• Proceedings of the Korean Society of Precision Engineering Conference
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• 1997.04a
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• pp.862-867
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• 1997

A 5-axis NC milling technology is presented on ruled surface. Problems in 5-axis NC machining are such as tool interference,tool collision and change of tool attitude,etc. The change of tool attitude causes rotation of cutter and variation of feedrate to overcut part surface. This poor control of tool attitude is the primary problem in multi-axis NC milling. This paper observes ruled surface for control of tool attitude. Ruled surface is composed of directrix and ruling, line of constant magnitude. Directrix corresponds to points on part surface and Ruling cutting tool. Trajectory of tool movement corresponds to ruled surface.

• Journal for History of Mathematics
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• v.25 no.4
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• pp.83-99
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• 2012

The conic sections are defined as algebraic expressions using the focus and the directrix in the high school curriculum. However it is difficult that students understand the conic sections without environment which they can manipulate the conic sections. To make up for this weak point, we have found the evidence for generating method of a conic section through a sundial and investigated the history of terms 'focus', 'directrix' and the tool of drawing them continuously.

• Honam Mathematical Journal
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• v.33 no.3
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• pp.335-340
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• 2011

We study some properties of tangent lines of conic sections. As a result, we establish a characterization of conic sections.

• Structural Engineering and Mechanics
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• v.76 no.3
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• pp.395-412
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• 2020

In this article, the values of internal force and deformation of a curved beam under any action with the firm or elastic supports are determined by using structural matrices. The article presents the general differential formulation of a curved beam in global coordinates, which is solved in an orderly manner using simple integrals, thus obtaining the transfer matrix expression. The matrix expression of rigidity is obtained through reordering operations on the transfer notation. The support conditions, firm or elastic, provide twelve equations. The objective of this article is the construction of the algebraic system of order twenty-four, twelve transfer equations and twelve support equations, which relates the values of internal force and deformation associated with the two ends of the directrix of the curved beam. This final algebraic system, expressed in matrix form, is divided into two subsystems: twelve algebraic equations of internal force and twelve algebraic equations of deformation. The internal force and deformation values for any point in the curved beam directrix are determined from these values in the initial position. The five examples presented show how to apply the matrix procedures developed in this article, whether they are curved beams with the firm or elastic support.

• The Pure and Applied Mathematics
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• v.19 no.4
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• pp.315-325
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• 2012

Let A and B denote a point, a line or a circle, respectively. For a positive constant $a$, we examine the locus $C_{AB}$($a$) of points P whose distances from A and B are, respectively, in a constant ratio $a$. As a result, we establish some equivalent conditions for conic sections. As a byproduct, we give an easy way to plot points of conic sections exactly by a compass and a straightedge.

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

• Communications of Mathematical Education
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• v.24 no.3
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• pp.731-744
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• 2010

Nowadays in school mathematics, the skill and method for solving problems are often emphasized in preference to the theoretical principles of mathematics. Students pay attention to how to make an equation mechanically before even understanding the meaning of the given problem. Furthermore they do not get to really know about the principle or theorem that were used to solve the problem, or the meaning of the answer that they have obtained. In contemporary textbooks the conic section such as circle, ellipse, parabola and hyperbola are introduced as the cross section of a cone. But they do not mention how conic section are connected with the quadratic equation or how these curves are related mutually. Students learn the quadratic equations of the conic sections introduced geometrically and are used to manipulating it algebraically through finding a focal point, vertex, and directrix of the cross section of a cone. But they are not familiar with relating these equations with the cross section of a cone. In this paper, we try to understand the quadratic curves better through the analysis of the discussion made in the process of the discovery and eventual development of the conic section and then seek for way to improve the teaching and learning methods of quadratic curves.