Well, the first part of this problem to consider, that makes it much more difficult, is that you are actually talking about the intersection of line segments, not lines themselves. This defeats the elementary proposition that two lines in a plane WILL intersect somewhere if their slopes are inequal.

However, there is a "brute-force" approach that will eliminate most of the possible pairs to consider in a pairwise approach: instead of looking at the segments pairwise to determine whether they intersect, look at the bounding rectangles of the segments pairwise to see if they overlap. Then, run the pairwise segment comparison only on this smaller subset.

For the purpose of this proposition, suppose that in some orthogonal, two-dimensional coordinate system (can you tell I really was a geometer at one point in my life?), the endpoints of a line segment are described by (a,b) and (p,q). Then the bounding rectangle of the line segment {(a,b),(p,q)} is the rectangle described by the point-set {(a,b),(a,q),(p,q),(p,b)}. It is a trivial exercise to demonstrate that if two line segments intersect, their bounding rectangles overlap; what we are actually interested in then is the contrapositive of that result: if their bounding rectangles do not overlap, two line segments do not intersect.

Spud Zeppelin * spud@spudzeppelin.com