• Journal of the Korean Mathematical Society
• /
• v.36 no.3
• /
• pp.621-631
• /
• 1999

In this paper we briefly describe the geometry of the Cayley hyperbolic plane and we show that every uniform lattice in quaternionic space cannot be deformed in the Cayley hyperbolic 2-plane. We also describe the nongeometric bending deformation by developing the theory of the Cartan angular invariant for quaternionic hyperbolic space.

• Honam Mathematical Journal
• /
• v.29 no.3
• /
• pp.481-493
• /
• 2007

We will construct several types of semi-regular tilings of a hyperbolic unit disk model by defining geometric features of the definition of distance in a hyperbolic plane, area of triangle, and isometry of inversions. We researched the method of regular tilings and semi-regular tilings of hyperbolic unit disk model and wrote an semi-regular tiling construction algorithm using Cabri2 program and Cinderella program. Lastly, We want to make a product related to traditional heritage cultural patterns using Photoshop, so we'll model the advertising photos of cites; Seoul, Gwangju.

• The KIPS Transactions:PartA
• /
• v.8A no.4
• /
• pp.489-498
• /
• 2001

A new method for local control of arbitrary scattered data interpolation in the hyperbolic plane is developed in this paper. The issue associated with local control is very critical in the interactive in the interactive design field. Especially the suggested method in this paper could be effectively applied to the interactive shape modeling of genus-N objects, which are constructed on the hyperbolic plane. Since the effects of the changed data affects only the limited area around itself, it is more convenient for end-users to design a genus-N object interactively. Therefore, by improving the global interpolation on the hyperbolic plane where the genus-N object is constructed, this research is aiming at the development and implementation of the local interpolation on the hyperbolic plane. It is implemented using the following process. First, for localizing the interpolating functions, the hyperbolic domain is tessellated into arbitrary triangle patches and the group of adjacent triangle patches of each data point is defined as a sub-domain. On each sub-domain, a weight function is defined. Last, by blending of three weight functions on the overlapped triangles, local MQ interpolation is completed. Consequently, it is compared with the global MQ interpolation using several sample data and functions.

• Journal of applied mathematics & informatics
• /
• v.39 no.3_4
• /
• pp.327-338
• /
• 2021

In this paper, we have introduced a new notion of recurrent structure Jacobi of real hypersurfaces in complex hyperbolic two-plane Grassmannians G^*₂(ℂ^m+2). Next, we show a non-existence property of real hypersurfaces in G^*₂(ℂ^m+2) satisfying such a curvature condition.

• Bulletin of the Korean Mathematical Society
• /
• v.60 no.2
• /
• pp.507-514
• /
• 2023

Suppose that a line passing through a given point P intersects a given circle 𝓒 at Q and R in the Euclidean plane. It is well known that |PQ||P R| is independent of the choice of the line as long as the line meets the circle at two points. It is also known that similar properties hold in the 2-sphere and in the hyperbolic plane. New proofs for the similar properties in the 2-sphere and in the hyperbolic plane are given.

• Bulletin of the Korean Mathematical Society
• /
• v.59 no.2
• /
• pp.351-360
• /
• 2022

We prove that a punctured-torus group of hyperbolic 4-space which keeps an embedded hyperbolic 2-plane invariant has a strictly parabolic commutator. More generally, this rigidity persists for a punctured-surface group.

• Korean Journal of Mathematics
• /
• v.30 no.4
• /
• pp.643-652
• /
• 2022

In this paper, we will describe affine homogeneous domains in the complex plane. For this study, we deal with the Lie algebra of infinitesimal affine transformations, a structure of the hyperbolic metric involved with affine automorphisms. As a consequence, an affine homogeneous domain is affine equivalent to the complex plane, the punctured plane or the half plane.

• Journal of the Chungcheong Mathematical Society
• /
• v.22 no.4
• /
• pp.771-776
• /
• 2009

Let (H, g) denote the upper half plane in $R^2$ with the Riemannian metric g := ($(dx)^2$ + $(dy)^2$)$/y^2$. First of all we get a necessary and sufficient condition for a diffeomorphism $\phi$ of (H, g) to be a harmonic map. And, we obtain the fact that if a diffeomorphism $\phi$ of (H, g) is a harmonic function, then the following facts are equivalent: (1) $\phi$ is a harmonic map; (2) $\phi$ is an affine transformation; (3) $\phi$ is an isometry (motion).

• Honam Mathematical Journal
• /
• v.22 no.1
• /
• pp.83-90
• /
• 2000

We establish a sufficient condition for a simple closed curve in the Euclidean plain $\mathbb{R}^2$, the unit sphere $S^2$ or the hyperbolic plane $H^2$ to be the boundary of a metric ball.

• Journal of the Chungcheong Mathematical Society
• /
• v.20 no.3
• /
• pp.203-210
• /
• 2007

Let X be a hyperbolic region in the complex plane C such that the hyperbolic metrix ${\lambda}_X(w){\mid}dw{\mid}$ exists. Let $R(X)=sup\{{\delta}_X(w):w{\in}X\}$ where ${\delta}_X(w)$ is the euclidean distance from w to ${\partial}X$. Here ${\partial}X$ is the boundary of X. A hyperbolic region X is called a Bloch region if R(X) < ${\infty}$. In this paper, we obtain lower bounds for the hyperbolic metric on Bloch regions in terms of the distance to the boundary.