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I found this map from this substack, World in motion:

Rohan Chabukswar and Kushal Mukherjee, a pair of researchers at the United Technologies Research Center in Cork, plotted the route in 2018 in response to a question posed by Live Science. They worked out that, if you set off from the coast of Pakistan, it would be possible to sail for nearly 20,000 miles – between Africa and Madagascar, narrowly missing the coast of South America – before hitting land again in the far north east of Russia.

And although it looks like a curve on a two-dimensional map, your theoretical boat would actually be sailing in a completely straight line.

sine-like wave headed from Pakistan, between Madagascar and Africa, below South America, to east Russia

It claims you can sail from north east Russia to Pakistan in a straight line. How is this true?

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3 Answers 3

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An issue of Projection of a 3D globe on a 2D surface. It becomes immediately obvious once you look at an actual 3D globe that the line is, indeed, straight.

enter image description here

The other side is just lots of Pacific Ocean.

If you want the 3D view animated, this YT video shows that.

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    If this surprises you, check out "non-Euclidian geometry". Where a triangle does not have an angle sum of 180 degrees.;-)
    – DevSolar
    Apr 25 at 14:08
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    I'm sure the answerer is already aware of this but the readers should note that even the above "proper 3D model" is still a projection and distorts countries: it's not a magic way of escaping the problems of map projections. The reason the straight line does actually appear straight in the above picture is that it goes through the centre. A great-circle (ie straight) path from, say, India to Paraguay on the above image would appear curved.
    – Dannie
    Apr 26 at 14:20
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    @Dannie: And I took some time to rotate the virtual globe so that the line from Pakistan to between the Cape and Antarktika would go (more or less) through the center. A picture of a globe would have been easier, but I didn't feel like setting up for a cringey selfie in front of the mirror. Or defacing my globe with a sharpie. (Why is it so difficult to leave "good enough" alone? Once demonstrated in this way, it should be obvious that the claim is true, with the minor errors in the approximation of a "real" globe done by Google Maps being insignificant.)
    – DevSolar
    Apr 26 at 14:24
  • Is this Google Maps or did you mean Google Earth? I can only get a sphere by zooming out in Earth, but not Maps. Can you actually get one on Maps?
    – terdon
    Apr 26 at 16:23
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    @terdon The sphere doesn't appear automatically when you zoom out (I'm sure it used to) but is available as an option at any zoom level - click "Layers", then "More", then "Globe View".
    – IMSoP
    Apr 26 at 19:32
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What they are saying is that, if you leave Kamchatka in the correct direction, you can sail to Pakistan without changing the rudder setting. It does not mean you will always follow the same compass heading, you emphatically will not.

The reason is that a "straight" line on a sphere is what's called a "great circle". If you cut a sphere with a plane that goes through the centre of the sphere, the line it traces on the surface is a great circle.

A great circle is considered a straight line, because it is the shortest distance between 2 points, as long as you can't leave the surface of the sphere. All great circles on a sphere are the same size. Meridians are great circles, but lines of latitude (except the equator) are not.

Airplanes flying long distance follow great circles, to minimise fuel use. A flight from Seattle to London will pass over Greenland; planes going to Sydney from Santiago sometimes (depending on winds) fly far enough south to make Antarctica visible. They fly a "straight" line around the earth. For more details, see Flight Paths and Great Circles – Why Are Great Circles the Shortest Flight Path?.

You may have noticed that the path of satellites, projected on the surface, follows a similar curve. See, for example, the path of the International Space Station. While the path of the ISS is not circular, compared to the radius of the earth, the difference is minimal. The ISS obviously does not ever deviate from a straight line around the earth, yet, around it goes. With the ISS, the curve shifts from one orbit to the next, because the earth rotates underneath. If you extend the curve on your map, you'll notice it gets you back to Kamchatka.

The real problem is that a sphere cannot be accurately mapped on a flat map. There are always distortions. It's those distortions that convert a straight line into what looks like a curve. Wikipedia has a good article on the various types of map projections and their pros and cons.

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    "you can sail to Pakistan without changing the rudder setting" -- that's not quite right in real world conditions due to currents, wind, etc. The "taut string on a globe" visualization is probably a better simplification of great circle arcs. Apr 25 at 20:15
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    "you can sail to Pakistan without changing the rudder setting" Great circle means not changing the rudder setting, but not changing the rudder setting doesn't mean great circle; it can mean any circle. "A great circle is considered a straight line, because it is the shortest distance between 2 points" Well, part of it is. Apr 26 at 2:02
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    @JohnBollinger there is no good reason to consider those lines straight. Great circles (and geodesics more generally), are the only way of generalising straight lines to curved surfaces
    – Tristan
    Apr 26 at 9:32
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    As a mathematician, let me weigh in. A great circle is a geodesic on a sphere, a notion that generalizes the notion of straight lines in non-curved (Euclidean) spaces. But no mathematician, mapmaker, or anyone else (e.g. a physicist) who has ever spent any time thinking about curved spaces would ever refer to a great circle as a “straight line” except in a way that’s clearly metaphorical, or when addressing a younger audience that you don’t want to intimidate with fancy words (even then you’d emphasize that this is an approximate term only).
    – Dan Romik
    Apr 26 at 19:41
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    … So, while this answer and the other ones are essentially correct, I don’t agree that a great circle is “considered a straight line”. It’s considered the closest analogue (and only meaningful analogue really) to a straight line that one can find on a sphere, and it shares some properties with straight lines, but it’s not considered a straight line in a literal sense. (See also the answer quoting Steven Strogatz, a well-known mathematician, who refers to great circles as the “straightest” paths, which makes it clear he also does not consider them to be “straight lines”.)
    – Dan Romik
    Apr 26 at 19:49
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Complementing the existing answers, quoting Professor Strogatz:

For example, when I was little, my dad used to enjoy quizzing me about geography. Which is farther north, he’d ask, Rome or New York City? Most people would guess New York, but surprisingly they’re at almost the same latitude, with Rome being just a bit farther north. On the usual map of the world (the misleading Mercator projection, where Greenland appears gigantic) it looks like you could go straight from New York to Rome by heading due east.

Yet airline pilots never take that route. They always fly northeast out of New York, hugging the coast of Canada. I used to think they were staying close to land for safety’s sake, but that’s not the reason. It’s simply the most direct route, if you take the earth’s curvature into account. The shortest path from New York to Rome goes past Nova Scotia and Newfoundland, then heads out over the Atlantic, and finally veers south of Ireland and across France for arrival in sunny Italy.

This kind of path on the globe is called an arc of a “great circle.” Like straight lines in ordinary space, great circles on a sphere contain the shortest paths between any two points. They’re called “great” because they’re the largest circles you can have on a sphere. Conspicuous examples include the equator and the longitudinal circles that pass through the north and south poles.

Another property that lines and great circles share is that they’re the straightest paths. That might sound strange — all paths on a globe are curved, so what do we mean by “straightest”? Well, some paths are more curved than others. The great circles don’t do any additional curving, above and beyond what they’re forced to do by following the surface of the sphere.

Here’s a way to visualize this. Imagine you’re riding a tiny bicycle on the surface of a globe, and you’re trying to stay on a certain path. If it’s part of a great circle, you won’t ever need to steer. That’s the sense in which great circles are “straight.” In contrast, if you try to ride along a line of latitude near one of the poles, you’ll have to keep turning the handlebars.

Source: Steven Strogatz, Think globally, The New York Times, March 21, 2010.


Bold is mine.

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    I wonder why Strogatz' bicycle is tiny.
    – gerrit
    Apr 27 at 7:36

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