• Journal of Integrative Natural Science
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• v.6 no.1
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• pp.46-52
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• 2013

In this thesis, we consider isoparametric curves of quadratic F-B$\acute{e}$zier curves. F-B$\acute{e}$zier curves unify C-B$\acute{e}$zier curves whose basis is {sint, cos t, t, 1} and H-B$\acute{e}$zier curves whose basis is {sinht, cosh t, t,1}. Thus F-B$\acute{e}$zier curves are more useful in Geometric Modeling or CAGD(Computer Aided Geometric Design). We derive the relation between the quadratic F-B$\acute{e}$zier curves and the quadratic rational B$\acute{e}$zier curves. We also obtain the geometric properties of isoparametric curve of the quadratic F-B$\acute{e}$zier curves at both end points and prove the continuity of the isoparametric curve.

• School Mathematics
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• v.16 no.4
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• pp.827-845
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• 2014

This study aims to investigate how the university educational programs about quadratic curves are operated in relation to the high school curriculum and what their effects may be, and the degree of understanding for the prospective and current teachers of the mathematical content knowledge about quadratic curves. To solve this research questions, we randomly selected three universities and one high school. Then we investigated the curricula of each department of mathematics education, compared them with the high school curricula, and conducted surveys of the professors' and students' conception on how much mathematical content knowledge they need to know about quadratic curves. The study resulted in the following conclusions. First, the curriculum on the subject of quadratic curves in the college of education is closely connected to the high school programs. This study's results showed that the college of education's curriculum includes a series of lectures regarding quadratic curves, and that within them, the mathematical content about quadratic curves associated with high school mathematics was thoroughly covered. Also, a large number of students who attended the lecture reported a significant increase in their understanding in regards to the quadratic curves. Second, it is strongly recommended to strengthen the connection between the college of education's curriculum and the actual high school education field. The prospective teachers think that there is a substantial need to learn about the quadratic curves because it is closely connected with the high school curriculum. But they find it challenging to put what they were taught into practical use in the high school education field, and feel that an improvement in this area is much needed. Third, it is necessary to promote, encourage and support the voluntary efforts to expand the range of the content knowledge in quadratic curves to cover the academic content associated with the high school mathematics.

• International Journal of CAD/CAM
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• v.9 no.1
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• pp.55-60
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• 2010

Quadratic B$\acute{e}$zier curves are important geometric entities in many applications. However, it was often ignored by the literature the fact that a single segment of a quadratic B$\acute{e}$zier curve may fail to fit arbitrary endpoint unit tangent vectors. The purpose of this paper is to provide a solution to this problem, i.e., constructing $G^1$ quadratic B$\acute{e}$zier curves satisfying given endpoint (positions and arbitrary unit tangent vectors) conditions. Examples are given to illustrate the new solution and to perform comparison between the $G^1$ quadratic B$\acute{e}$zier cures and other curve schemes such as the composite geometric Hermite curves and the biarcs.

• Communications of Mathematical Education
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• v.31 no.2
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• pp.103-121
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• 2017

Taxicab geometry was a typical non-Euclid geometry for mathematically gifted. Most educational material related quadratic curves on taxicab geometry for mathematically gifted served them to inquire the graph of the curves defined by focis and constant. In this study, we provide a shape of quadratic curves on taxicab geometry by applying three definitions(geometric algebraic definition, eccentricity definition, conic section definition).

• 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.

• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.861-873
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• 2005

In this paper we approximate a cylindrical helix by bi-conic and bi-quadratic Bezier curves. Each approximation method is $G^1$ end-points interpolation of the helix. We present a sharp upper bound of the Hausdorff distance between the helix and each approximation curve. We also show that the error bound has the approximation order three and monotone increases as the length of the helix increases. As an illustration we give some numerical examples.

• Journal of the Korean Mathematical Society
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• v.38 no.6
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• pp.1179-1190
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• 2001

Quadratic invariant curve is one of the simplest nonlinear invariant curves and was considered by C. T. Ng and the author in order to study the one-dimensional nonlinear dynamics displayed by a second order delay differential equation with piecewise constant argument. In this paper a functional equation derived from the problem of invariant curves is discussed. Using a different method from what C. T. Ng and the author once used, we define solutions piecewisely and give results in the remaining difficult case left in C. T. Ng and the authors work. A problem of analytic extension given in their work is also answered negatively.

• Journal of Electrical Engineering and Technology
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• v.11 no.1
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• pp.1-8
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• 2016

It is well known that the behavior of PV curves is similar to a quadratic function. This is used in some papers to approximate PV curves and calculate the maximum-loading point by minimum number of power flow runs. This paper also based on quadratic approximation of the PV curves is aimed at completing previous works so that the computational efforts are reduced and the accuracy is maintained. To do this, an iterative method based on a quadratic function with two constant coefficients, instead of the three ones, is used. This simplifies the calculation of the quadratic function. In each iteration, to prevent the calculations from diverging, the equations are solved on the assumption that voltage magnitude at a selected load bus is known and the loading factor is unknown instead. The voltage magnitude except in the first iteration is selected equal to the one at the nose point of the latest approximated PV curve. A method is presented to put the mentioned voltage in the first iteration as close as possible to the collapse point voltage. This reduces the number of iterations needed to determine the maximum-loading point. This method is tested on four IEEE test systems.

• Korean Journal of Mathematics
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• v.22 no.3
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• pp.567-581
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• 2014

Over the years, there has been much research conducted on quadratic curves and the set of points with the generalized notion of eccentricity in a plane with metrics such as taxicab distance or Chinese-checker distance. On the other hand, polar taxicab distance has been newly proposed on the polar coordinate system, a type of curvilinear coordinate system, to overcome the limitation of pre-existing metrics in terms of describing curved routes. Previous study has looked into the fundamental properties of this metric. From this point of view, we study the quadratic curves and the set of points with the generalized notion of eccentricity in a plane with polar taxicab distance.

• Journal of the Korean School Mathematics Society
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• v.24 no.1
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• pp.151-172
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• 2021

In this study, I investigated prospective mathematics teachers' perceptions of activities using Wax-paper, a hands-on material (manipulatives), and GeoGebra, a technological tool, in teaching quadratic curves. Twenty prospective mathematics teachers in the Mathematics Education Department of a local university participated in a survey on their perception of the use of hands-on materials and technological tools in teaching quadratic curves. According to the results of this study, prospective mathematics teachers generally preferred the use of technological tools for learning and teaching quadratic curves. Additionally, mathematics teachers thought that the tool helped students develop intuitive thinking through visualizing quadratic curves, enabling the exploration of various mathematical properties, assisting the comprehension of various concepts, and increasing students' interest levels. However, they were concerned about the immature use of technological tools by students or teachers, and recognized that the advantages and disadvantages of using hands-on material and technological tools were complementary. Based on these findings, it is suggested that hands-on material and technological tools should be used complementally in mathematics classes, and the development and dissemination of class materials that are not affected by students' or teachers' ability to use technological tools is important.