Crossing without slopes
Two line segments either cross, touch, or miss each other. Computing slopes invites division by zero for vertical lines and rounding trouble. The robust approach uses the orientation test on the endpoints, so the whole decision is signed area comparisons.
The general case
Label the segments and test how each segment's endpoints sit relative to the other:
- Compute the orientation of the first segment with each endpoint of the second.
- Compute the orientation of the second segment with each endpoint of the first.
- If the two orientations differ in sign on both checks, the segments straddle each other and therefore intersect.
This straddling rule captures every proper crossing where the segments pass through each other's interior.
Collinear and touching cases
When an orientation comes out zero, the points are collinear and you must check whether an endpoint actually lies on the other segment, using a bounding box style range check. This handles overlaps and shared endpoints, the cases that trip up naive code. Treating these explicitly keeps the test correct on the boundary, not just in the clean middle case.
Key idea
Two segments cross when each straddles the other in the orientation tests, with collinear zeros handled by an explicit on segment range check.