Determining Intersection of Two Line Segments in the Same Plane

In computational geometry, determining whether two line segments intersect is a fundamental problem. The core solution leverages the properties of vector cross products to detect the relative positional relationship between segments.

To determine if two line segments intersect, we first exclude obviously non-intersecting cases through: Rapid Rejection Test: If the minimum bounding rectangles of the two segments don't overlap, the segments cannot intersect. This step quickly filters out clearly separated cases through simple coordinate comparisons. Cross Product Test: This is the key intersection detection step. By calculating vector cross products, we determine whether the segments "straddle" each other. Specifically, if segments AB and CD intersect, point C and D must lie on opposite sides of AB, while point A and B must lie on opposite sides of CD. The implementation involves checking if the cross products (AB×AC) and (AB×AD) have opposite signs, and similarly for (CD×CA) and (CD×CB).

Critical edge cases require special handling in code implementation: Collinear Case: When segments lie on the same line, additional checks are needed for overlap detection by comparing coordinate ranges Endpoint Coincidence: When an endpoint of one segment lies exactly on the other segment, this counts as intersection Parallel Non-Collinear: Such segments never intersect due to their constant separation

This method achieves high computational efficiency, requiring only a few vector operations (typically 4-6 cross product calculations) to determine intersection status, making it suitable for most geometric computing applications including computer graphics and collision detection algorithms.