For a personal project, I\'d need to find out if two cubic Bézier curves intersect. I don\'t need to know where: I just need to know if they do. However, I\'d need to do it
I'm by no way an expert on this kind of thing, but I follow a nice blog that talks a lot about curves. He has link to two nice articles talking about your problem (the second link has an interactive demonstration and some source code). Other people may have much better insight into the problem but I hope this helps!
http://cagd.cs.byu.edu/~557/text/ch7.pdf