Interval subdivision checks whether Bézier curves intersect by replacing a difficult global comparison with progressively smaller, bounded sections. It first splits each curve at points where horizontal or vertical tangents occur, making every section monotone. section 15.4.4
A practical intersection procedure is:
- Find tangent points parallel to the coordinate axes and split the curves there. section 15.4.4
- Construct axis-aligned bounding boxes for the monotone sections and discard section pairs whose boxes do not overlap. section 15.4.4
- Approximate overlapping sections with line segments, then halve the intervals and repeat until the approximation error is acceptable. section 15.4.4
Symmetric quadratic curves
For symmetric quadratic Bézier curves, initial bounding boxes are easy to obtain. If they overlap, split at axis-parallel tangents, compare only overlapping section pairs, and iterate until segment approximation meets the error threshold.
