arXiv subjects

Dong-Soo Kim

Publications and source records attributed to Dong-Soo Kim.

8 recordsLinked to original sources

arXiv2015

Centroid of triangles associated with a curve

Archimedes showed that the area between a parabola and any chord $AB$ on the parabola is four thirds of the area of triangle $ΔABP$, where P is the point on the parabola at which the tangent is parallel to the chord $AB$. Recently, this property of parabolas was proved to be a characteristic property of parabolas. With the aid of this characterization of parabolas, using centroid of triangles associated with a curve we present two conditions which are necessary and sufficient for a strictly locally convex curve in the plane to be a parabola.

math.DG↗

arXiv2015

Center of gravity and a characterization of parabolas

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.

math.DG↗

arXiv2014

Cheng-Yau operator and Gauss map of surfaces of revolution

We study the Gauss map $G$ of surfaces of revolution in the 3-dimensional Euclidean space ${\mathbb{E}^3}$ with respect to the so called Cheng-Yau operator $\square$ acting on the functions defined on the surfaces. As a result, we establish the classification theorem that the only surfaces of revolution with Gauss map $G$ satisfying $\square G=AG$ for some $3\times3$ matrix $A$ are the planes, right circular cones, circular cylinders and spheres.

math.DG↗

arXiv2014

Area of triangles associated with a curve II

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. In this article, we consider whether this property and similar ones characterizes parabolas. As a result, we present three conditions which are necessary and sufficient for a strictly convex curve in the plane to be an open part of a parabola.

math.DG↗

arXiv2014

Area of triangles associated with a curve

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. In this article, we study some properties of $U$ and $T$ for strictly convex plane curves. As a result, we establish a characterization for parabolas.

math.DG↗

arXiv2013

On the Archimedean characterization of parabolas

Archimedes knew that the area between a parabola and any chord $AB$ on the parabola is four thirds of the area of triangle $ΔABP$ where P is the point on the parabola at which the tangent is parallel to $AB$. We consider whether this property (and similar ones) characterizes parabolas. We present five conditions which are necessary and sufficient for a strictly convex curve in the plane to be a parabola.

math.DG↗

arXiv2013

Ellipsoids and elliptic hyperboloids in the Euclidean space ${\Bbb E}^{n+1}$

We establish some characterizations of elliptic hyperboloids (resp., ellipsoids) in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$, using the $n$-dimensional area of the sections cut off by hyperplanes and the $(n+1)$-dimensional volume of regions between parallel hyperplanes. We also give a few characterizations of elliptic paraboloids in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$.

math.DG↗

arXiv2012

Some characterizations of spheres and elliptic paraboloids II

We show some characterizations of hyperspheres in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$ with intrinsic and extrinsic properties such as the $n$-dimensional area of the sections cut off by hyperplanes, the $(n+1)$-dimensional volume of regions between parallel hyperplanes, and the $n$-dimensional surface area of regions between parallel hyperplanes. We also establish two characterizations of elliptic paraboloids in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$ with the $n$-dimensional area of the sections cut off by hyperplanes and the $(n+1)$-dimensional volume of regions between parallel hyperplanes. For further study, we suggest a few open problems.

math.DG↗