R -functions in mathematical modeling of geometric objects with symmetry
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R-FUNCTIONS IN MATHEMATICAL MODELING OF GEOMETRIC OBJECTS WITH SYMMETRY K. V. Maksimenko-Sheikoa† and T. I. Sheikoa†
UDC 517.95+518.517
Normalized equations for regular n o -gons translated n d times in a regular n b -gon are set up based on coordinate transformations that allow translating geometric objects on segments. Such approaches are especially important in solving automation problems in solid modeling, geometric design, and boundary-value problems of mathematical physics. Keywords: symmetry types, R-functions, normalized boundary equations, regular polygons, translation. PROBLEM STATEMENT People have borrowed the symmetry phenomenon observed in nature (molecules, crystals, flora, and fauna) and transferred it into architecture, construction, and mechanical engineering not only for aesthetic reasons but also for rationality, equilibrium, stability, reliability, lightness, and cheapness of buildings, engineering structures, machines, mechanisms, and other structures and products. All this has been reflected in geometry, algebra, analysis, and other branches of mathematics and has stimulated studying the laws of symmetry and methods of their description and application and formulating new problems. Since antiquity, symmetry in a broad sense has been considered as an equivalent of balance and harmony. The theories based on symmetry laws in natural sciences may be important, and the ways symmetry is applied may be distinguished by a subtle plan and elaborated details [1]. Weyl said: “As far as I see, all a priori statements in physics have their origin in symmetry.” [2]. Reflection is a type of symmetry best known and frequently encountered in nature (Fig. 1a). The concept of mirror symmetry is of fundamental importance for the mathematical theory of symmetry; however, its role in science is incomparably wider. An example is the relationship, discovered by Pasteur, between the rotation of the polarization plane of light, solutions of some chemical substances, and absence of mirror symmetry in the crystals precipitating from these solutions. The effect observed by Pasteur is as follows: if both the crystal and the solution rotate the polarization plane of light, then individual molecules of the substance cannot possess mirror symmetry. Repeating any geometric object (GO) in space with certain spatial frequency results in symmetry known as translation (Fig. 1b). The symmetry in a kaleidoscope is formed by two mirrors inclined at an appropriate angle. It is possible to create a configuration possessing rotational and mirror symmetries (Fig. 1c), which means that the pattern will not change upon rotation by a certain angle around the axis running through the center. If the rotation angle is 90o , then the complete revolution by 360o needs four turns one by one. In this case, the axis is called tetragonal. If the angle of rotation is equal to 60o , then the axis is called hexagonal, etc. There are geometric objects with rotational symmetry and without planes of mirror symmetry. A whirligig (Fig. 1d) is an example o
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