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Following a curve

I am totally new to image recognition and image analysis and have a project concerning detecting a smooth curve on a uniform background (white/black/any color) and then be able to follow this curve until its end. The curve may cross itself but the path that should be recognized should follow along the "smoothness" of the curve. Also, I want to be able to recognize and extract positions (x:s and y:s, i.e. pixel position I guess) of where the curve might turn left or right.
Since I am new to the area, which seem to be large enough to just become confused, I´m not sure of were to begin reading so any literature or tutorials or whatever is highly appreciated. What I need is some kind of starting point (of course, if anyone know if it is already done somewhere that even better!), what libraries or software to use. Anything is helpful!