Areas associated with a Strictly Locally Convex Curve

DOI QR코드

DOI QR Code

Kim, Dong-Soo;Kim, Dong Seo;Kim, Young Ho;Bae, Hyun Seon

  • 투고 : 2014.11.07
  • 심사 : 2015.04.03
  • 발행 : 2016.06.23

초록

Archimedes showed that for a point P on a parabola X and a chord AB of X parallel to the tangent of X at P, the area S of the region bounded by the parabola X and chord AB is four thirds of the area T of triangle ${\Delta}ABP$. It is well known that the area U formed by three tangents to a parabola is half of the area T of the triangle formed by joining their points of contact. Recently, the first and third authors of the present paper and others proved that among strictly locally convex curves in the plane ${\mathbb{R}}^2$, these two properties are characteristic ones of parabolas. In this article, in order to generalize the above mentioned property $S={\frac{4}{3}}T$ for parabolas we study strictly locally convex curves in the plane ${\mathbb{R}}^2$ satisfying $S={\lambda}T+{\nu}U$, where ${\lambda}$ and ${\nu}$ are some functions on the curves. As a result, we present two conditions which are necessary and sufficient for a strictly locally convex curve in the plane to be an open arc of a parabola.

키워드

triangle;area;parabola;strictly locally convex curve;plane curvature

참고문헌

  1. J. -S. Baek, D. -S. Kim and Y. H. Kim, A characterization of the unit sphere, Amer. Math. Monthly, 110(9)(2003), 830-833. https://doi.org/10.2307/3647802
  2. A. Benyi, P. Szeptycki and F. Van Vleck, Archimedean properties of parabolas, Amer. Math. Monthly, 107(2000), 945-949. https://doi.org/10.2307/2695591
  3. A. Benyi, P. Szeptycki and F. Van Vleck, A generalized Archimedean property, Real Anal. Exchange, 29(2003/04), 881-889.
  4. B.-Y. Chen, D.-S. Kim and Y. H. Kim, New characterizations of W-curves, Publ. Math. Debrecen., 69/4(2006), 457-472.
  5. W. A. Day, Inequalities for areas associated with conics, Amer. Math. Monthly, 98(1)(1991), 36-39. https://doi.org/10.2307/2324034
  6. M. P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Englewood Cliffs, NJ, 1976.
  7. D.-S. Kim, A characterization of the hypersphere, Honam Math. J., 27(2)(2005), 267-271.
  8. D.-S. Kim, 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
  9. D.-S. Kim, and S. H. Kang, A characterization of conic sections, Honam Math. J., 33(3)(2011), 335-340. https://doi.org/10.5831/HMJ.2011.33.3.335
  10. D.-S. Kim and D. S. Kim, 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
  11. D.-S. Kim, D. S. Kim and Y. H. Kim, 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. D.-S. Kim, W. Kim, Y. H. Kim and D. H. Park, 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. D.-S. Kim and Y. H. Kim, A characterization of space forms, Bull. Korean Math. Soc., 35(4)(1998), 757-767.
  14. D.-S. Kim and Y. H. Kim, A characterization of ellipses, Amer. Math. Monthly, 114(1)(2007), 66-70. https://doi.org/10.1080/00029890.2007.11920393
  15. D.-S. Kim and Y. H. Kim, 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. D.-S. Kim and Y. H. Kim, Some characterizations of spheres and elliptic paraboloids, Linear Algebra Appl., 437(1)(2012), 113-120. https://doi.org/10.1016/j.laa.2012.02.013
  17. D.-S. Kim and Y. H. Kim, Some characterizations of spheres and elliptic paraboloids II, Linear Algebra Appl., 438(3)(2013), 1356-1364. https://doi.org/10.1016/j.laa.2012.08.024
  18. D.-S. Kim and Y. H. Kim, On the Archimedean characterization of parabolas, Bull. Korean Math. Soc., 50(6)(2013), 2103-2114. https://doi.org/10.4134/BKMS.2013.50.6.2103
  19. D.-S. Kim, Y. H. Kim and D. W. Yoon, 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. D.-S. Kim, J. H. Park and Y. H. Kim, Some characterizations of parabolas, Kyungpook Math. J., 53(1)(2013), 99-104. https://doi.org/10.5666/KMJ.2013.53.1.99
  21. D.-S. Kim and K.-C. Shim, 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. D.-S. Kim and B. Song, A characterization of elliptic hyperboloids, Honam Math. J., 35(1)(2013), 37-49. https://doi.org/10.5831/HMJ.2013.35.1.37
  23. J. Krawczyk, On areas associated with a curve, Zesz. Nauk. Uniw.Opol. Mat., 29(1995), 97-101.
  24. B. Richmond and T. Richmond, How to recognize a parabola, Amer. Math. Monthly, 116(10)(2009), 910-922. https://doi.org/10.4169/000298909X477023
  25. S. Stein, Archimedes. What did he do besides cry Eureka?, Mathematical Association of America, Washington, DC, 1999.

과제정보

연구 과제 주관 기관 : National Research Foundation of Korea(NRF)