Document

# Maximum Overlap Area of a Convex Polyhedron and a Convex Polygon Under Translation

## File

LIPIcs.SoCG.2023.61.pdf
• Filesize: 0.78 MB
• 16 pages

## Acknowledgements

This paper is the result of the MIT SPUR 2022, a summer undergraduate research program organized by the MIT math department. The authors would like to thank the faculty advisors David Jerison and Ankur Moitra for their support and the math department for providing this research opportunity. We thank the anonymous referees for providing helpful comments that increased the quality of this paper.

## Cite As

Honglin Zhu and Hyuk Jun Kweon. Maximum Overlap Area of a Convex Polyhedron and a Convex Polygon Under Translation. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 61:1-61:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SoCG.2023.61

## Abstract

Let P be a convex polyhedron and Q be a convex polygon with n vertices in total in three-dimensional space. We present a deterministic algorithm that finds a translation vector v ∈ ℝ³ maximizing the overlap area |P ∩ (Q + v)| in O(n log² n) time. We then apply our algorithm to solve two related problems. We give an O(n log³ n) time algorithm that finds the maximum overlap area of three convex polygons with n vertices in total. We also give an O(n log² n) time algorithm that minimizes the symmetric difference of two convex polygons under scaling and translation.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
##### Keywords
• computational geometry
• shape matching
• arrangement

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Hee-Kap Ahn, Peter Brass, and Chan-Su Shin. Maximum overlap and minimum convex hull of two convex polyhedra under translations. Comput. Geom., 40(2):171-177, 2008. URL: https://doi.org/10.1016/j.comgeo.2007.08.001.
2. Hee-Kap Ahn, Siu-Wing Cheng, Hyuk Jun Kweon, and Juyoung Yon. Overlap of convex polytopes under rigid motion. Comput. Geom., 47(1):15-24, 2014. URL: https://doi.org/10.1016/j.comgeo.2013.08.001.
3. Hee-Kap Ahn, Siu-Wing Cheng, and Iris Reinbacher. Maximum overlap of convex polytopes under translation. Comput. Geom., 46(5):552-565, 2013. URL: https://doi.org/10.1016/j.comgeo.2011.11.003.
4. Hee-Kap Ahn, Otfried Cheong, Chong-Dae Park, Chan-Su Shin, and Antoine Vigneron. Maximizing the overlap of two planar convex sets under rigid motions. Comput. Geom., 37(1):3-15, 2007. URL: https://doi.org/10.1016/j.comgeo.2006.01.005.
5. David Avis, Prosenjit Bose, Thomas C. Shermer, Jack Snoeyink, Godfried Toussaint, and Binhai Zhu. On the sectional area of convex polytopes. In Communication at the 12th Annu. ACM Sympos. Comput. Geom., page C. Association for Computing Machinery, New York, NY, 1996.
6. Mark de Berg, Olivier Devillers, Marc van Kreveld, Otfried Schwarzkopf, and Monique Teillaud. Computing the maximum overlap of two convex polygons under translations. In International Symposium on Algorithms and Computation, pages 126-135. Springer, 1996.
7. Bernard Chazelle. An optimal algorithm for intersecting three-dimensional convex polyhedra. SIAM J. Comput., 21(4):671-696, 1992. URL: https://doi.org/10.1137/0221041.
8. Bernard Chazelle. Cutting hyperplanes for divide-and-conquer. Discrete Comput. Geom., 9(2):145-158, 1993. URL: https://doi.org/10.1007/BF02189314.
9. Bernard Chazelle. An optimal convex hull algorithm in any fixed dimension. Discrete Comput. Geom., 10(4):377-409, 1993. URL: https://doi.org/10.1007/BF02573985.
10. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to algorithms. MIT Press, Cambridge, MA, third edition, 2009.
11. Greg N. Frederickson and Donald B. Johnson. Generalized selection and ranking: sorted matrices. SIAM J. Comput., 13(1):14-30, 1984. URL: https://doi.org/10.1137/0213002.
12. Hyuk Jun Kweon and Honglin Zhu. Maximum overlap area of a convex polyhedron and a convex polygon under translation, 2023. URL: https://doi.org/10.48550/ARXIV.2301.02949.
13. Nimrod Megiddo. Linear programming in linear time when the dimension is fixed. J. Assoc. Comput. Mach., 31(1):114-127, 1984. URL: https://doi.org/10.1145/2422.322418.
14. Juyoung Yon, Sang Won Bae, Siu-Wing Cheng, Otfried Cheong, and Bryan T. Wilkinson. Approximating convex shapes with respect to symmetric difference under homotheties. In 32nd International Symposium on Computational Geometry, volume 51 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 63, 15. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2016.
X

Feedback for Dagstuhl Publishing