Abstract
In this paper, a new computational algorithm is proposed for accurately determining the Smallest Enclosing Circle (SEC) of a finite point set (P) in plane. The set P that we are concerned here contains more than two non-collinear points which is a typical case. The algorithm basically searches for three particular points from P that forms the desired SEC of the set P. The SEC solution space of arbitrary P is uniform under this algorithm. The algorithmic mechanism is simple and it can be easily programmed. Our analysis proved that algorithm is robust and our empirical study verified its effectiveness. The computational complexity of the algorithm is found to be O(nlogn).
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References
Weckenmann, A., Eitzert, H., Garmer, M., Weber, H.: Functionality-oriented evaluation and sampling strategy in coordinate metrology. Precis. Eng. 17(4), 244–252 (1995)
Huang, X., Gu, P.: CAD-model based inspection of sculptured surfaces with datum. Int. J. Prod. Res. 36(5), 1351–1367 (1998)
Elzinga, D.J., Hearn, D.W.: Geometrical solutions for some minimax location problems. Transp. Sci. 6(4), 379–394 (1972)
Oommen, B.J.: A learning automaton solution to the stochastic minimum spanning circle problem. IEEE Trans. Syst. Man Cybern. 16(4), 598–603 (1986)
Chakraborty, R.K., Chaudhuri, P.K.: Note on geometrical solution for some minimax location problems. Transp. Sci. 15(2), 164–166 (1981)
Oommen, B.J.: An efficient geometric solution to the minimum spanning circle problem. Oper. Res. 35(1), 80–86 (1987)
Chrystal, G.: On the problem to construct the minimum circle enclosing N given points in the plane. Proc. Edinburgh Math. Soc. 3, 30–33 (1885)
Li, X., Shi, Z.: The relationship between the minimum zone circle and the maximum inscribed circle and the minimum circumscribed circle. Precis. Eng. 33(3), 284–290 (2009)
Jywe, W.Y., Liu, C.H., Chen, C.K.: The min-max problem for evaluating the form error of a circle. Measurement 26(4), 777–795 (1999)
Shunmugam, M.S., Venkaiah, N.: Establishing circle and circular-cylinder references using computational geometric techniques. Int. J. Adv. Manuf. Technol. 51(1), 261–275 (2010)
Gadelmawla, E.S.: Simple and efficient algorithms for roundness evaluation from the coordinate measurement date. Measurement 43(2), 223–235 (2010)
Li, X., Shi, Z.: Development and application of convex hull in the assessment of roundness error. Int. J. Mach. Tools Manuf. 48(1), 135–139 (2008)
Lei, X., Zhang, C., Xue, Y., Li, J.: Roundness error evaluation algorithm based on polar coordinate transform. Measurement 44(2), 345–350 (2011)
Nair, K.P.K., Chandrasekaran, R.: Optimal location of a single service center of certain types. Naval Res. Logist. Q. 18, 503–510 (1971)
Chen, M.C., Tsai, D.M., Tseng, H.Y.: A stochastic optimization approach for roundness measurements. Pattern Recogn. Lett. 20(7), 707–719 (1999)
Goch, G., Lübke, K.: Tschebyscheff approximation for the calculation of maximum inscribed/minimum circumscribed geometry elements and form deviations. CIRP Ann. Manuf. Technol. 57(1), 517–520 (2008)
Anthony, G.T., Anthony, H.M., Bittner, B., Butler, B.P., Cox, M.G., Drieschner, R., et al.: Reference software for finding Chebyshev best-fit geometric elements. Precis. Eng. 19(1), 28–36 (1996)
Zhou, P.: An algorithm for determining the vertex of the convex hull. J. Beijing Inst. Technol. 13(1), 69–72 (1993)
Graham, R.L.: An efficient algorithm for determining the convex hull of a finite planar set. Info. Proc. Lett. 1, 132–133 (1972)
Zhou, P.: Computational geometry algorithm design and analysis, 4th edn, pp. 79–81. Tsinghua University Press, Beijing (2011)
Klein, J.: Breve: a 3D environment for the simulation of decentralized systems and artificial life. In: Proceedings of Artificial Life VIII, the 8th International Conference on the Simulation and Synthesis of Living Systems. The MIT Press (2002)
Acknowledgements
The authors appreciate the financial support from Doctor Scientific Research Startup Project of Hanshan Normal University (No. QD20140116). This work is also partly supported by the 2013 Comprehensive Specialty (Electronic Information Science and Technology) Reform Pilot Projects for Colleges and Universities granted by the Chinese Ministry of Education (No. ZG0411) and the Education Department of Guangdong Province in China (No. [2013]322).
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Li, X., Ercan, M.F. (2016). An Algorithm for Smallest Enclosing Circle Problem of Planar Point Sets. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2016. ICCSA 2016. Lecture Notes in Computer Science(), vol 9786. Springer, Cham. https://doi.org/10.1007/978-3-319-42085-1_24
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