WALCOM 2010: WALCOM: Algorithms and Computation pp 252-262

# On Finding a Better Position of a Convex Polygon Inside a Circle to Minimize the Cutting Cost

• Syed Ishtiaque Ahmed
• Md. Mansurul Alam Bhuiyan
• Masud Hasan
• Ishita Kamal Khan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5942)

## Abstract

The problem of cutting a convex polygon P out of a planar piece of material Q (P is already drawn on Q) with minimum total cutting cost is a well studied problem in computational geometry that has been studied with several variations such as P and Q are convex or non-convex polygons, Q is a circle, and the cuts are line cuts or ray cuts. In this paper, we address this problem without the restriction that P is fixed inside Q and consider the variation where Q is a circle and the cuts are line cuts. We show that if P can be placed inside Q such that P does not contain the center of Q, then placing P in a most cornered position inside Q gives a cutting cost of 6.48 times the optimal. We also give an O(n 2)-time algorithm for finding such a position of P, a problem that may be of independent interest. When any placement of P must contain the center of Q, we show that P can be cut of Q with cost 6.054 times the optimal.

## Keywords

Polygon cutting line cut cutting cost most cornered position cornerable and non-cornerable polygon

## References

1. 1.
Ahmed, S.I., Hasan, M., Islam, M.A.: Cutting a cornered convex polygon out of a circle. Journal of Computers (to appear), http://203.208.166.84/masudhasan/cut.pdf
2. 2.
Ahmed, S.I., Hasan, M., Islam, M.A.: Cutting a convex polyhedron out of a sphere. In: 7th Japan Conference on Computational Geometry and Graphs, JCCGG 2009 (2009); Also in 4th Annual Workshop on Algorithms and Computation (WALCOM 2010), Dhaka, Bangladesh, February 10-12, 2010, http://arxiv.org/abs/0907.4068
3. 3.
Bereg, S., Daescu, O., Jiang, M.: A PTAS for cutting out polygons with lines. In: Chen, D.Z., Lee, D.T. (eds.) COCOON 2006. LNCS, vol. 4112, pp. 176–185. Springer, Heidelberg (2006)
4. 4.
Bhadury, J., Chandrasekaran, R.: Stock cutting to minimize cutting length. Euro. J. Oper. Res. 88, 69–87 (1996)
5. 5.
Daescu, O., Luo, J.: Cutting out polygons with lines and rays. International Journal of Computational Geometry and Applications 16, 227–248 (2006)
6. 6.
Demaine, E.D., Demaine, M.L., Kaplan, C.S.: Polygons cuttable by a circular saw. Computational Geometry: Theory and Algorithms 20, 69–84 (2001)
7. 7.
Dumitrescu, A.: An approximation algorithm for cutting out convex polygons. Computational Geometry: Theory and Algorithms 29, 223–231 (2004)
8. 8.
Dumitrescu, A.: The cost of cutting out convex n-gons. Discrete Applied Mathematics 143, 353–358 (2004)
9. 9.
Jaromczyk, J.W., Kowaluk, M.: Sets of lines and cutting out polyhedral objects. Computational Geometry: Theory and Algorithms 25, 67–95 (2003)
10. 10.
Overmars, M.H., Welzl, E.: The complexity of cutting paper. In: SoCG 1985, pp. 316–321 (1985)Google Scholar
11. 11.
Tan, X.: Approximation algorithms for cutting out polygons with lines and rays. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 534–543. Springer, Heidelberg (2005)
12. 12.
Toussaint, G.: Solving geometric problems with the rotating calipers. In: MELECON 1983, Athens, Greece (1983)Google Scholar

## Authors and Affiliations

• Syed Ishtiaque Ahmed
• 1
• Md. Mansurul Alam Bhuiyan
• 1
• Masud Hasan
• 1
• Ishita Kamal Khan
• 1
1. 1.Department of Computer Science and EngineeringBangladesh University of Engineering and TechnologyDhakaBangladesh