Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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CENTROIDS AND SOME CHARACTERIZATIONS OF CATENARIES

  • Kim, Dong-Soo (Department of Mathematics Chonnam National University) ;
  • Moon, Hyung Tae (Department of Mathematics Chonnam National University) ;
  • Yoon, Dae Won (Department of Mathematics Education and RINS Gyeongsang National University)
  • Received : 2016.06.09
  • Accepted : 2016.09.27
  • Published : 2017.07.31
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Abstract

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 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 show that one of ${\bar{x}}_L={\bar{x}}_A$ and ${\bar{y}}_L=2{\bar{y}}_A$ characterizes the family of catenaries among nonconstant $C^2$ functions. Furthermore, we show that among nonconstant and nonlinear $C^2$ functions, ${\bar{y}}_L/{\bar{x}}_L=2{\bar{y}}_A/{\bar{x}}_A$ is also a characteristic property of catenaries.

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References

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