
Finding the longest straight line you could sail without hitting land - zeristor
https://www.technologyreview.com/s/611012/computer-scientists-have-found-the-longest-straight-line-you-could-sail-without-hitting/
======
mcrider
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](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.

~~~
otras
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](http://www.souloftheworld.com/genesis.html)
[1]
[http://www.souloftheworld.com/work.html](http://www.souloftheworld.com/work.html)

~~~
digikata
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.

[https://en.wikipedia.org/wiki/World_Geodetic_System](https://en.wikipedia.org/wiki/World_Geodetic_System)

------
OscarCunningham
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).

~~~
jfries
> 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."

~~~
OscarCunningham
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](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.

~~~
GavinMcG
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.

~~~
OscarCunningham
>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.

~~~
mjevans
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.)

------
ColinWright
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 ...

~~~
gonvaled
> 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
...

~~~
ColinWright
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.

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

[https://nwhyte.livejournal.com/2929721.html](https://nwhyte.livejournal.com/2929721.html)

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

Central America (?)

------
mjsweet
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.

~~~
NKosmatos
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.

~~~
kijin
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.

------
larkeith
"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?

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

~~~
larkeith
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.

~~~
Luc
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.

------
state_less
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?

[https://www.windy.com/-57.136/-71.191?-37.370,-80.244,3,m:3V...](https://www.windy.com/-57.136/-71.191?-37.370,-80.244,3,m:3Vaecv)

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

~~~
jacquesm
That sounds like a lot of fun :)

~~~
chime
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.

~~~
bufferoverflow
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.

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

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

~~~
murbard2
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!

~~~
logfromblammo
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.

~~~
andbberger
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...

~~~
logfromblammo
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.

~~~
murbard2
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.

~~~
logfromblammo
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.

~~~
andbberger
wat

------
205guy
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/](https://beyondhorizons.eu/lines-of-sight/)

Furthest points from the sea/land:
[https://en.wikipedia.org/wiki/Pole_of_inaccessibility](https://en.wikipedia.org/wiki/Pole_of_inaccessibility)

Latitude and longitude integer degree intersections around the world:
[http://confluence.org/](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](http://confluence.org/antipodes.php)

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

------
rwmj
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".

~~~
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.

~~~
dahart
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.

~~~
arnarbi
> 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.

[http://wordpress.mrreid.org/wp-
content/uploads/2011/07/antar...](http://wordpress.mrreid.org/wp-
content/uploads/2011/07/antarctic-circle.jpg)

~~~
dahart
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.

~~~
arnarbi
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.

------
grkvlt
FYI, the PDF of the source paper is
[https://arxiv.org/pdf/1804.07389.pdf](https://arxiv.org/pdf/1804.07389.pdf)

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

~~~
cup-of-tea
That's pedantic. Most people will realise that "straight line" means constant
bearing.

~~~
mikeash
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.

~~~
cup-of-tea
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](https://en.wikipedia.org/wiki/Rhumb_line)

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

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

[https://www.google.com/maps/dir/Jinjiang,+Quanzhou,+Fujian,+...](https://www.google.com/maps/dir/Jinjiang,+Quanzhou,+Fujian,+Chine/Sagres,+Portugal)

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

[http://www.gcmap.com/mapui?P=prm-JJN](http://www.gcmap.com/mapui?P=prm-JJN)

~~~
macintux
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).

------
cozzyd
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#...](https://geographiclib.sourceforge.io/html/greatellipse.html#gevsgeodesic)

------
GW150914
Weird, they talk about great circles yet insist on “straight line” rather than
geodesic terminology. Still a good read though.

------
Ranlot
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:

[https://ranlot.shinyapps.io/coastlinetrip/](https://ranlot.shinyapps.io/coastlinetrip/)

------
OrganicMSG
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.

------
mindB
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.

~~~
kxyvr
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.

~~~
mindB
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.

------
noetic_techy
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.

------
jhanschoo
Further reading on the branch-and-bound technique:
[https://en.wikipedia.org/wiki/Branch_and_bound](https://en.wikipedia.org/wiki/Branch_and_bound)

------
xs
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?

~~~
bufferoverflow
Going east is only straight at the equator.

------
julienchastang
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.

~~~
SeoxyS
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 :).

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

~~~
enriquto
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.

~~~
Freak_NL
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.

~~~
cookingrobot
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/](http://mrgris.com/projects/merc-
extreme/)

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

~~~
cup-of-tea
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.

------
votepaunchy
The Earth is not a sphere.

[https://en.m.wikipedia.org/wiki/Figure_of_the_Earth](https://en.m.wikipedia.org/wiki/Figure_of_the_Earth)

~~~
wlll
It's a sphere in human scale.

~~~
kgwgk
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).

~~~
wlll
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](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 :)

