• East Asian mathematical journal
• /
• v.34 no.2
• /
• pp.219-238
• /
• 2018

The purpose of this paper is to reinterpret and visualize Pappus' methods for trisecting the angle by utilizing the Nicomedes' conchoid and Apollonius' symptom of a hyperbola. In particular, we reinterpret the Pappus' three results which are the methods of hyperbola and circle, the trisection of the arc and focus and directrix of the hyperbola by 3 steps(analysis, construction, and proof) in the current middle school curriculum of Mathematics. Moreover, we visualize the construction of an hyperbola which is represented by means of an eccentricity.

• Journal of Educational Research in Mathematics
• /
• v.12 no.1
• /
• pp.81-93
• /
• 2002

This study is on the introduction of the natural logarithm by the quadrature of the hyperbola. In School mathematics curriculum, Logarithm is introduced formally. But in that introduction, students could't know the meaning of the natural logarithm and e well. Historically, natural logarithm is related to the quadrature of the hyperbola. So in this study we consider the introduction of the natural logarithm by the means of quadrature of the hyperbola and the significance of the introduction.

• Journal of the Korean Institute of Navigation
• /
• v.13 no.1
• /
• pp.1-10
• /
• 1989

Nowadays Hyperbolic Navigation System-LORAN, DECCA, OMEGA, OMEGA-is available on the ocean, and Spherical Navigation System, GPS (Global Positioning System) is operated partially. Hyperbolic Navigation System has the blind area near the base line extention because divergence rate of hyperbola is infinite theoretically. The Position Accuracy is differ from the cross angle of LOP although each LOP has the same error of quantity. GDOP(Geometric Dilution of Precisoin) is used to estimate the position accuracy according to the cross angle of LOP and LOP error. Hyperbola and ellipse are crossed at right angle everywhere. Hyperbola and ellipse are used to LOP in Rectangular Navigation System. The equation calculating the GDOP of rectangular Navigation System is induced and GDOP diagram is completed in this paper. A scheme that can improve the position accuracy in the blind area of Hyperboic Navigation System using the Rectangular Navigation System is proposed through the computer simulation.

• Geophysics and Geophysical Exploration
• /
• v.1 no.1
• /
• pp.8-18
• /
• 1998

In exploration seismology, the Kirchhoff hyperbola has been successfully used to migrate reflection seismo-grams. The mathematical basis of Kirchhoff hyperbola has not been clearly defined and understood for the application of prestack or poststack migration. The travel time from the scatterer in the subsurface to the receivers (exploding reflector model) on the surface can be a kinematic approximation of Green's function when the source is excited at position of the scatterer. If we add the travel time from the source to the scatterer in the subsurface to the travel time of exploding reflector model, we can view this travel time as a kinematic approximation of the partial derivative wavefield with respect to the velocity or the density in the subsurface. The summation of reflection seismogram along the Kirchhoff hyperbola can be evaluated as an inner product between the partial derivative wavefield and the field reflection seismogram. In addition to this kinematic interpretation of Kirchhoff hyperbola, when we extend this concept to shallow refraction seismic data, the stacking of refraction data along the straight line can be interpreted as a measurement of an inner product between the first arrival waveform of the partial derivative wavefield and the field refraction data. We evaluated the Kirchhoff hyperbola and the straight line for stacking the refraction data in terms of the first arrival waveform of the partial derivative wavefield with respect to the velocity or the density in the subsurface. This evaluation provides a firm and solid basis for the conventional Kirchhoff migration and the straight line stacking of the refraction data.

• Bulletin of the Korean Mathematical Society
• /
• v.50 no.4
• /
• pp.1243-1259
• /
• 2013

In this paper, we study the root system of rank 2 hyperbolic Kac-Moody algebras. We give some sufficient conditions for the existence of imaginary roots of square length $-2k(k{\in}\mathbb{Z}_{>0}$. We also give several relations between the integral points on the hyperbola $\mathfrak{h}$ to show that the value of the symmetric bilinear form of any two integral points depends only on the number of integral points between them. We also give some generalizations of Binet formula and Catalan's identity.

• Proceedings of the Korean Geotechical Society Conference
• /
• 2005.03a
• /
• pp.924-931
• /
• 2005

In this study, we intended to compare and examine several settlement management methods by analyzing measurement results of a site of the industrial complex at ${\bigcirc}{\bigcirc}$ province. We predicted and analyzed the amount of final settlement by using generally used final settlement methods as like Hyperbola method, Hoshino methods and Asaoka method. And then, We compared the predicted results with that of measurement. On the basis of comparison of the three methods, Hyperbola method was the most convenient and accurate method of the three methods and if a sufficient time was given enough after embankment construction, the use of Hoshino method was possible. In the case of the Asaoka methods, it was possible to know that it had an approaching tendency to the measured one with increasing time interval spent on analysis. Therefore, in order to predict settlement behavior more accurately it is needed to understand their advantages and shortcomings sufficiently and pay attention to application to the real site.

• Geomechanics and Engineering
• /
• v.15 no.6
• /
• pp.1219-1225
• /
• 2018

In this study, a comprehensive laboratory experimental programme was conducted on expansive soil with a high swelling potential to study the influence of different additive materials on swelling pressure and index properties. Lime, sand, multifilament fiber and fibrillated fiber were used for stabilization of expansive soil. Lime, sand and fibers were respectively added to the expansive soil at 0-7%, 0-80%, 0-0.5%. On each mixture that was prepared by the proportions mentioned above, Atterberg limits, compaction, and swelling pressure tests were conducted. From the result of these experiments, the swelling pressure-time relation could be replaced by a rectangular hyperbola established to facilitate the prediction of ultimate percent swelling with a few initial data points. The best type of additive and its optimum ratio for engineering purposes could be estimated rapidly by this approach.

• The Mathematical Education
• /
• v.53 no.3
• /
• pp.313-327
• /
• 2014

'Tangent' is one of the most important concepts in the middle and high school mathematics, especially in dealing with calculus. The concept of tangent in the current textbook consists of the ways which make use of discriminant or differentiation. These ways, however, do not present dynamic view points, that is, the concept of variation. In this paper, after applying 'Roberval's way of finding tangent using vectors in terms of kinematics to parabola, ellipse, circle, hyperbola, cycloid, hypocycloid and epicycloid, we will identify that this is the tangent of those curves. This trial is the educational link of mathematics and physics, and it will also suggest the appropriate example of applying vector. We will also help students to understand the tangent by connecting this method to the existing ones.

• The Pure and Applied Mathematics
• /
• v.20 no.3
• /
• pp.137-148
• /
• 2013

In this paper, we study the Lie-generalized Fibonacci sequence and the root system of rank 2 symmetric hyperbolic Kac-Moody algebras. We derive several interesting properties of the Lie-Fibonacci sequence and relationship between them. We also give a couple of sufficient conditions for the existence of the integral points on the hyperbola $\mathfrak{h}^a:x^2-axy+y^2=1$ and $\mathfrak{h}_k:x^2-axy+y^2=-k$ ($k{\in}\mathbb{Z}_{>0}$). To list all the integral points on that hyperbola, we find the number of elements of ${\Omega}_k$.

• The Pure and Applied Mathematics
• /
• v.22 no.1
• /
• pp.25-34
• /
• 2015

In this paper we characterize the isogonal and isotomic conjugates of conic. Every conic can be expressed by a quadratic rational B$\acute{e}$zier curve having control polygon $b_0b_1b_2$ with weight w > 0. We show that the isotomic conjugate of parabola and hyperbola with respect to ${\Delta}b_0b_1b_2$ is ellipse, and that the isotomic conjugate of ellipse with the weight $w={\frac{1}{2}}$ is identical. We also find all cases of the isogonal conjugate of conic with respect to ${\Delta}b_0b_1b_2$. Our characterizations are derived easily due to the expression of conic by the quadratic rational B$\acute{e}$ezier curve in standard form.