# Algorithms for Reporting and Counting Geometric Intersections

```@article{Bentley1979AlgorithmsFR,
title={Algorithms for Reporting and Counting Geometric Intersections},
author={Jon Louis Bentley and Thomas Ottmann},
journal={IEEE Transactions on Computers},
year={1979},
volume={C-28},
pages={643-647}
}```
• Published 1 September 1979
• Computer Science, Mathematics
• IEEE Transactions on Computers
An interesting class of "geometric intersection problems" calls for dealing with the pairwise intersections among a set of N objects in the plane, These problems arise in many applications such as printed circuit design, architectural data bases, and computer graphics. Shamos and Hoey have described a number of algorithms for detecting whether any two objects in a planar set intersect. In this paper we extend their work by giving algorithms that count the number of such intersections and…
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## References

SHOWING 1-6 OF 6 REFERENCES
Geometric intersection problems
• Computer Science, Mathematics
17th Annual Symposium on Foundations of Computer Science (sfcs 1976)
• 1976
An O(N log N) algorithm is given to determine whether any two intersect and use it to detect whether two simple plane polygons intersect and to show that the Simplex method is not optimal.
Fast algorithms for LSI artwork analysis
• H. Baird
• Art, Computer Science
DAC '77
• 1977
A novel IC mask analysis algorithm is described which does not restrict the representation of artwork, permits a wide range of functions, avoids pathologies, and achieves good runtime and tolerable main memory demands, even for LSI applications.
Eine neue Klass von ausgeglichenen Binarbaumen
• Angewandte Informatik, vol. 9, pp. 395-400, 1976.
• 1976
Four-dimensional binary search trees as a means to speed up associative searches in design rule verification of integrated circuits
• J. Des. Autom. and Fault-Tolerant Comput., vol. 2, pp. 241-247, July 1978.
• 1978