How to follow a line while moving in (x,y) plane?

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In short what i want is to follow a line like the guy in this video:
https://www.youtube.com/watch?v=iHiasfAE63k
He talked a little bit about his algorithm in the description and in the comment section, but he wasn't that clear, since This type of following the line has great applications, i thought you guys might help me shed some light on it.

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