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It is well known that the area U of the triangle formed by three tangents to a parabola X is half of the area T of the triangle formed by joining their points of contact. Recently, it was proved that this property is a characteristic one of parabolas. That is, among strictly locally convex $C^{(3)}$ curves in the plane $\mathbb{R}^2$ parabolas are the only ones satisfying the above area property. In this article, we study strictly locally convex curves in the plane $\mathbb{R}^2$. As a result, generalizing the above mentioned characterization theorem for parabolas we present some conditions which are necessary and sufficient for a strictly locally convex $C^{(3)}$ curve in the plane to be an open part of a parabola.
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참고문헌 (13)
참고문헌 신청
Day, W. A. / 1991 / Inequalities for areas associated with conics / Amer. Math. Monthly 98(1):36~39
do Carmo, M. P. / 1976 / Differential Geometry of Curves and Surfaces / Prentice-Hall
김동수 / 2011 / A CHARACTERIZATION OF CONIC SECTIONS / 호남수학학술지 33(3):335~340
김동수 / 2015 / Area of triangles associated with a curve II / 대한수학회보 52(1):275~286
Kim, D.-S. / 2007 / A characterization of ellipses / Amer. Math. Monthly 114(1):66~70
do Carmo, M. P. / 1976 / Differential Geometry of Curves and Surfaces / Prentice-Hall
김동수 / 2011 / A CHARACTERIZATION OF CONIC SECTIONS / 호남수학학술지 33(3):335~340
김동수 / 2015 / Area of triangles associated with a curve II / 대한수학회보 52(1):275~286
Kim, D.-S. / 2007 / A characterization of ellipses / Amer. Math. Monthly 114(1):66~70
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