DOI QR코드

DOI QR Code

Center of Gravity and a Characterization of Parabolas

KIM, DONG-SOO;PARK, SOOKHEE;KIM, YOUNG HO

  • Received : 2014.10.29
  • Accepted : 2014.12.11
  • Published : 2015.06.23

Abstract

Archimedes determined the center of gravity of a parabolic section as follows. For a parabolic section between a parabola and any chord AB on the parabola, let us denote by P the point on the parabola where the tangent is parallel to AB and by V the point where the line through P parallel to the axis of the parabola meets the chord AB. Then the center G of gravity of the section lies on PV called the axis of the parabolic section with $PG=\frac{3}{5}PV$. In this paper, we study strictly locally convex plane curves satisfying the above center of gravity properties. As a result, we prove that among strictly locally convex plane curves, those properties characterize parabolas.

Keywords

Archimedes;center of gravity;area;parabolic section;locally strictly convex curve;curvature

References

  1. Bae J.-S., Kim D.-S. and Kim Y. H., A characterization of the unit sphere, Amer. Math. Monthly, 110(9)(2003), 830-833. https://doi.org/10.2307/3647802
  2. Benyi A., Szeptycki P. and Van Vleck F., Archimedean properties of parabolas, Amer. Math. Monthly, 107(2000), 945-949. https://doi.org/10.2307/2695591
  3. Benyi A., Szeptycki P. and Van Vleck F., A generalized Archimedean property, Real Anal. Exchange, 29(2003/04), 881-889.
  4. Chen B.-Y., Kim D.-S. and Kim Y. H., New characterizations of W-curves, Publ. Math. Debrecen., 69/4(2006), 457-472.
  5. do Carmo, M. P., Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, NJ, 1976.
  6. Kim D.-S., A characterization of the hypersphere, Honam Math. J., 27(2)(2005), 267-271.
  7. Kim D.-S., Ellipsoids and elliptic hyperboloids in the Euclidean space $E^{n+1}$, Linear Algebra Appl., 471 (2015), 28-45. https://doi.org/10.1016/j.laa.2014.12.014
  8. Kim D.-S. and Kang S. H., A characterization of conic sections, Honam Math. J., 33(3)(2011), 335-340. https://doi.org/10.5831/HMJ.2011.33.3.335
  9. Kim D.-S. and Kim D. S., Centroid of triangles associated with a curve, Bull. Korean Math. Soc., 52(2)(2015), 571-579. https://doi.org/10.4134/BKMS.2015.52.2.571
  10. Kim D.-S., Kim D. S., Bae H. S. and Kim H.-J., Area of triangles associated with a strictly locally convex curve, Honam Math. J., 37(1)(2015), 41-52. https://doi.org/10.5831/HMJ.2015.37.1.41
  11. Kim D.-S., Kim D. S. and Kim Young Ho, On triangles associated with a curve, Bull. Korean Math. Soc., 52(3)(2015), 925-933. https://doi.org/10.4134/BKMS.2015.52.3.925
  12. Kim D.-S., Kim W., Kim Y. H. and Park D. H., Area of triangles associated with a curve II, Bull. Korean Math. Soc., 52(1)(2015), 275-286. https://doi.org/10.4134/BKMS.2015.52.1.275
  13. Kim D.-S. and Kim Y. H., A characterization of space forms, Bull. Korean Math. Soc., 35(4)(1998), 757-767.
  14. Kim D.-S. and Kim Y. H., A characterization of ellipses, Amer. Math. Monthly, 114(1)(2007), 66-70.
  15. Kim D.-S. and Kim Y. H., New characterizations of spheres, cylinders and W-curves, Linear Algebra Appl., 432(11)(2010), 3002-3006. https://doi.org/10.1016/j.laa.2010.01.006
  16. Kim D.-S. and Kim Y. H., Some characterizations of spheres and elliptic paraboloids, Linear Algebra Appl., 437(2012), 113-120. https://doi.org/10.1016/j.laa.2012.02.013
  17. Kim D.-S. and Kim Y. H., Some characterizations of spheres and elliptic paraboloids II, Linear Algebra Appl., 438(2013), 1356-1364. https://doi.org/10.1016/j.laa.2012.08.024
  18. Kim D.-S. and Kim Y. H., On the Archimedean characterization of parabolas, Bull. Korean Math. Soc., 50(2013), 2103-2114. https://doi.org/10.4134/BKMS.2013.50.6.2103
  19. Kim D.-S., Kim Y. H. and Yoon D. W., On standard imbeddings of hyperbolic spaces in the Minkowski space, C. R. Math. Acad. Sci. Paris, Ser. I, 352(2014), 1033-1038. https://doi.org/10.1016/j.crma.2014.09.003
  20. Kim D.-S., Park J. H. and Kim Y. H., Some characterizations of parabolas, Kyungpook Math. J., 53(1)(2013), 99-104. https://doi.org/10.5666/KMJ.2013.53.1.99
  21. Kim D.-S. and Shim K.-C., Area of triangles associated with a curve, Bull. Korean Math. Soc., 51(3)(2014), 901-909. https://doi.org/10.4134/BKMS.2014.51.3.901
  22. Kim D.-S. and Song B., A characterization of elliptic hyperboloids, Honam Math. J., 35(1)(2013), 37-49. https://doi.org/10.5831/HMJ.2013.35.1.37
  23. Krawczyk J., On areas associated with a curve, Zesz. Nauk. Uniw. Opol. Mat., 29(1995), 97-101.
  24. Richmond B. and Richmond T., How to recognize a parabola, Amer. Math. Monthly, 116(2009), 910-922. https://doi.org/10.4169/000298909X477023
  25. Stein S., Archimedes. What did he do besides cry Eureka?, Mathematical Association of America, Washington, DC, 1999.

Cited by

  1. CENTROIDS AND SOME CHARACTERIZATIONS OF PARALLELOGRAMS vol.31, pp.3, 2016, https://doi.org/10.4134/CKMS.c150165
  2. Various centroids and some characterizations of catenary rotation hypersurfaces vol.42, pp.13036149, 2018, https://doi.org/10.3906/mat-1703-61

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)