### Applied Mathematics, Mechanics and Physics

### Аuthors

^{1}

^{*},

^{2}

^{**},

^{3}

^{***}

1. Fatih University, 34500 Buyukcekmece, Istanbul, Turkey

2. Ufa State University of Economics and Service, 145, Chernyshevskogo st., Ufa, 450078, Russia

3. Shizuoka University, 3-5-1, Johoku, Naka-ku, Hamamatsu Shizuoka, 432 Japan

*e-mail: rushanziatdinov@yandex.ru, ziatdinov@fatih.edu.tr

**e-mail: dizain55@yandex.ru

***e-mail: tmkmiur@ipc.shizuoka.ac.jp

### Abstract

The article deals with some of the families of planar curves with monotone curvature functions and their application in the geometric modeling and aesthetic design. The curves of this type include pseudospirals, aesthetic curves and superspirals, the curvature of which is given by the Gaussian hypergeometric function satisfying strict monotonicity under certain restrictions imposed on the parameters, as well as Class A Bezier curves, the function of curvature of which is monotone, and a detailed analysis of which has shown that with the degree of the polynomial increased, the curve converges to a logarithmic spiral. It is noted that controlling of the monotonicity of curvature function of Bezier curves and B-splines of n> 2 order requires a more in-depth analysis and development of appropriate algorithms. The descriptive part of this article provides an example of modeling the surface of the car body with a help of aesthetic splines.For the first time in the area of geometric modeling there is performed an aesthetic analysis and evaluation of the structure and plastic properties of curves with monotone curvature function from the standpoint of the laws of technical aesthetics. This analysis is based on the theoretical and methodological principles of aesthetic appraisal of form through art in the historical dynamics of its origin, formation and development in different cultures. The paper sets forth the idea of consciousness dependence on the psychological content of the formal signs of perceived reality and the possibility of their direct impact on human being to model the world-view and mould the certain personality type. In this regard, the objective character of the results represented in this paper and objectified in the formulas and visualizations is based on the integrative principle of «Dialectical Unity of Science and Art» as the two forms of social consciousness inseparably integrated into the system of public, spiritual and material production. In addition to the above, we make a conclusion that the union of science and art creates the conditions for formation of humanistic way of thinking of man supporting the unity of beauty and benefit of form in the spiritual and material space of his life as a leading ethical principle.

Appraisal part of this paper includes a provision of design as an activity designed to unite the benefit and beauty in the harmonious form embodied in design rules and which for the individual are objective criteria for comfortable object perception of all its constituent features as a whole.

### Keywords:

spiral, pseudospiral, aesthetic curve, superspiral, multispiral, monotonicity of curvature, high-quality curve, aesthetic design, spline, computer aided geometric design, plastic, tension, attraction, structure, aesthetic measure, shape modeling, composition### References

- Farin G. Curves and Surfaces for CAGD, Morgan Kaufmann, 2001, 5th edition.
- Yoshida N., Saito T. Interactive aesthetic curve segments, The Visual Computer, 2006, vol.22, no 9, pp. 896-905.
- Levien R., Sequin C. Interpolating splines: which is the fairest of them all? Computer-Aided Design and Applications, 2009, vol 6, no 1, pp. 91-102.
- Savelov A.A. Planar curves, Moscow, 1960, 294 p.

- Ziatdinov R. Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function, Computer Aided Geometric Design, 2012, vol. 29, no. 7, pp. 510-518.
- Miller K.S., Samko S.G. Completely monotonic functions. Integral Transforms and Special Functions, 2001, vol. 12, no. 4, pp. 389-402.
- Farin G. Class A Bezier curves, Computer Aided Geometric Design, 2006, vol. 23, no. 7, pp. 573-581.
- Yoshida N., Hiraiwa T., Saito T. Interactive Control of Planar Class A Bezier Curves using Logarithmic Curvature Graphs, Computer-Aided Design & Applications, 2008, vol.5, nos.1-4, pp. 121-130.
- Ziatdinov R., Yoshida N., Kim T. Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions, Computer Aided Geometric Design, 2012, volume 29, issue 2, pp. 129-140.
- Ziatdinov R., Yoshida N., Kim T. Fitting G2 multispiral transition curve joining two straight lines, Computer Aided Design, 2012, vol. 44, no. 6, Jun, pp. 591-596.
- Dankwort C.W., Podehl G. A new aesthetic design workflow: results from the European project FIORES. In: CAD Tools and Algorithms for Product Design, Springer-Verlag, Berlin, Germany, 2000, pp. 16-30.
- Harada T., Mori N., Sugiyama K. Curves physical characteristics and self-affine properties, Computer-Aided Design and Applications, 1995, vol. 42, no. 3, pp. 30-40 (Japanese).
- Harada H. Study of quantitative analysis of the characteristics of a curve, Forma, 1997, vol. 12 no. 1, pp. 55-63.
- Miura K.T. A general equation of aesthetic curves and its self-affinity, Computer Aided Design and Applications, 2006, vol. 3, nos. 1-4, pp. 457-464.
- Miura K., Sone J., Yamashita A., Kaneko T. 8th International Conference on Humans and Computers (HC2005), Aizu-Wakamutsu, Japan, 2005, pp. 166-171.
- Ziatdinov Rushan, Miura Kenjiro T. On the variety of planar spirals and their applications in computer aided design, European Researcher, 2012, vol. 27, no. 8-2, pp. 1227-1232.
- Kim M.-J., Kim M.-S. and Shin S.Y. SIGGRAPH 95 Proceedings of the 22nd annual conference on computer graphics and interactive techniques, 1995, pp. 369-376.
- Miura K.T. Unit Quaternion Integral Curve: A New Type of Fair Free-Form Curves, Computer Aided Geometric Design, 2000, vol.17, no.1, pp. 39-58.
- Shoemake K. Animating Rotation with Quaternion Curves, Computer Graphics 1985, vol. 19, issue 3, pp. 245-254.
- Stancu D.D. Approximation of functions by a new class of linear polynomial operators, Revue Roumaine de Mathematiques Pures et Appliquees, 1968, vol. 13, pp. 1173-1194.
- Lupas A. A q-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca, 1987, pp. 85-92.
- Videnskii V.S. Nekotorye aktualnye problemy sovremennoi matematiki i matematicheskogo obrazovaniya. Gertsenovskie chteniya - 2008, St.-Petersburg, RGPU imeni A.I. Gertsena, pp. 134-146.
- Arnkheim R. Iskusstvo i vizualnoe vospriyatie (Art and visual perception), Moscow, Progress, 1974, 392 p.
- Bulatov M.S. Geometricheskaya garmonizatsiya v arkhitekture Srednei Azii IX-XV vv. (Geometrical harmonization in architecture of Central Asia of the IX-XV centuries), Moscow, Nauka, 1978, 380 p.
- Belyaev A.A. Estetika (AestheticsSingular aesthetics Plural aesthetics), Moscow, Politizdat, 1989, 447 p.

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