本研究の目的は, Delossの作図不能問題の解法にCAD機能を利用することにある.それは, 作図の理論的根拠の説明とその観点からのCAD作図例さらに作図精度測定からなる.関連して, 現代図学におけるCAD表示画面上で作図に必要な, Y=X^1/3関数曲線の新たな創成法を述べる.又, CAD作業画面上での円周長さπdの直延とその線分積算法を示す.CAD作図の誤差は, πd値で4.7×10^-8, 稜の長さで-4.3×10^-5, 正六面体積の計算で-1.3×10^-4であった.提示した作図方法の精度の向上と他の3-D図形への適用は今後の課題である.

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In learning of trigonometric ratios in high school mathematics "Mathematics I", many text books define at first a tangent, next a sine and cosine. A reason to define in this order would come from an educational care since students are familiar with a notion of tangent as a slope, for example, in their daily life. However, in the process of the study, the sine and cosine play a central role in trigonometric ratios and functions. For example, the Sine Theorem and the Cosine Theorem are the important topics in "Mathematics I", and in fact the tangent is simply expressed as the ratio of sine to cosine. Then, we can propose a problem: To consider trigonometric ratios and functions from a viewpoint of tangent. We study this problem by using an intersection of a line through the origin and a line x=k in the orthogonal coordinate plane. We show Tangent Addition Theorem without using Sine and Cosine Addition Theorems. A derivative of tangent function tanx is obtained as 1+ tan^2x by using the Tangent Addition Theorem. A geometric interpretation is given for the property: an integration of a rational function of the sine and cosine functions of x is reduced to an integration of rational function of t by a substitution tan(x/2)=t. As a result, we see that some properties of trigonometric ratios and functions are naturally obtained from a viewpoint of tangent or line x=k.

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この研究の目的は, 関節可動域表示の可動範囲に適用する3次元関節角度の新しい方法を提案することである.この方法は関節可動域 (ROM) を2次元の角に類似する関節立体角と呼ぶ容積的な概念に写す.基本的アイデアは関節立体角が関節可動範囲に対応した単位球面領域の面積として同じように定義できることである.この領域の境界曲線は関節角度をパラメータとして用いる円弧か円と定義した.したがって関節ベクトルに関するベクトル式を球面3角法による関節立体角度のために用いた.この方法の利用例をコンピュータで示す.機能障害の有無でその可動範囲の差異を理解するときこれは有益と考えられる.

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Originally an airplane is the vehicle which moves toward its nose direction. But strictly speaking, the direction of flight path and that of nose are not the same due to the existence of the angle of attack α and the sideslip angle β (except due to the drift angle). Then, is it possible to control flight attitude (nose direction) and flight path separately by controlling these aerodynamic angles (which are controlled through elevator and rudder)?
We begin with this question and first study qualitatively the relation between "flight attitude and path." And then, their quantitative relations, that is, the relations of attitude angles and path angles (these are EULER angles) are derived in explicit and easily applicable forms by means of both algebraic and geometric methods.

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　　 Our consideration about ‘the problem on the sum of angles’ began in the following ‘Problem e’.  Nishikawa et al. (2014) insisted that the ‘Problem e’ would be valuable teaching materials in junior and senior high school mathematics classes.

　　　 Problem e:  Find the sum of angle α and β in the figure in which three  congruent squares are placed side by side.  

　　 This article deepened Nishikawa et al. (2014) and its purposes are as follows:

　　　 (1)  To make the similar problems of the ‘Problem e’ and to raise the value as the teaching materials of the problem more.

　　　 (2)  To consider the ‘Problem e’ from the viewpoint of inversion and to explore the possibility as teaching materials.

　　 In Section 2, we made the similar problems that had a different condition from the ‘Problem e’, and gave three kinds of elementary geometrical solutions in different ways of viewing sums of angles i.e. neighboring two angles, an external angle and expression of the product.  As a result, it was revealed that these similar problems could become superior teaching materials.

　　 Through the consideration in Section 2, we reached an idea that it was a general way to view the problem on sum of angles from the viewpoint of inversion.  The idea is reflected in theorem 3.3 in Section 3 as the relationship of sum of angles with respect to a point on circle of inversion and a pair of points of the inversion. It would be said that the figure of theorem 3.3 is ‘the general figure of the sum of angles’.  The result of theorem 3.3 is also viewed as a kind of extension of ‘The relationships between inscribed angles and its central angle in a circle’.

　　 In Section 4, we considered a property in ‘the general figure of the sum of angles’.  In Section 4.1, we made it clear that the figure was closely related to Apollonius’s circle and we came to the conclusion that these facts would be interesting teaching materials in high school mathematics classes.  In Section 4.2, we obtained the proposition 4.4--4.7 by using the properties of inversion.  The proposition 4.7 can be used in constructing one point of the inversion from the other.

　　 There are many contents in this paper which give us a clear image of making students acquire some kind of mathematical ability.  In this sense, we conducted content-study of ‘the problem on sum of angles’ for the development of a teaching material.  In addition, the consideration of this paper clarified the core of the problem that everyone had overlooked by the elegant solution, by various answers to the similar problems and the generalization based on it.  Therefore, we insist that the contents of this paper have values as a teaching material of teacher’s training for both pre-service and in-service teachers.

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　　Kawasaki (2001) used the ‘Problem e’ below, to clarify the nature of geometrical sense.

　　　Problem e : Find α + β in the figure in which three congruent squares are placed side by side.

　　Independently, we found the formula Tan ⁻¹ 1/2+Tan ⁻¹ 1/3=π/4 (Euler’s formula) in Izumi (1961), and we recognized that the problem e is a geometrical version of Euler’s formula.　We felt that the problem e which is related to Euler’s formula has interesting geometrical aspects and the value in mathematics education.　In this article, we consider the nature of the problem e and its usefulness and possibilities as teaching material in junior and senior high school mathematics classes.

　　In Section 2, we considered a variety of solutions to the problem e and classified them by learning contents.　As a result, we came to the conclusion that the problem e can be used variously both in junior and senior high school mathematics classes and will be a valuable teaching material.

　　In solving the problem e geometrically,“the viewpoints of the sum of the measure of angles”are the key to the success. We considered “the sum of angles” from three viewpoints :

　　Viewpoint 1 : Adjoining two angles. This is the most natural way.　The examples of the solutions to the problem e from this viewpoint are shown in 2.1.1.

　　Viewpoint 2 : Reducing to the fact “the measure of an external angle is equal to the sum of the measures of two inner angles which aren’t adjacent the external angle”.　We recognized that this viewpoint is very useful. In fact, this viewpoint is a fundamental idea in this article.　The examples of the solutions to the problem e from this viewpoint are shown in 2.1.2 and 2.2.3.　We discussed in detail the geometrical proofs of Clausen’s formula which is more complicated than Euler’s formula in 3.2 of Section 3. Viewpoint 3 : Direct and effective way which doesn’t use the movement of angles to consider “the sum of the measure of angles”.　In viewpoint 1 and 2, we need to “move the angle adequately”.　We found the way from

　　viewpoint 3 which dosen’t need to move the angle by developing the way in 2.2.3.　We formulated this way as the lemma in 3.3 and we showed the proofs of the formulae which are more complicated than Euler’s formula.

　　In Section 4, we developed some problems which are related to the problem e from the viewpoints 2 and 3. As a result of the consideration in this article, the viewpoints and ways concerning the problem e will be a valuable teaching material that makes junior and senior high school students enhance their interest in geometry.

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In this paper, the stability analysis of an active noise control system using single adaptive filter with an external injection noise and an acceleration method that makes use of the property of this noise are discussed. The convergence performance of this adaptive algorithm is analyzed from the viewpoint of a property of the Hessian matrix. As a result, it is proved that the Hessian matrix is positive definite even when the power density spectrum of the noise source signal and the external injection noise are different. Furthermore, by selecting the power density spectrum of the external injection noise to be the inverse property of that of the noise source, it is proved that the system with the white noise source and the white injection noise can be realized. Finally, these theoretical considerations are verified by computer simulation.

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New integrals of fundamental solution of three-dimensional Laplace equation are derived by using Gauss' divergence theorem. These are useful for boundary element method. One of the integrals is single layer potential of a constant triangular element and others are single and double layer potential of a linear triangular element. There are two advantages of these integrals. First, coordinate transformation and subdivision of a triangular element are not necessary to evaluate these integrals. Second, it is possible to evaluate formulas of single and double layer potential effectively, because the formula of double layer potential is related to the formula of single layer potential. The validity of these integrals is confirmed by comparing with numerical integration of fundamental solution over a triangular element by using Lachat algorithm. The effective gradient formulas are derived by differentiation of these integrals analytically. Present integrals can be applied for not only collocation BEM but also Galerkin BEM and fast multipole BEM.

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The purpose of this study was to investigate the effect of arm swing on the generation of twisting from somersault. By using a human body model composed of three geometric solids, mechanism of generation of twisting during a somersault was confirmed theoretically. Then, quantitative calculation was done based on the model. In this model a performer initially took a symmetrical position with both arms above the head, and then swung down one arm laterally. To examine the validation of the model, several twist-somersaults were analyzed by a 3-dimensional cinematography (DLT method) and compared with the model. Two male proficient gymnasts were selected as subjects and were asked to perform forward twist-somersaults of 1/2, 1 and 3/2 revolutions from a vaulting horse. The results were summarized as follows: 1) The angular velocities of the twist and somersault, moment of inertia, and the tilting of a body's principal axis of inertia from its original position calculated from the model were compared with the measured data. The mean relative errors between the data from simulation model and the measured were less than 4.1%. 2) An asymmetrical arm swing could generate a twist about the longitudinal axis of the body during a somersault. Large angular velocity of the somersault before the change in the form and large swing angle of arm were effective for the generation of twisting. 3) The direction of the twist depended upon direcitons of the initial somersault and the arm swing.

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