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Funnel Algorithm - Apex Confusion

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Hello, I've started researching into the funnel algorithm. I'm not sure what to check for to see when the apex is needed to be changed to the next vertex. In Douglas Jon Demyen's thesis, he mentions that "If the apex is popped off the deque in this way, the next vertex to be considered becomes the new apex..." So, why is the apex in the deque? Why does it need to be popped out? How do i check for the cases when the apex needs to be changed. Thing is, i've been playing around with the funnel algorithm here and i'm able to work out what will happen before the simulation plays, except i can't always predict when the apex will change.

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