• East Asian mathematical journal
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• v.34 no.4
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• pp.403-427
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• 2018

The purpose of this paper is to reinterpret and visualize the medieval Arab's studies on the geometric methods of solving the cubic equation by utilizing Apollonius' symptom of the parabola. In particular, we investigate the results of $Kam{\bar{a}}l$ $al-D{\bar{i}}n$ ibn $Y{\bar{u}}nus$, Alhazen, Umar al-$Khayy{\bar{a}}m$ and $Al-T{\bar{u}}s{\bar{i}}$ by 4 steps(analysis, construction, proof and examination) which are called the complete solution in the constructions. This paper is available in the current middle school curriculum through dynamic geometry program(Geogebra).

• East Asian mathematical journal
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• v.37 no.4
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• pp.401-426
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• 2021

In order to propose ways to implement mathematical connection between algebra and geometry, this study reinterpreted and visualized the Khayyam's geometric method solving the cubic equations using two conic sections and the Al-Kāshi's method of constructing of angle trisection using a cubic equation. Khayyam's method is an example of a geometric solution to an algebraic problem, while Al-Kāshi's method is an example of an algebraic a solution to a geometric problem. The construction and property of conics were presented deductively by the theorem of "Stoicheia" and the Apollonius' symptoms contained in "Conics". In addition, I consider connections that emerged in the alternating process of algebra and geometry and present meaningful Implications for instruction method on mathematical connection.

• East Asian mathematical journal
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• v.37 no.4
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• pp.499-521
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• 2021

This research is devoted to investigate Omar Khayyām's geometric solution for the cubic equation using conic sections in the Medieval Islam as a useful alternative connecting logic geometry with analytic geometry at a secondary school. We also introduce Omar Khayyām's 25 cases classification of the cubic equation with all positive coefficients. Moreover we study 6 cases with 4 terms of 25 cubic equations and in particular we reinterpret geometric methods of solving in 2015 secondary Mathematics curriculum and visualize them by means of dynamic geometry software.

• Journal of Educational Research in Mathematics
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• v.27 no.1
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• pp.89-112
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• 2017

In this study we have analyzed processes of generalization in which students have geometrically solved cubic equation $x^3+ax=b$, regarding geometrical solution of cubic equation $x^3+4x=32$ as examples. The result of this research indicate that students could especially re-interpret the geometric solution of the given cubic equation via dynamically understanding the variables in dynamic geometry environment. Furthermore, participants could simultaneously re-interpret the given geometric solution and then present a different geometric solutions of $x^3+ax=b$, so that the level of generalization could be improved. In conclusion, the study could provide useful pedagogical implications in school mathematics that the dynamic geometry environment performs significant function as a means of students-centered exploration when understanding variables and enhancing the level of generalization in problem solving.

• School Mathematics
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• v.18 no.3
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• pp.589-609
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• 2016

In this study, researchers have modernly reinterpreted geometric solving of cubic equations presented by an arabic mathematician, Omar Khayyam in medieval age, and have considered the pedagogical significance of geometric solving of the cubic equations using two conic sections in terms of analytic geometry. These efforts allow to analyze educational application of mathematics instruction and provide useful pedagogical implications in school mathematics such as 'connecting algebra-geometry', 'induction-generalization' and 'connecting analogous problems via analogy' for the geometric approaches of cubic equations: $x^3+4x=32$, $x^3+ax=b$, $x^3=4x+32$ and $x^3=ax+b$. It could be possible to reciprocally convert between algebraic representations of cubic equations and geometric representations of conic sections, while geometrically approaching the cubic equations from a perspective of connecting algebra and geometry. Also, it could be treated how to generalize solution of cubic equation containing variables from geometric solution in which coefficients and constant terms are given under a perspective of induction-generalization. Finally, it could enable to provide students with some opportunities to adapt similar solving procedures or methods into the newly-given cubic equation with a perspective of connecting analogous problems via analogy.

• Journal for History of Mathematics
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• v.36 no.1
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• pp.1-17
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• 2023

Paper folding is a versatile tool that can be used not only as a mathematical model for analyzing the geometric properties of plane and spatial figures but also as a visual method for finding the real roots of polynomial equations. The historical evolution of origami's geometric and algebraic techniques has led to the discovery of definitions and properties that can enhance one's cognitive understanding of mathematical concepts and generate mathematical interest and motivation on an emotional level. This paper aims to examine the history of origami geometry, the utilization of origami for solving polynomial equations, and the process of determining the real roots of quadratic, cubic, and quartic equations through origami techniques.

• International Journal of CAD/CAM
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• v.9 no.1
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• pp.103-109
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• 2010

Recently, various conformal geometric methods have been presented for non-rigid surface matching and registration. This work proposes to improve the robustness of conformal geometric methods to the boundaries by incorporating the symmetric information of the input surface. We presented two symmetric conformal mapping methods, which are based on solving Riemann-Cauchy equation and curvature flow respectively. Experimental results on geometric data acquired from real life demonstrate that the symmetric conformal mapping is insensitive to the boundary occlusions. The method outperforms all the others in terms of robustness. The method has the potential to be generalized to high genus surfaces using hyperbolic curvature flow.

• 한국전산유체공학회:학술대회논문집
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• 2011.05a
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• pp.6-9
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• 2011

In this study, the two-phase incompressible flow in two-dimensional channel considering the effect of surface tension is simulated using an improved level-set method. Quadratic element is used for solving the continuity and Navier-Stokes equations to avoid using an additional pressure equation, and Crank-Nicholson scheme and linear element are used for solving the advection equation of the level set function. Direct approach method using geometric information is implemented instead of the hyperbolic-type partial differential equation for the reinitializing the level set function. The benchmark test case considers various arrays of defomable droplets under different flow conditions in straight channel. The deformation and migration of the droplets are computed and the results are compared very well with the existing studies.

• Journal of Educational Research in Mathematics
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• v.26 no.4
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• pp.715-730
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• 2016

The purpose of this study is to analyze Descartes's point of view on the mathematical connection of algebra and geometry which help comprehend the traditional frame with a new perspective in order to access to unsolved problems and provide useful pedagogical implications in school mathematics. To achieve the goal, researchers have historically reviewed the fundamental principle and development method's feature of analytic geometry, which stands on the basis of mathematical connection between algebra and geometry. In addition we have considered the significance of geometric solving of equations in terms of analytic geometry by analyzing related preceding researches and modern trends of mathematics education curriculum. These efforts could allow us to have discussed on some opportunities to get insight about mathematical connection of algebra and geometry via geometric approaches for solving equations using the intersection of curves represented on coordinates plane. Furthermore, we could finally provide the method and its pedagogical implications for interpreting geometric approaches to cubic equations utilizing intersection of conic sections in the process of inquiring, solving and reflecting stages.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.263-273
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• 2013

A mean-value result, saying that the difference quotient of a differentiable function in a real interval is a mean value of its derivatives at the endpoints of the interval, leads to the functional equation $$\frac{f(x)-F(y)}{x-y}=M(g(x),\;G(y)),\;x{\neq}y$$, where M is a given mean and $f$, F, $g$, G are the unknown functions. Solving this equation for the arithmetic, geometric and harmonic means, we obtain, respectively, characterizations of square polynomials, homographic and square-root functions. A new criterion of the monotonicity of a real function is presented.