Center of Gravity and a Characterization of Parabolas
0
조회
0
다운로드
12
인용
0
참고문헌
키워드
초록
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 = 3 5PV . 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.
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 = 3 5PV . 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.
참고문헌 (25)
- Bae J. -S., "A characterization of the unit sphere", Amer. Math. Monthly, 110(9), pp.830~833 (2003)
- B´enyi ´A., "Archimedean properties of parabolas", Amer. Math. Monthly, 107, pp.945~949 (2000)
- B´enyi ´A., "A generalized Archimedean property", Real Anal. Exchange, 29(4), pp.881~889 (2003)
- Chen B. -Y., "New characterizations of W-curves", Publ. Math. Debrecen, 69(4), pp.457~472 (2006)
- do Carmo., "Differential Geometry of Curves and Surfaces" (1976), Prentice-Hall
- 김동수, "A characterization of the hypersphere", 호남수학학술지, 27(2), pp.267~271 (2005), 호남수학회
- Kim D. -S., "Ellipsoids and elliptic hyperboloids in the Euclidean space En+1", Linear Algebra Appl, 471, pp.28~45 (2015)
- 김동수, "A CHARACTERIZATION OF CONIC SECTIONS", 호남수학학술지, 33(3), pp.335~340 (2011), 호남수학회
- 김동수, "CENTROID OF TRIANGLES ASSOCIATED WITH A CURVE", 대한수학회보, 52(2), pp.571~579 (2015), 대한수학회
- 김동수, "Area of triangles associated with a strictly locally convex curve", 호남수학학술지, 37(1), pp.41~52 (2015), 호남수학회
- 김동수, "ON TRIANGLES ASSOCIATED WITH A CURVE", 대한수학회보, 52(3), pp.925~933 (2015), 대한수학회
- 김동수, "Area of triangles associated with a curve II", 대한수학회보, 52(1), pp.275~286 (2015), 대한수학회
- Kim D. -S. ., "A characterization of space forms", Bull. Korean Math. Soc, 35(4), pp.757~767 (1998)
- Kim D. -S. ., "A characterization of ellipses", Amer. Math. Monthly, 114(1), pp.66~70 (2007)
- Kim D. -S. ., "New characterizations of spheres, cylinders and W-curves", Linear Algebra Appl, 432(11), pp.3002~3006 (2010)
- Kim D. -S. ., "Some characterizations of spheres and elliptic paraboloids", Linear Algebra Appl, 437, pp.113~120 (2012)
- Kim D. -S. ., ""Some characterizations of spheres and elliptic paraboloids II", Linear Algebra Appl, 438, pp.1356~1364 (2013)
- 김동수, "On the Archimedean characterization of parabolas", 대한수학회보, 50(6), pp.2103~2114 (2013), 대한수학회
- Kim D. -S., "On standard imbeddings of hyperbolic spaces in the Minkowski space", C. R. Math. Acad. Sci. Paris, Ser. I, 352, pp.1033~1038 (2014)
- 김동수, "Some Characterizations of Parabolas", Kyungpook Mathematical Journal, 53(1), pp.99~104 (2013), 경북대학교 자연과학대학 수학과
- 김동수, "Area of triangles associated with a curve", 대한수학회보, 51(3), pp.901~909 (2014), 대한수학회
- 김동수, "A CHARACTERIZATION OF ELLIPTIC HYPERBOLOIDS", 호남수학학술지, 35(1), pp.37~49 (2013), 호남수학회[DOI]
- Krawczyk J., "On areas associated with a curve", Zesz. Nauk. Uniw. Opol. Mat, 29, pp.97~101 (1995)
- Richmond B. ., "How to recognize a parabola", Amer. Math. Monthly, 116, pp.910~922 (2009)
- Stein S., "Archimedes. What did he do besides cry Eureka?" (1999), Mathematical Association of America