator, pick any one point on the equator and rotate it around that point to get a new great circle.

So in order to stay sailing along 66° south, you’d have to have your steering turned constantly just a little bit south.

> It’s confusing because the latitude line 66° south looks like a straight line on a globe.

On the contrary, latitude 66° on a globe looks to me like a fairly tight circle.


It sure does, you’re right. So does any “straight line” great circle too, so that isn’t super helpful. The equator has the same circle projection that 66° does.

Look at it from the side and it looks straight. If you’re sitting in the plane of 66°, the projection is straight.

I was trying to be supportive of @dbatten while explaining. It’s easy to get confused about what straight means on a sphere, since nothing is actually straight.

May apologies if I sounded knee-jerky. The intention was to show that a picture was a better answer to @dbatten's very legitimate comment.

The idea is to demonstrate that a "straight" line on a surface needs to be viewed along a normal to that surface at the point of the line you are concerned with, assuming the definition of "straight" is "don't have to turn when travelling along line on the surface". That makes great circles look straight, and non-great circles not.

>> larkeith: You would have to keep turning slightly South to maintain a bearing parallel to the lines of latitude - the only one that's "straight" (it curves downward, but not North or South) is the Equator.

> dbatten: Source? My gut tells me this is not true, but I'm willing to be convinced.

Consider any point on a sphere, pick a direction (which might be along a line of latitude) and visualise the plan defined by that vector and the centre of the sphere. The intersection of the plane with the surface of the sphere is a Great Circle, and that is where you would go if you didn't apply turn to the rudder.

The parallel of latitude defines a plane that does not go through the centre of the sphere (unless it's the equator) and so isn't a "straight line". If you want to stay on the parallel of latitude then you will deviant from the Great Circle, and that's why you need to apply rudder to stay on the parallel instead of the Great Circle.

Does that help?

This is all pretty obvious once you've done some spherical geometry, but can be completely opaque to anyone who hasn't.

The problem is that the proposed solution doesn't "count" for the purposes of this experiment, which uses a great circle definition of "straight". The easiest way to visualise a great circle, is to place a piece of string over a globe between your origin and destination and pull it tight, that will track a great circle route between the two (and how a flight flying straight between Europe and the US west coast will take off on an pretty northern bearing, and land on a ditto southern, despite not actually going over the north pole and Europe and the US being located east/west of each other).

If you tried to place the string around Antarctica and pull it tight, it'd slip off in the southern direction, which represents the parent's explanation that you'd be steering south (from a great circle course) to keep straight along a latitude.

Get a ball and a piece of string. Put the string on any two points of the ball and pull it tight. You'll observe that the path the string follows is part of an "equator" of the ball.

No line of latitude is straight, except arguably the equator.

Think about the line of latitude at 89.9999°. It will trace a circle around the pole of a few meters radius. 60° south is the same such circle, only bigger.

FYI, the PDF of the source paper is https://arxiv.org/pdf/1804.07389.pdf

Isn't "straight line" somewhat arbitrary? How about "the longest circle you could sail without hitting land" or some other reasonably smooth shape?

I was just thinking this. If it is allow to sail over the same places more than once, they must be a route that is just a circle. Maybe along one of the southern parallels?

That's pedantic. Most people will realise that "straight line" means constant bearing.

From the maps they show, it’s pretty clearly a great circle (the obvious meaning for “straight line” on the surface of a sphere), not constant bearing.

Whoops, turns out I didn't actually understand what "constant bearing" meant. What I actually meant was "no rudder/steering" which is exactly what people would imagine as travelling in a straight line.

A constant bearing actually gives a Rhumb line: https://en.wikipedia.org/wiki/Rhumb_line

Don't apologize. Your use is perfectly correct, you're obviously not referring to compass bearing.

Obvious perhaps, in a mathematically idealised system that's idealised according to axioms that are unstated ... a non-equitorial latitudinal line on a sphere is straight.

Walk (or drive, or sail) without turning and you’ll follow a great circle.

Stay perpendicular to North/South and you follow a parallel.

None of these things is of course possible IRL.

Staying perpendicular to north/south requires turning, unless you’re on the equator.

Or listing (ie leaning); still it seems straight, YMMV.

Consider the degenerate case where you’re ten feet away from the North Pole. You’ll need to go in a tight circle. The situation at more moderate latitudes is the same, just less extreme. It only seems straight because we’re usually at moderate latitudes and the rate of turning is low.

What sort of constant bearing - magnetic bearings will vary along a map-based bearing. There's an arbitrary choice involved. We're probably not looking at changes in sea-level that put us off line - is a necessarily idealised system in which the question makes sense.

But, unless you are passing over a pole, a great circle doesn't use a constant bearing.

Life is arbitrary. Your question is equally interesting, except for the fact that you haven't offered up an answer to it.

Yes, "straight" is misguiding.

Google maps seems unable to find a route between Jinjiang, Quanzhou, Fujian, China and Sagres, Portugal :


If anyone is interested in a 7000 mile trek, this is the approximate path of the land route:


Thanks. Maybe I'll try that over lunch next week.

I was curious one day after reading yet another comment contrasting driving in Europe vs driving in the U.S., distance-wise, and I decided to measure Seattle->Key West and compare that to Europe. Decided that the equivalent would be Paris->China (almost, I think the terminating point was eastern Kazakhstan).

I wonder if the answer changes if you consider actual geodesics instead of great circles (which are geodesics on a sphere, but the Earth is not a sphere). Even great ellipses (closer in length to actual geodesics) can deviate laterally from the geodesic by kilometers. See e.g. https://geographiclib.sourceforge.io/html/greatellipse.html#...

Weird, they talk about great circles yet insist on “straight line” rather than geodesic terminology. Still a good read though.

Looks like another cool app: Click somewhere on the coastline and it'll take you to what's directly on the other side of the sea. Results may be surprising:


This is cool. Though as you take into account more of the detail of the shape of the surface, your straight line can get a lot longer. I would be interested to see one that takes into account the large scale ocean surface topography, to see if that can change it by much.

I thought the solution set had to be convex or concave for branch and bound to work as an optimization algorithm, and this one clearly isn't?

EDIT: Nope. It's just the objective function that needs to be convex, not the constraints.

Technically, you need neither. Branch and bound works by solving a series of relaxations with fixed integer variables in order to better search the discrete space. Now, imagine we're solving a minimization problem. Finding a feasible solution to the problem may be difficult, but if we were to find one, discrete variables and all, we'd have an upper bound on how good the solution could be. Simply, it's feasible, but not optimal, so the objective value is higher. Now, since integer variables are hard to work with, we can relax them into continuous. For example, instead of a binary variable that's {0,1}, we could relax it into a continuous variable bounded between 0 and 1, [0,1]. If we do this relaxation and find the globally optimal solution, then we have a lower bound as to how good the solution is. In branch and bound, if we can show that the lower bound for one branch is not as good as the upper bound of another branch, we don't have to explore that branch.

Now, convexity comes into play because it allows us to correctly determine these lower bounds because we can guarantee a globally optimal solution to the relaxed problem. If we lack convexity, it's hard to find such solutions. Does that mean branch and bound requires convexity? No. You can still perform branch and bound using locally optimal solutions and even though you don't have a guaranteed global bound, there's relatively good information about where we should search next. This can lead to good, but not provably globally optimal solutions.

But the article was making a strong claim that this was the provably optimal solution. In this case it is, because the objective function (in the way I'm assuming they formulated the problem) is convex.

Its always surprised me that you cant get to the farthest eastern reaches of Russia by car.

Its such a massive amount of territory they hold in the east, you could easily start another country as large as China or the USA out there.

Further reading on the branch-and-bound technique: https://en.wikipedia.org/wiki/Branch_and_bound

Why not start at the most southern point of South America and go due east? Maybe even 0.01 degree South. Couldn't you do loops around the Earth before hitting Antarctica?

Going east is only straight at the equator.

I once flew from Chicago to Beijing and the thing that astonished me is that you are over land the entire flight except in the vicinity of the Bering Strait.

If your plane flies the shortest great-circle path, you'll fly over the arctic sea, which is definitely water, though sometimes solidified. IAH or DFW would be better examples :).

And what better way of showing it than a Mercator map!

The maps on the page are not Mercator, but what's the problem with Mercator? It is a perfectly reasonable projection, with some nice properties.

Ideally they'd show a map with a projection based on having the line drawn on it as a straight horizontal or vertical line in the middle of the map. So basically, a Mercator projection with the line being illustrated as its 'equator'.

That would require a custom rendering for each line though.

I like this idea, and they're not in the paper. (The paper shows a globe rotated so the paths are straight, but not a flat mercator projected map).

It should be possible to make one with this tool: http://mrgris.com/projects/merc-extreme/

They have these illustrations in the paper. https://arxiv.org/abs/1804.07389

Given the length of the line over water, that would be a somewhat bizarre map.

See page 9/10 of the source paper.

A single map using that projection calls a lot of attention to the quirks of projecting a sphere onto a plane. A different projection or multiple maps centered on segments of the line would call attention to the straight path you'd take as you graze landmasses.

They need to account for currents as well otherwise a straight line wouldn't be straight at all

Why are these comments full of people trying to make what is an interesting and easy to understand problem into an insanely complicated problem? This is hacker news. We're not actually going to sail around the world. It's interesting because someone used a computer to find an answer.

Absolutely. Particularly under sail, where top speed is typically going to be under 15mph (often under 8mph), a 2mph side current can impact course over ground quite a bit over a day or two when you are only doing 200 miles a day.

From that page:

> the Earth deviates from spherical by only a third of a percent, sufficiently close to treat it as a sphere in many contexts

It's a sphere in human scale.

What do you mean by “human scale”? It may be a sphere in much larger scales (~1000km) but I’d say it’s far from being a sphere in human scale (~1m).

It's based on context. So something like a scale that makes sense when going about your average human business. For instance, the sun is 150 million miles from the sun. It's not, it's slightly less, but for "normal" human activity (eating, sleeping, sailing in straight lines, programming web apps, etc. etc.) it doesn't matter, it's close enough.

So for the earth (as someone else pointed out):

> the Earth deviates from spherical by only a third of a percent, sufficiently close to treat it as a sphere in many contexts (https://en.m.wikipedia.org/wiki/Figure_of_the_Earth)

For most of what most of us do with our everyday lives it's good enough, to the extent that pointing out that "the earth is not actually a sphere" just sounds pedantic.

It did occur to me that sailing in a straight line could be interpreted as an actual straight line, but the surface of the ocean is spherical(fine, -ish) and so, modulo waves, we might actually be measuring an infinitely small distance on the earth that was sailable (Ah ha, got you all!) but realised that jokes about derivatives, despite the crowd, weren't going to be all that amusing :)

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