A more efficient convex hull algorithm

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Before I start, I would like to say that this is for a programming project of mine I'm doing but I figured my question is only about the part involving math so here I am. So in Grahams Scan algorithm, you have to find the point with the lowest y coordinate and start with that. But why not just start from the point with the lowest polar angle?In Grahams Scan, you need to sort the points based on their polar angle so you are not doing anything extra in the algorithm I mentioned. P.S: Although finding the lowest item in a set is real fast, in HUGE cases it takes A LOT of time.