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They only identify the Reddit user as "kepleronlyknows", but in the Reddit thread [1] he links to his video visualization [2], where his user name is Patrick Anderson.

Before he made the video visualization using Google Earth, he actually went to the library and used a string on a real globe. [3]

1: https://www.reddit.com/r/MapPorn/comments/15mwai/the_longest...

2: http://www.youtube.com/watch?v=VpQwuGueeoA

3: https://imgur.com/a/great-circle-2hYFM

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Adding: The two paths respectively are about 80% and 28% of the earth's circumference.
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The Portugal to (approximately) Macao one is quite interesting, thanks for the links!
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Kudos to whoever built tiled clouds background on the gcmap site.
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I opened Apple Maps in globe mode and I still have trouble visualizing the water path.
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Does it help to point out that the water path looks suspiciously like the day/night line on a map?

Like this: https://svs.gsfc.nasa.gov/5477/

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yes!! why is that ? :)
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The day/light line is a great circle. The "straight line paths" are great-circle segments.
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It's going South and then trying to close the loop from behind. It gets interrupted by land before the loop closes.

Hope it makes sense.

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Open the linked paper and scroll down, they have images of the path on a globe as well as the flat map view.
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