Finding intersections of numerically implemented 1-dimensional curves on a 2-dimensional plane
http://en.wikipedia.org/wiki/Sweep_line_algorithm

(Or open any standard book on Computational Geometry).

For practical implemented algorithms, see CGAL http://www.cgal.org/

If your main problem is numerical issues, then this can be challenging in practice. A lot of practical work on this problem was done by Dan Halperin in Tel-Aviv University.