Draw a straight line between A and B on a globe. Draw a straight line between the same A and B on a flat map. (In both cases, “straight line" means the shortest distance between those points). Will the two lines match? Meaning, will they trace out the same path, pass through the same points, in both cases?

The surprising answer: probably not. Try it yourself. Pick Mumbai and New York, let’s say. On an atlas I have, the line between them passes through Oman, Saudi Arabia, Egypt, Libya, Algeria and Morocco before crossing the Atlantic to end in New York City (NYC). You’re probably nodding your head—if you know your geography, that must seem about right. But on a globe I also have, the straight line goes through Pakistan, Afghanistan, Turkmenistan, Kazakhstan, Russia, the three Scandinavian countries and Iceland, then flirts with the southern tip of Greenland and the coast of north-east Canada before touching down in NYC.

Totally different! But here’s something that might surprise you even more. The straight line on the atlas is about 13,000km-long. The straight line on the globe? About 8,000km. Would you have imagined such a drastic difference?

You probably know why this is so. On a globe like the Earth, the shortest path between any two points lies on what’s called a “great circle". When you project a globe onto a flat surface to make an atlas, distortions are inevitable. (One reason that you probably think of Greenland as much larger than India, though actually it is only about two-thirds India’s size). In particular, one of these distortions is that straight lines on the globe are curved when laid flat. As a corollary, shortest distances on a map turn out significantly longer when traced on a globe. 

This mention of straight lines on the globe got some people thinking about two intriguing problems: what’s the longest distance you can travel in a straight line on Earth so that you’re on land the whole time—or at least, without crossing a major body of water? And similarly, what’s the longest straight line you can travel on water?

On a globe, that’s a straight line. The longest straight line on land, from China to Portugal.
On a globe, that’s a straight line. The longest straight line on land, from China to Portugal.

The first of these is hard to answer, because water is everywhere in some form or the other. Leave out the oceans, fine. But is the Caspian Sea a “major body of water"? What about Lake Baikal in Irkutsk? The Salton Sea in California? The talao, or large pond, near Bandra Station? What about the Brahmaputra? Mithi River in Mumbai? 

In 2010, a Canadian researcher, Guy Bruneau, proposed such a straight line. It stretches from Liberia to the east coast of China. Crucially, it crosses from Africa into the Middle East and Asia via the tiny isthmus that separates the Red Sea from the Mediterranean. Bruneau commented: “A 13,589.31 km walk in straight line! It crosses 9 time zones and 18 countries and territories. The road goes like this: Liberia, Côte d’Ivoire, Burkina Faso, Ghana, Burkina Faso again, Niger, Chad, Libya, Egypt, Israel, the West Bank, Jordan, Iraq, Iran, Turkmenistan, Uzbekistan, Tajikistan, Afghanistan, Tajikistan again and finally China." (See the line here on a flat projection of that part of the Earth’s surface, that tries to account for the Earth’s curvature: is.gd/joXbba)

If a straight line between two points on the atlas is 13,000km, what about on a globe? About 8,000km, a drastic difference-

The line skims past the Mediterranean Sea, the Gulf of Suez and the Caspian Sea, so it seems plausible that it doesn’t cross a major body of water. But wait a minute…in 1869, the Suez Canal sliced through that isthmus. And about 250km east of there, the line crosses the Dead Sea. Are either the canal or that sea a major body of water?

Opinions about that might vary, of course. The second question—the longest straight line on water—may be less contentious. In May, Rohan Chabukswar and Kushal Mukherjee published a paper (Longest Straight Line Paths on Water or Land on the Earth) that responded to a 2012 Reddit post claiming to have found the longest straight line on water.

Their conclusion? Yes indeed: there is such a straight line you can draw on a globe. It starts on the coast of Pakistan west of Karachi, plunges about south-west and between Africa and Madagascar, dances between a finger of Antarctica and Tierra del Fuego, then streaks north-west across the Pacific to make landfall on Russia’s Kamchatka Peninsula. Between Pakistan and Russia on this line, only water. (Seawater, at that).

Of course, on a map (see image 1) that looks nothing like a straight line. It resembles instead the outline of a gracefully curved bowl. But if you plot it on a globe, you’ll see for yourself: it’s a better-than-75%-complete great circle, a straight line over 32,000km-long. That is, if you set out from that point on the Pakistan coast in a boat in a nearly south-west direction and make sure the boat never deviates from that straight line as it barrels through the waves, you’d circle the globe and finally wash ashore on Kamchatka, never having touched land all the way. The paper that Chabukswar and Mukherjee wrote confirms the Reddit claim and shows that this is the longest such straight line you can mark out on our planet.

But what’s fascinating is how Chabukswar and Mukherjee went about this task. Think about it for a moment: how would you search for a great circle that includes a large water-only segment? 

To begin with, there is an infinite number of great circles you can draw on the globe. No way to check infinite circles, so the first task is to choose a finite number of them that is a reasonable approximation to the whole globe. Chabukswar and Mukherjee used an available model of the Earth’s surface that grids it with a separation of one arcminute (1/60th of a degree; that is, there are 21,600 arcminutes in the 360 degrees of a full circle). This corresponds to a distance of just under 2km at the equator. Finite for sure, but as the pair writes, that still amounts to “233,280,000 great circles to consider". (I’ll spare you the arithmetic that produced that number).

When you project a globe like the Earth onto a flat surface to make an atlas, distortions are inevitable-

For each point on the grid, the model tells us its altitude, meaning how much above or below sea level it is. The authors assumed that if a point was above sea level, it was on land; below would be water. They write: “Although this is not strictly true in general, in the absence of actual data it is the closest approximation to real relief that is freely available." Each great circle will have 21,600 such points to check for the kind of path we’re interested in—which leaves us with “a staggering 5,038,848,000,000 points to verify". (That’s better than 5 trillion). 

How staggering is that? A computer verifying a million points a second would still take two months to complete the job. Nevertheless, Chabukswar and Mukherjee actually tried to programme a computer to do this exhaustively. But they had to give up. Instead, they then turned to a well-known computer science technique called “branch-and-bound". In essence, branch-and-bound manages to decrease the number of points to be checked by finding ways to skip those that won’t produce better results than we have already—in this case, a longer water-only path. (I’ll also spare you these details). 

The result was spectacular. “The algorithm returned the longest path"—the Pakistan to Kamchatka cruise—“in about 10 minutes of computation…on a standard laptop."

Not two months, but 10 minutes. 

Chabukswar and Mukherjee also set their algorithm to work on the first question, about the longest straight line on land. With their assumption about the altitude of a given point, they did not have to make decisions about major bodies of water. Sure enough, the algorithm found a path (this search took 45 minutes). It stretches from Portugal through several European countries, then Russia, Kazakhstan and Mongolia to end, like Bruneau’s line, on the coast of China. At 11,241km, it isn’t as long as Bruneau’s path — but then it does not cross bodies of water like the Dead Sea. Which is why Chabukswar and Mukherjee call it the “longest drivable straight line path on Earth." (See image 2). Their water-only path is, you guessed it, the “longest sailable straight line path on Earth".

What’s the point of all this? Well, there is the challenge of working out the algorithm. In doing so, mathematicians and computer scientists also identify the algorithm’s limitations and what will make it work better. Three limitations, here. The authors’ assumption about altitude “unfortunately discounts, among other features, highland rivers and low-lying plains." The one arcminute resolution means there are possible paths—in those one arcminute gaps—that the algorithm may miss altogether. And the Earth is not perfectly spherical, so great circles may not necessarily be straight lines. 

Any future work will have to account and correct for these limitations—itself another challenge. But Chabukswar and Mukherjee put this in perspective by pointing out that addressing all this “does seem to be an overkill for what is essentially a recreational problem".

Not, let it be said, that a recreational problem cannot teach you things. I’d like to sail or drive those lines, recreationally, and see what I learn.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners.
His Twitter handle is @DeathEndsFun

Image courtesy: Longest Straight Line Paths on Water or Land on the Earth, by Rohan Chabukswar and Kushal Mukherjee, arXiv:1804.07389 [math.HO]

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