Intersections of two lines

I am interested in finding all the intersection points of two x,y defined curves (several hundred x,y points each). Since many of the intersections fall between the actual data points, I have been unsuccessful using vlookup. I suspect that some preprocessing using an interpolation function may be necessary, but is currently beyond my understanding.

As a corollary to the previous post, I am interested in finding the intersection points of where the same string of data points crosses over itself. Again a data string is made up of several hundred x,y points and may cross over itself multiple times.