Areas associated with a Strictly Locally Convex Curve

• Journal title : Kyungpook mathematical journal
• Volume 56, Issue 2,  2016, pp.583-595
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2016.56.2.583
Title & Authors
Areas associated with a Strictly Locally Convex Curve
Kim, Dong-Soo; Kim, Dong Seo; Kim, Young Ho; Bae, Hyun Seon;

Abstract
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 $\small{{\Delta}ABP}$. 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 $\small{{\mathbb{R}}^2}$, these two properties are characteristic ones of parabolas. In this article, in order to generalize the above mentioned property $\small{S={\frac{4}{3}}T}$ for parabolas we study strictly locally convex curves in the plane $\small{{\mathbb{R}}^2}$ satisfying $\small{S={\lambda}T+{\nu}U}$, where $\small{{\lambda}}$ and $\small{{\nu}}$ 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.
Keywords
triangle;area;parabola;strictly locally convex curve;plane curvature;
Language
English
Cited by
References
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.

2.
A. Benyi, P. Szeptycki and F. Van Vleck, Archimedean properties of parabolas, Amer. Math. Monthly, 107(2000), 945-949.

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.

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.

9.
D.-S. Kim, and S. H. Kang, A characterization of conic sections, Honam Math. J., 33(3)(2011), 335-340.

10.
D.-S. Kim and D. S. Kim, Centroid of triangles associated with a curve, Bull. Korean Math. Soc., 52(2)(2015), 571-579.

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.

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.

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.

15.
D.-S. Kim and Y. H. Kim, New characterizations of spheres, cylinders and W-curves, Linear Algebra Appl., 432(11)(2010), 3002-3006.

16.
D.-S. Kim and Y. H. Kim, Some characterizations of spheres and elliptic paraboloids, Linear Algebra Appl., 437(1)(2012), 113-120.

17.
D.-S. Kim and Y. H. Kim, Some characterizations of spheres and elliptic paraboloids II, Linear Algebra Appl., 438(3)(2013), 1356-1364.

18.
D.-S. Kim and Y. H. Kim, On the Archimedean characterization of parabolas, Bull. Korean Math. Soc., 50(6)(2013), 2103-2114.

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.

20.
D.-S. Kim, J. H. Park and Y. H. Kim, Some characterizations of parabolas, Kyungpook Math. J., 53(1)(2013), 99-104.

21.
D.-S. Kim and K.-C. Shim, Area of triangles associated with a curve, Bull. Korean Math. Soc., 51(3)(2014), 901-909.

22.
D.-S. Kim and B. Song, A characterization of elliptic hyperboloids, Honam Math. J., 35(1)(2013), 37-49.

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.

25.
S. Stein, Archimedes. What did he do besides cry Eureka?, Mathematical Association of America, Washington, DC, 1999.