대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제33권1호
  • /
  • Pages.237-245
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  • 2018
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  • 1225-1763(pISSN)
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  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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VARIOUS CENTROIDS AND SOME CHARACTERIZATIONS OF CATENARY CURVES

  • Bang, Shin-Ok (Department of Mathematics Chonnam National University) ;
  • Kim, Dong-Soo (Department of Mathematics Chonnam National University) ;
  • Yoon, Dae Won (Department of Mathematics Education and RINS Gyeongsang National University)
  • 투고 : 2017.02.11
  • 심사 : 2017.05.11
  • 발행 : 2018.01.31
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초록

For every interval [a, b], we denote by $({\bar{x}}_A,{\bar{y}}_A)$ and $({\bar{x}}_L,{\bar{y}}_L)$ the geometric centroid of the area under a catenary curve y = k cosh((x-c)/k) defined on this interval and the centroid of the curve itself, respectively. Then, it is well-known that ${\bar{x}}_L={\bar{x}}_A$ and ${\bar{y}}_L=2{\bar{y}}_A$. In this paper, we fix an end point, say 0, and we show that one of ${\bar{x}}_L={\bar{x}}_A$ and ${\bar{y}}_L=2{\bar{y}}_A$ for every interval with an end point 0 characterizes the family of catenaries among nonconstant $C^2$ functions.

키워드

참고문헌

  1. V. Coll and M. Harrison, Two generalizations of a property of the catenary, Amer. Math. Monthly 121 (2014), no. 2, 109-119. https://doi.org/10.4169/amer.math.monthly.121.02.109
  2. M. J. Kaiser, The perimeter centroid of a convex polygon, Appl. Math. Lett. 6 (1993), no. 3, 17-19. https://doi.org/10.1016/0893-9659(93)90025-I
  3. B. Khorshidi, A new method for nding the center of gravity of polygons, J. Geom. 96 (2009), no. 1-2, 81-91. https://doi.org/10.1007/s00022-010-0027-1
  4. D.-S. Kim and D. S. Kim, Centroid of triangles associated with a curve, Bull. Korean Math. Soc. 52 (2015), no. 2, 571-579. https://doi.org/10.4134/BKMS.2015.52.2.571
  5. D.-S. Kim, W. Kim, K. S. Lee, and D. W. Yoon, Various centroids of polygons and some characterizations of rhombi, Commun. Korean Math. Soc. 32 (2017), no. 1, 135-145. https://doi.org/10.4134/CKMS.c160023
  6. D.-S. Kim and Y. H. Kim, On the Archimedean characterization of parabolas, Bull. Korean Math. Soc. 50 (2013), no. 6, 2103-2114. https://doi.org/10.4134/BKMS.2013.50.6.2103
  7. D.-S. Kim, Y. H. Kim, and S. Park, Center of gravity and a characterization of parabolas, Kyungpook Math. J. 55 (2015), no. 2, 473-484. https://doi.org/10.5666/KMJ.2015.55.2.473
  8. D.-S. Kim, Y. H. Kim, and D. W. Yoon, Some characterizations of catenary rotation surfaces, submitted.
  9. D.-S. Kim, K. S. Lee, K. B. Lee, Y. I. Lee, S. Son, J. K. Yang, and D. W. Yoon, Centroids and some characterizations of parallelograms, Commun. Korean Math. Soc. 31 (2016), no. 3, 637-645. https://doi.org/10.4134/CKMS.c150165
  10. D.-S. Kim, H. T. Moon, and D. W. Yoon, Centroids and some characterizations of catenaries, Commun. Korean Math. Soc., to appear.
  11. S. G. Krantz, A matter of gravity, Amer. Math. Monthly 110 (2003), no. 6, 465-481. https://doi.org/10.1080/00029890.2003.11919985
  12. S. G. Krantz, J. E. McCarthy, and H. R. Parks, Geometric characterizations of centroids of simplices, J. Math. Anal. Appl. 316 (2006), no. 1, 87-109. https://doi.org/10.1016/j.jmaa.2005.04.046
  13. E. Parker, A property characterizing the catenary, Math. Mag. 83 (2010), 63-64. https://doi.org/10.4169/002557010X485120
  14. S. Stein, Archimedes. What did he do besides cry Eureka?, Mathematical Association of America, Washington, DC, 1999.