mathematical design & technical aesthetics 1(1)

ISSN 2306-1405

Institute of Design & National Arts, Ufa State University of Economics and Service, Russia

Department of Computer & Instructional Technologies,Fatih University, Istanbul, Turkey

National Centre for Computer Animation, Bournemouth University, UK

Moscow State Pedagogical University & International Academy of Pedagogical Education, Moscow, Russia

Department of Fine Art, Kazan State University of Architecture and Engineering, Kazan, Russia

National Centre for Computer Animation, Bournemouth University, UK

Department of System Analysis, National Research Nuclear University «MEPhI», Moscow, Russia

Ufa College of Arts, Ufa, Russia

Department of Fine Art, Kazan State Universityof Architecture and Engineering, Kazan, Russia

MATHEMATICAL DESIGN & TECHNICAL AESTHETICS

№1/2013

ISSN 2306-1405VOLUME 1

ISSUE 1

Scientific journalThe journal has been published since 2013

Rushan Ziatdinov, Rifkat Nabiyev.Editors’ note.

Rushan Ziatdinov, Rifkat Nabiyev, Kenjiro T. Miura.MC-curves and aesthetic measurements for pseudospiral curve segments.

Emmogulsum T. Ardashirova. Global problems of «man-art-science» triad in the methodology of prognostic studies of Leonardo da Vinci.

Albina F. Basharova, Konstantin S. Ivshin. Contemporary principles of 3D-modelling in industrial design education.

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Founding editors

Rifkat I. Nabiyev

Rushan Ziatdinov

Editorial board

Alexander Pasko

Emmogulsum T. Ardashirova

Iskander V. Rafikov

Valery Adzhiev

Victor V. Pilyugin

Industrial advisory board

Ilshat H. Nabiyev

Ilyas I. Rafikov

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Our official sponcor is JSC «RARITEK», Naberezhnye Chelny, Russia

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Rushan ZiatdinovDepartment of Computer & Instructional Technologies,Fatih University,34500 Büyükçekmece, Istanbul, TurkeyE-mail: [email protected], [email protected] URL: http://www. ziatdinov-lab.com/

Rifkat I. NabiyevDepartment of Fine Art and Costume Art, Faculty of Design and National Culture,Ufa State University of Economics and Service, 450068 Ufa, RussiaE-mail: [email protected]: http://www.nabiyev.mathdesign.ru/

science and art, as a condition of the technical progress of humanity and spiritual development of the personality.

The main interests of the journal are novel discoveries and developments in mathematical design and their applications, as well as in technical aesthetics, including but not constrained to the following topics.Curves and surfaces in CAGD;• Geometric and topological methods for shape

and solid modelling;• Computer aided design of aesthetic surfaces;• Aesthetic measures for curves and surfaces;• Aesthetic measures for designing objects;• Industrial and scientific applications;• Computer art;• Improvement of the scientific means of

The journal “Mathematical Design & Technical Aesthetics” publishes original papers of advanced scientific value and survey papers, as well as short communications in all realms of CAGD, geometric modelling, computer aided aesthetic and ergonomic design, technical aesthetics, art criticism and plastic arts.

One of the cordial goals of this journal is integrating

Editors’ note

ON THE AIMS AND SCOPE OF THE JOURNAL “MATHEMATICAL DESIGN & TECHNICAL AESTHETICS”

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personality and promotion of the best world design examples;

• Aesthetics as a methodology of art and its role in art studies development.The audience consists of designers, artists,

applied mathematicians, computational scientists and engineers.

For manuscript submission and other issues contact the editorial assistant at:

[email protected]

technical aesthetics as design methodologies;• Problems of domain-spatial environmental

humanization by means of design with reliance on human values and ideals;

• Questions about the dialectic unity of science and art as conditions for the improvement of scientific tools and art as means of design expressiveness;

• Training of specialists in the various directions of design and ergonomics;

• Upbringing of the design culture of society as a progressive indicator of the erudition of the

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MC-CURVES AND AESTHETICMEASUREMENTS FOR PSEUDOSPIRAL CURVE SEGMENTS

Rushan ZiatdinovDepartment of Computer & Instructional Technologies,Fatih University,34500 Büyükçekmece, Istanbul, TurkeyE-mail: [email protected], [email protected] URL: http://www. ziatdinov-lab.com/

Rifkat I. NabiyevDepartment of Fine Art and Costume Art, Faculty of Design and National Culture,Ufa State University of Economics and Service, 450068 Ufa, RussiaE-mail: [email protected]: http://www.nabiyev.mathdesign.ru/

Kenjiro T. MiuraDepartment of Information Science and Technology,Graduate School of Science and Technology, Shizuoka University,3-5-1, Johoku, Naka-ku, Hamamatsu Shizuoka, 432 JapanE-mail: [email protected]: http://ktm11.eng.shizuoka.ac.jp/profile/ktmiura/welcome.htmlя



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Abstract. This article studies families of curves with monotonic curvature function (MC-curves) and their applications in geometric modelling and aesthetic design. Aesthetic analysis and assessment of the structure and plastic qualities of pseudospirals, which are



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curves with monotonic curvature function, are conducted for the first time in the field of geometric modelling from the position of technical aesthetics laws. The example of car body surface modelling with the use of aesthetics splines is given.

Keywords: MC-curve; spiral; pseudospiral; aesthetic curve; superspiral; multispiral; curvature monotonicity; high-quality curve; aesthetic design; spline; computer-aided geometric design; plastics; tension; gravity; structure; aesthetic measurement; shape modelling; composition.

1. IntroductionAesthetic appeal of industrial products is

a very important factor for their successful promotion in the market. Most of the curves and surface profiles used within traditional CAD/CAM systems [1] have polynomial or rational parametric form and do not meet high aesthetic requirements [2]. One of their disadvantages is the difficulty of controlling the monotonicity of the curvature function.

2. Curves with monotonic curvature

functionComputer-aided geometric design refers to

monotone-curvature curves as fair curves [3]. Unfortunately, this attitude is based only on well-known geometric principles, and the laws of technicaяl aesthetics are not taken into account. Therefore, in this work we prefer to avoid this term, using “MC-curve” (monotone-curvature curve) instead.

MC-curves include spirals with monotonic curvature function (Euler spiral, Nielsen’s spiral, logarithmic spiral, involutes of a circle), pseudospirals [4] and so-called log-aesthetic curves [2], actually a linear reparameterization of pseudospirals. The curves comprise the superspiral [5] family, curvature function of which is given by the Gaussian hypergeometric function satisfying conditions of strict monotonicity with several limitations applied to its parameters [6]. Recently, class A Bézier curves with monotonic curvature function were proposed [7]; detailed analysis has shown, however, that when the polynomial degree is increased the curve restricts to a logarithmic

Fig.1. New conceptual design1 of a car created by log-aesthetic curves, multispirals and kinematic spiral surfaces1Created by Prof. Takashi Hada and Tomonobu Nishikawa (Faculty of Design, Shizuoka University of Art and Culture, Japan).



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spiral [8]. Controlling the Bézier and B-spline curve’s (for polynomial degree n>2) curvature function’s monotonicity requires deeper analysis and the development of corresponding algorithms. Curves satisfying the monotonicity of curvature function are widely used in car surface design, bevel design, development of transition curves in highways and railroad design, and font design [9-10], and are a crucial element of aesthetic design [11]. Figure 1 shows a new conceptual design of a car created with aesthetic curves and multispirals [10].

3. Mathematical models of log-aesthetic curves

From the perspective of aesthetic design and potential applications, the log-aesthetic curves1 are the most interesting; their mathematical theory has been elaborated in a number of studies [12-15] [2] [9]. By studying properties of multiple attractively shaped curves’ characteristics in real and virtual-world objects, Toshinobu Harada and his research team discovered that their curvature functions were linear or near-linear, depending on the natural parameter (arc length) on the logarithmic curvature graph (LCG) [12-13]. Natural equations of these curves are as follows [8]:

(1)where α is the shape parameter and λ is the

scaling factor. The Gauss-Kronrod numerical integration method has been used for curve segment computation in [2]. In [9], parametric equations for (1) in terms of incomplete gamma functions were found; the obtained equations allowed the computation of curve segments with a high degree of accuracy. A computation experiment staged in [2] yielded drawable regions for the control point, which is determined by the directions of unit tangent vectors in the initial and end points of the aesthetic curve segment. It was found that the Euler (α = -1) and Nielsen’s (α = 0) spirals, as 1 The term log-aesthetic curve (LAC) is used in foreign literature as suggested by Prof. Carlo H. Sequin from the University of California (USA).

well as the involutes of a circle (α = 2), have limited drawable regions for the control point, determined by the directions of unit tangent vectors in the initial and end points; the curve segment which fits with tangent directions therefore does not always exist. Nevertheless, judging by the computation experiment carried out in [2], the two-point Hermite interpolation problem always has a solution for spirals with the shape parameter 0 ≤ α ≤ 1, but this hypothesis has not been analytically proven. Yoshida and Saito [2] studied the behaviour of reflection lines on ruled spiral surfaces and came to the conclusion that when kinematic surfaces are rotated they lack reflection line oscillations, which is one of the features of their high quality.

Some examples of application of C1 class aesthetic splines can be found in [16].

4. Unit Quaternion Integral CurvesThe class of a Unit Quaternion Curves q(s) in

the SO(3) rotation group was first introduced in [17]. This study also proposed a method enabling transformation of a curve, determined by a sum of basis functions, into its unit quaternion analogue in SO(3). Given a spline curve in R3, the spline curve is reformulated in a cumulative basis form and the corresponding quaternion curve is constructed by converting each vector addition into the quaternion multiplication. For example, using the method proposed in [17] for splines such as Bézier, Hermite and B-splines, unit quaternion curves can be derived, many differential properties of which are invariant.

Unit quaternion integral curve (QI-curve) is defined as:

(2)where is the arc length and is the arbitrary

unit vector [18]. In this case, the quaternions, especially unit quaternions, are useful to describe rotations and are used to control the direction of the tangent vector, which adds some efficiency and simplicity in the design of aesthetic curve shapes. Quaternion coordinates are considered ideal for orientation interpolation of objects [19].

Due to the fact that the QI-curve is defined



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by a tangent vector, which is controlled by unit quaternion curve, the arc length being its natural parameter, a more convenient manipulation of its curvature function is possible than using the polynomial parametric curves in CAD/CAM systems.

5. Historical, theoretical and methodological foundations of science and art integration in the context of aesthetic analysis of pseudospiral curve segments

Since ancient times, the ability to cause sensual empathy and a positive state of mind, and to encourage creative activities, was a matter of learning not only in art, but also many scientific disciplines. History suggests that the search for ways to improve living space has been inspired by knowledge of the world and born of positive spiritual incentives; in broad terms, enjoyment of life is conditional upon the integration of precise scientific disciplines with forms of arts.

Note that this is not to be understood narrowly. These are powerful mechanisms that have formed a certain style of human thinking and modelled human perception of the world, connected to a large extent with the interests of developing certain personality types: ascetic, dogmatic, scholastic, humanistic, etc. Knowledge of the mechanisms influencing the psyche of an individual through art allows the simulation of consciousness, as well as the content approved in the attributes of form through a specific attitude expressed in the formal signs of the results of aesthetic understanding of the world. There is an example from history: “In Ancient Egypt, where one of the means of expressing ruling class ideology was architecture, the chief architect, in addition to his duties, carried out the duties of vizier, keeper of the seal or high priest” [24]. The great monuments of any culture are the first examples illustrating the means of influence on human consciousness, and thus on human mentality. United by their components, these monuments were intended to assert the power of the rulers and establish the norms of peoples’ spiritual life.

Thus, art in all cultural and historical periods has had a clearly specified nature, since the

psychological potential of its instruments was a strategic mechanism for forming moral and ethical behavioural standards, which determined the content of human spiritual ideals.

It is no coincidence that information could be brought to the human consciousness most effectively if form gave rise to interest and stimulated the perception of its characteristics. That is, this form had to suggest a specific idea to the perceiver as quickly as possible. It is possible to arouse an immediate effect of interest using human psychological mechanisms. The form must have “useful” qualities which can help to actualize the possibilities of effective influence on the psycho-emotional sphere of an individual.

This is about the beauty of form as an expression of the highest indicator of the unity and integrity of all of its components.

In their own way, beauty or disharmony of form in human consciousness causes an instinctive reflex. By comparison with the ethical sphere of personality, we can discover an interrelation in the nature of this reaction: the human reaction to moral aspects can be positive or negative. Such categories as “measure”, “beauty” and “harmony”, used in evaluating the external quality of form, are also relevant in evaluating an individual’s moral qualities.

Methodologically it is important to note that consideration of the question of evaluating the beauty of form was historically based on the union of Art and Science, allowing analyses to be made using two mutually complementary perspectives. This dialectic of interaction between two forms of social consciousness makes it possible to exclude unreasonable declarative conclusions and solve problems in well-reasoned and demonstrative forms.

In view of the above, we emphasize that the best minds of Ancient Greece accepted the idea of achieving harmony in Art by means of exact sciences. Thus, “under the influence of Pythagorean philosophy, which proclaimed ‘that all things are regulated and can be learned by the strength of numbers’, the Canon of Polykleitos arose in sculpture (5th century BC), and the Grid Plan developed in urban planning”. Socrates was the first to formulate the thesis that



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the beautiful is not just the sum of beautiful parts: “conformity and subordination are required to create artistic unity” [24]. Plato proved the importance of harmonizing the proportions of form, influenced by the discovery of irrationality

( 2 ) in Ancient Greek mathematics. It is worth noting that this discovery negated the importance of the mathematical atomism of the Pythagorean School, as its original idea was based on idealistic philosophy.

In this regard, for example, Abu Nasr Al-Farabi, one of the greatest Eastern scientists, while acknowledging the Pythagorean theory of music, rejected the teaching of the Pythagorean connection between music and the movement of the stars as nothing more than a baseless invention. Ibn Sina said about this that he did not seek to establish a link between the state of the sky, the behaviour of the soul, and musical intervals, for “it is the custom of those who cannot distinguish one science from the other” [24].

We should note, however, that, in its purest form, proportioning based on the golden section is rare in the practice of fine art, design, and architecture. Of greater importance for the achievement of aesthetic value and reasonability is to measure the consistency of the “golden section” with other means of expression.

Also of methodological significance in the context of the integration of science and art is the work of the famous Roman architect Vitruvius, who came to identify certain numerical relationships in proportion to the human body, thus creating a scientific basis for ergonomic systems of proportioning. These later gained acceptance in the fine arts and architecture under the name of “The Vitruvian Man”. The Italian Renaissance provides an example of cultivating ideas for the unity of science and art in order to nurture a new type of socio-cultural identity, an ethical relationship to the world that is determined by the depth and breadth of true knowledge. It was no accident that such a cultural setting for the disclosure and realization of human artistic potential found itself logically embodied in the outstanding people of the Renaissance, and above all in the

work of Leonardo da Vinci, who is known as the pioneer of design.

Thus, in connection with the above, it is possible to come to a definitive conclusion that the union of science and art creates the conditions for nurturing a humanistic style of thought in man which affirms the unity of beauty and usefulness in the spiritual and material space of life, as a leading ethical principle.

6. Aesthetic analysis of pseudospiral curve segments

Let us consider the features of the curves (1) presented in the figures, and endeavour to describe their aesthetic properties from the point of view of technical aesthetics laws.

Aesthetic judgement is a way to establish the aesthetic value of an object, to be aware of the result of aesthetic perception, usually fixed in judgements like “It’s beautiful”, “It’s ugly”, and so on [25]. In this sense, addressing the evaluation of curves’ aesthetic properties should be done in the framework of human nature in terms of aesthetic perception, where latent semantic depth of form can be revealed.

When performing aesthetic evaluation, an important indicator of curves’ beauty is harmony. This covers both substantive and formal features of the object and is rated as the highest form of its organization, order, and structural integrity.

It is crucial to detect the mechanisms of making a form integral and harmonious in the totality of its qualities, free from a gustative approach and an idealistic interpretation of beauty. Here we should define the extent of how a person’s subjective aesthetic mode of evaluating the formal indices is related to its objective aesthetic value, taking into consideration the relativity aspect in the estimation of an object value. It presupposes that in the framework of one system its aesthetic characteristics might constitute value, while in the framework of the other the value is constituted by its usefulness. The subjective criteria of aesthetic evaluation should be formed on the basis of objective beauty factors. Such a principle will to a huge degree define the objectivity of the



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judgements made by a person, acting as a recipient in various spacial, material and cultural circumstances of interaction with other objects.

It should be noted that by the concept of “form” we understand the unity of the external and intrinsic properties of an object. In an aesthetic assessment of those properties of form which result in a corresponding human emotional response, one has to identify the objective laws of the design process which are intended to provide integrated and harmonious contours. At the same time one must take into account that, in the design of an object, the form must also comply with the requirements of aesthetic, functional and economic value. In this regard a design item should contain no unjustifiable random elements. This is particularly relevant with decoration which may conceal underlying discordancy and create the effect of false beauty. Such compensations for imperfect form result in a loss of self-sufficiency – an impairment of structure. Here only substantial unity of the decorative attributes as part of the shape itself can provide the aesthetic, ergonomic, economic and social effects which ultimately result in positive value-judgements of the item.

Assessment of the appropriateness of the aesthetic features of curves should be based on the actual properties expressed in the plastic characteristics of those curves. It is noteworthy that applying certain theoretical design elements to them allows the identification of the propriety of their use (Figs. 5, 6).

With regard to the assessment of the aesthetic properties of form, the main issue is to identify those mechanisms in shaping a design object which provide a structural balance of composition and integrity related to perceptions of the attributes of form, since any violation of such structural relationships leads to a deformation of the entire compositional structure.

Since human perception involves visual judgement, compositional laws are perceived by an individual as objectively active conditions for the comfortable perception of an object, taking all its properties as a whole. The objectivity of those conditions is realized through the

psychological mechanisms of human perception and their corresponding reactions to specific stimuli: visual, tactile, auditory, etc.

These mechanisms are considered within the framework of perception regularities inherent in all persons due to the common functioning principles of the higher nervous system. Thus, it is known that the process of human reaction within visual perception is transformed into visual judgement “which is not the result of intellectual activity since the latter takes place after the process of perception has been completed” [23]. It logically follows that “the characteristics of objects perceptible to the eye constitute an essential, intrinsic part of the visual process as such” [23].

One such characteristic is “intensity” as a quality of the object’s formal properties and a characteristic of the dynamic interaction of internal “forces,” one of these being “attraction.” Tension, in turn, “has its value and direction – the criteria which can be treated as a psychological force” [23]. The value of intensity depends on the nature of the combination of active “forces”, represented as the interaction of the object’s attributes resulting in certain qualitative changes between the elements of the composition.

At the same time, the arrangement itself is defined by guides or axes that can be any curves along which the elements are structured and grouped with each other. These theoretical guides may be characterized as representing significant factors for harmonization of relations between composite elements, as well as prerequisites to optimize the properties for formal features of facilities.

Observe the two graphs with curves (1) in Fig. 2. The principle for “structural interaction of curves with coordinate axes” is used as the basis for analysis of curves’ aesthetic properties. The curve on the left graph is more homogeneous and stable than the one on the right graph. The laconism of the curve is stipulated by a more visible connection with the coordinate axis: the distance of the upper top is smaller in comparison with the curve on the right example; the attractive force to the axis is more strongly expressed. The feeling of larger stability and integrity is evidence of reasonability



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and utility, which are transformed into aesthetic reasonability at the level of sensory perception and are characterized by the estimations “more beautiful” and “more pleasant”. The low curves in Fig. 2 demonstrate the maximum stress ranges, marked with arrows.

Fig. 3 shows three plots with examples illustrating the stress range of each curve. Comparing the charts, it is possible to conclude that middle curve (b) is the most compliant with terms of feasibility. It is almost a hemisphere (characterized by a symmetrical shape); hence, it is concise and self-sufficient.

Options (a) and (c) lack the integrity inherent to the curve in example (b). Development of these curves is not finished, with the logical support of another object (required to compensate for their instability with respect to coordinate axes).

Fig. 4 shows an object perception pattern

Fig. 2. Comparative aesthetic measurements for

Euler spiral (α = 1) (a) and Nielsen’s spiral (α = 0) (b).

(depending on change of its properties). The figure shows a curve (g) (Euler (Cornu) spiral, (α = –1)) with different options of thickness. Upon increase of the line thickness, the visual weight of the curve varies (thus affecting the plastic features). Fig. 4 (e) shows a critical thickness, which does not interfere with feasibility of the curve plastics. Increase in thickness of the curve results in visual deformation of its plastics: the curve is visually compressed. Reduction in thickness of the curve results in loss of structural integrity.

Figure 5 shows various decorative interpretations of the curve Euler (Cornu) spiral (α = –1) shown in Fig. 2 (a). The curve is taken as a path along which various forms were distributed and grouped (using such composition tools as “rhythm” and “colour”). In these examples, the decorative character of



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the curve is its leading feature, which does not meet the requirement of unity in function and form (as in design).

Figures 6 and 7 evidence the existence of technical object form-generation, which reveals properties of analysed curves.

7. ConclusionPlane curves with monotone function of

curvature are widely applied in computer geometric design and computer graphics. The field of their application is quite broad, ranging from font design to simulation of aircrafts’ and ballistic rockets’ surfaces.

The performed aesthetic analysis and evaluation of plastic properties of presented

curves from the viewpoint of technical aesthetics vividly show the practicability of using tools for mathematic modelling of geometrical forms. Complex industrial products utilizing functions of knowledge-intensive design involving mathematical methods should embody unity of usefulness and beauty as a condition of harmonization of natural and artificial matters. This principle may not be implemented without formation of geometrical structures and their analysis from the viewpoint of aesthetic practicability, which is dialectically integrated in any matter or phenomenon.

Aesthetic analysis does not cover all aspects of the problem in question and offers very

Fig. 3. Comparative aesthetic analysis of plastic properties of the curves: involutes of a circle (α = 2) (a) quasi-circle (α = 10) (b) and logarithmic spiral (α = 1) (c).



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promising prospects for further development; for example, this method can be used for curve estimation in a 3D Cartesian coordinate system.

Thus, our analysis demonstrates the ability to implement the classical concept of science and art combined to produce an integrated approach to geometrical shaping, realizing its

full potential both for design and engineering purposes.

In our opinion, development of algorithms to generate and control the curvature for class A Bézier curves, as basic functions containing generalized Bernstein polynomials, i.e., Stancu- [20], Lupas- [21] or Vidensky-type

Fig. 4. Object perception depending on change of its features.

Fig. 5. Ornamental interpretation of Euler (Cornu) spiral (α = -1)

based on structuring and grouping of geometrical primitives along the

spiral’s curve.



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linear operators [22], seems to offer great opportunities. We will address these issues in our further work.

References

[1] Farin, G., 2001. Curves and Surfaces for CAGD, Morgan Kaufmann, 5th edition.

[2] Yoshida, N., Saito, T., 2006. Interactive aesthetic curve segments. The Visual Computer 22 (9), 896–905.

[3] Levien, R., Séquin, C., 2009. Interpolating splines: which is the fairest of them all? Computer-Aided Design and Applications 4, 91–102.

[4] A. A. Savelov, 1960. Planar curves, GIFML: Moscow.

[5] Ziatdinov, R., 2012. Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function. Computer Aided Geometric Design 29(7): 510-518.

[6] Miller, K.S., Samko, S.G., 2001. Completely monotonic functions. Integral Transforms and Special Functions 12 (4), 389–402.

[7] Farin, G., 2006. Class A Bézier curves.

Computer Aided Geometric Design 23 (7), 573–581.

[8] N. Yoshida , T. Hiraiwa, T. Saito, 2008. Interactive Control of Planar Class A Bezier Curves using Logarithmic Curvature Graphs, Computer-Aided Design & Applications 5(1-4), 121-130.

[9] Ziatdinov, R., Yoshida, N., Kim, T. 2012. Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions. Computer Aided Geometric Design 29 (2), 129–140.

[10] Ziatdinov, R., Yoshida, N., Kim, T., 2012. Fitting G2 multispiral transition curve joining two straight lines. Computer Aided Design 44 (6), 591–596.

[11] Dankwort, C.W., Podehl, G., 2000. A new aesthetic design workflow: results from the European project FIORES. In: CAD Tools and Algorithms for Product Design. Springer-Verlag, Berlin, Germany, pp. 16–30.

[12] Harada, T., Mori, N., Sugiyama, K., 1995. Curves’ physical characteristics and self-affine properties, Design Research 42 (3), 30-40 (in Japanese).

[13] Harada, H., 1997. Study of quantitative analysis of the characteristics of a curve, Forma

Fig. 6. Norman Bel Geddes. Rapid intercity bus.



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12(1), 55-63.[14] Miura, K.T., 2006. A general equation of

aesthetic curves and its self-affinity. Computer Aided Design and Applications 3 (1–4), 457–464.

[15] Miura, K., Sone, J., Yamashita, A., Kaneko, T., 2005. Derivation of a general formula of aesthetic curves. In: 8th International Conference on Humans and Computers (HC2005). Aizu-Wakamutsu, Japan, pp. 166–171.

[16] Rushan Ziatdinov, Kenjiro T. Miura, 2012. On the variety of planar spirals and their applications in computer aided design, European Researcher 27(8-2), 1227-1232.

[17] Kim, M.-J., Kim, M.-S. and Shin, S.Y.,1995. A general construction scheme for unit quaternion curves with simple high order derivatives, in: SIGGRAPH ‘95 Proceedings of the 22nd annual conference on computer graphics and interactive techniques, 369–376.

[18] Miura, K.T., 2000. Unit Quaternion Integral Curve: A New Type of Fair Free-Form Curves, Computer Aided Geometric Design 17(1), 39-58.

[19] Shoemake, K., 1985, Animating Rotation with Quaternion Curves, Computer

Graphics 19, 245-254.[20] D.D. Stancu., 1968. Approximation of

functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pur. Appl. 13, 1173 - 1194.

[21] Lupas, A., 1987. A q-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on Numerical and Statistical Calculus, Preprint 9, 85-92.

[22] В.С. Виденский, 2008. Замечание о рассмотренных А. Лупасом рациональных положительных операторах // Некоторые ак-туальные проблемы современной математи-ки и математического образования («Герце-новские чтения - 2008»). СПб.:РГПУ имени А.И. Герцена, 2008. С. 134-146.

[23] Арнхейм Р. Искусство и визуальное восприятие. – М.: Прогресс, 1974. – 392 с.

[24] Булатов М.С. Геометрическая гармо-низация в архитектуре Средней Азии IX-XV вв. (историко-теоретическое исследование). – М.: Наука, 1978. – 380 с.

[25] Эстетика: Словарь/ Под общ. ред. А.А. Беляева и др. – М.: Политиздат, 1989. – 447 с.

Fig. 7. Norman Bel Geddes. Model of streamlined “car of the future”,

1933.



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Ключевые слова: ренессансная концеп-ция «варьета», идеал человека с мыслящим мировоззрением в контексте взаимосвязи искусств и наук, вариативность прочтения методологии творчества Леонардо да Винчи в свете гносеологии и герменевтики искусст-ва.

Синтез искусств и наук в отражении «вечных» тем жизни человека, природы, вселенной

Искусство и наука в своей исторической подвижности и преемственности «вечных» тем и проблематики неустанно сохраняли ведущую тенденцию развития - это обра-щенность к человеку и глобальным пробле-мам жизни на земле.

GLOBAL PROBLEMS OF «MAN-ART-SCIENCE» TRIAD IN THE METHODOLOGYOF PROGNOSTIC STUDIES OF LEONARDO DA VINCI

Emmogulsum T. ArdashirovaMoscow State Pedagogical University & International Academy of Pedagogical Education, Moscow, Russia

Эта ориентация получает особую трактов-ку, когда мы обращаемся к классическому научно-художественному наследию, в кото-ром можно найти параллели с актуальными проблемами современности в их истоках.

Ёмкое и проникновенное суждение выда-ющегося художника-исследователя эпохи Высокого Возрождения Леонардо да Винчи относительно мастерства и техники изобра-зительного искусства, живописного портрета, что живописец должен писать две главные

Fig. 8. Автопортрет.Ок. 1510 - 1515 г.г.

Сангина.




19

20



21

вещи: человека и представление его души, обретает глубокую образную значимость и для осознания социально-нравственного смысла искусства и науки: не погубить чело-века, не погубить душу… .

В социальной и духовной фазе своей эво-люции искусство и наука в лице их выдаю-щихся лучших представителей отстаивали высокую философскую идею сохранение и утверждение идеала человека с мыслящим мировоззрением.

Неслучайно именно в эпоху Возрождения возникло представление и об интеллиген-ции: « интеллигенты… все те, кто выраба-тывает линии и знания, беря на себя роль

носителей критического разума, историче-ского, нравственного и иного самосознания и самоизменения общества»1

Известно, что Леонардо да Винчи очень резко выступал против средневековых схо-ластических и дуалистических толкований человека, пытаясь строить духовную куль-туру человека на твердом фундаменте ес-тествознания, искусства, технических наук, пронизывая этот сложный синтез культурной гуманистической значимостью.

Как известно, крупными учеными мира раз-ных времен на международных конферен-циях, симпозиумах ООН, ЮНЕСКО наряду с множеством проблем, в том числе и взаимо-действия искусства и науки, центром внима-1 Баткин Л.М. Итальянские гуманисты: стиль жизни и стиль мышления. – М.: Наука, 1978. – с. 17.

Fig. 10. Рисунок цветущей лилии. 1480 – 1485 г.г. с. 169.

Fig. 9. Джоконда.



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ния которых является человек, обсуждалась не угасавшая проблема – сохранение мы-слящей личности в противовес возможности и допустимости «формовки людей» как ан-тиценностное явление, пагубно влияющее на прогресс искусства, науки, образования, культуры, и в целом на человеческое разви-тие и существование.

Ведь, как в своё время подчеркивал ав-тор книги «Формовщики людей» В. Пакард «возможности манипулирования человеком настолько велики, что мы обязаны взять на себя социальную ответственность за это».2

В решении этой глобальной проблемы формирования мыслящей, творческой лич-ности, безусловно, огромная роль отводит-ся сфере образования, включая школьное и

2 Фрагменты из этой книги опубликованы в журнале «Иностранная литература» (1981. №12).

высшее, научному предвидению в воспита-тельной работе.

Напомним, что еще в 1971 году академи-ком П.Л. Капицей были остро поставлены проблемы творческого воспитания и образо-вания современной молодежи.

Проблема реализации духовно-творческих потенций человека во многом была решена именно в Эпоху Ренессанса на основе синте-за искусств и наук, утвержденном в научно-художественном опыте Леонардо да Винчи, ставшим классическим примером в истории человечества, истории науки.

«Открытие» духовно-творческих возмож-

Fig. 11. Ветка ежевики. Ок. 1505 г.

Fig. 12. Мадонна Бенуа.


23


1624



25

ностей человека решалось, как правило, с помощью методов искусств и наук, образу-ющих некоторую «дополнительность», аб-солютизирующая многообразие творческих, внутренних потенций человека.

В основе творческой деятельности такого типа личности лежала категория «вырьета» (разнообразие) и как воплощенная варьета в жизни был сам Леонардо да Винчи.

Очевидно следует рассматривать концеп-цию «вырьета» в научно-художественной деятельности Леонардо да Винчи как гносе-ологический прием добывания истин в раз-ного рода науках и искусствах, а также как

стиль мышления художника – исследовате-ля – философа.

Новый взгляд на союз искусства и науки, новый характер концепции гуманизма приво-дит Леонардо да Винчи к утверждению типа личности, имеющей емкое разностороннее мышление, мыслящее мировоззрение.

Построение духовной культуры человека как важнейшая заслуга Леонардо да Вин-чи было обосновано им не просто благими пожеланиями, благими стремлениями, а ре-ально выстраивалась на методологическом фундаменте множества открытий естествен-ных и гуманитарных наук, центром внимания которых был человек, когда изменение чело-века соотносилось с изменением общества, культуры, цивилизации и наоборот измере-

Fig. 14. Рисунок дерева.Перо, итальянский карандаш.

Fig. 13. Святая Анна с Марией и младенцем Христом.



ние новой науки, нового искусства мысли-лось с изменением нового человека и нового общества.

Многообразие и сложные проблемы, ко-торые встают в современной действитель-ности, перед современной образовательной сферой требуют соотнесение их с более ши-роко понятыми социальными целями, гума-нистическими идеалами, с этическими цен-ностями общества как целого.

Социальная значимость методологических позиции Леонардо да Винчи на искусство, науку, технику не утратила своей глубины и составляет неисчерпаемый резерв и для преобразования методологии вузовского об-разования.

Имея в виду многообразный круг неотлож-ных проблем, которые должны волновать представителей искусства и науки вузовской среды, где каждый из нас мог бы сказать:«каждый, в ком сидит Рафаэль, должен иметь возможность беспрепятственно раз-виваться» (К. Маркс и Ф. Энгельс); «меня мучает, что в каждом человеке, быть может, убит Моцарт» (Антуан де Сент Экзьюпери).

Леонардо открывает внутренний мир че-ловека в синтезе искусства и науки систем-ными силами, и как никто другой, тесно свя-

Fig. 16. Мадонна Литта.

Fig. 15. Кусты. Ок. 1508 – 1509 г.г.


26


27



28

Fig. 17. Тайная вечеря. 1495 – 1498 г.г.

Fig. 18. Этюд для фрески «Тайная вечеря» (Иуда Искариот). Ок. 1495 г.

зывая искусство и науку, придавая при этом каждой из сфер универсальное значение.

Так, например, портретное искусство Лео-нардо да Винчи в равной мере бесценно для науки и для искусства.

Проблема выражения внутреннего мира человека в его внешнем эмоциональном проявлении – движении, жесте, позе, ми-мике, взгляде – развивается в стройную теорию, своего рода «грамматику языка» духовного состояния человека, в познании которого Леонардо интересовало все, на-чиная с развития в утробе матери, кончая строением скелета и черепа.

Изображение человека, в частности, в набросках к «Тайной вечере» было своего рода научно-художественным исследова-



Fig. 19. Голова Христа. Ок. 1495 г.

29

нием жизни людей, каждого изображенного лица отдельно взятого, где пластическая анатомия, психология характера, рисунок, математика и композиция, изучение и рас-крытие этической стороны жизни составля-ют единое целое.

Исследовательский дух пронизывает ка-ждое выражение лица и положение рук как глубинное стремление добиться воплоще-ния той или иной этической мысли, этических свойств души: любовь, страдание, страх… .

В изображении ветки, дерева Леонардо да Винчи также подходит и как уче-ный и как художник, поэтому его зарисовки использу-ются и в современных учебниках ботаники,

Fig. 20. Этюд для фрески «Тайная вечеря» (Апостол Иаков Старший)

и наброски архитектуры. Ок. 1495 г.



30

живописи, рисунка.Леонардо разрабатывает методику на ос-

нове законов науки и искусства: он описы-вает законы филлотаксии (управляющие расположением листьев на стебле) и законы гелиотропизма (описание влияния солнца и гравитации на растения) и одновременно со-ставляет зарисовки и указания как рисовать.

Научное наблюдение и методические ука-зания к рисунку сохраняют его научное ви-дение: …чем дерево и его ветки толще, тем они темнее; … если ветка расположена на фоне двух веток, то самые яркие ее части кажутся самыми светлыми и многие другие, изложенные в «Трактате о живописи».

Глубокий научный ум и живописное виде-

Fig. 21. Голова Христа. Рисунок Леонардо для фрески «Тайная вечеря».

Fig. 22. Этюд для фрески «Тайная вечеря» (Апостол Варфоломей). Ок. 1495г.



31

ние явлений природы в сочетании с фило-софскими размышлениями оборачивались в его творчестве созданием не только ше-девров искусства, но и шедевров человече-ской мысли, предсказывающие проблемы будущего искусства, науки, природы и чело-веческой жизни на земле в грядущем мире, сопровождающие его социальные катаклиз-мы, экологические бедствия, духовно-нрав-ственные искания человека (в хаотичном мире), которые так остро осознаются и в наши дни.

Дух кассандры живет не только в техниче-ских изобретениях Леонардо да Винчи, но и в произведениях, отражающие глобальные проблемы этики человеческой жизни на зем-

Fig. 23. Этюд для фрески «Тайная вечеря» (Апостол Филипп).

Ок. 1495 г.

Fig. 24. Этюд младенца Иисуса для картины «Мадонна с Младенцем и Святой Ан-ной».

Ок. 1501 – 1510 г.г.


ле, к примеру, апокалипсическая серия «По-топ» из 10-ти мотивов.

Глубокое понимание Леонардо того, что человек часть бесконечного мироздания природы породило многостороннюю трак-товку предназначения человека на земле, его эстетического и нравственного отноше-ния к жизни человека и природы, вселенной.

Апокалипсическая серия «Потоп» предста-ет как форма изобразительно-философского размышления над широким комплексом во-просов диалектики жизни и смерти челове-ка и природы, призывая к осмыслению: что ждет человека и природу как духовно-мате-риальную нишу человека в будущем, в гря-дущем мире? Самоуничтожение человека и природы как этика расправы за содеянные человеком разрушения духовных и природ-но-материальных ценностей, составляющие эстетическую ценность человеческого бы-

тия?... .В серии рисунков «Потоп» Леонардо да

Винчи демонстрирует мастерство в переда-че движений, закругленных форм, как рисо-вать ливень, струи дождя, яростный ветер, вывороченные деревья, динамику потопа в соединении с устрашающим философским выводом: приговор человечеству – конец света… .

Гуманистическое предназначение ме-тодологии прогнозирующих исследова-ний Леонардо да Винчи

«Возвращение» к классическому опыту объединения и переплетения искусств и наук Леонардо, центром внимания которых является духовный мир человека и природа, вселенная, это своего рода «возвращение» к тем истокам концепции мира и личности, которые не менее, если не более остаются существенными и для искусства, науки, об-


32

Fig. 25. Рисунок ребенка. Ок. 1499 г.


разования XXI века, также нуждающегося в своих титанах мысли и духа.

В соответствии с герменевтической кон-цепцией существенное значение имеет ва-риативность прочтения множества смыслов методологии исследовательского творчест-ва Леонардо, построенной на диалектике искусства и науки, ядро которой составляют универсальные проблемы бытия, объединя-ющие прошлое и будущее человечества.

Переосмысление методологии прогнози-рующих исследований Леонардо в области искусства, науки, техники обретают сегодня особую актуальность:

- ориентирует на выявление возможностей исследовательского духа человека – «варь-ета», духовно-творческих потенций индиви-дуальности, не обезличивая и не унифици-руя ее склад мышления;

- показывает возможности методологиче-

ских и методических приемов исследования проблем искусства и науки в создании ше-девров искусства и научных трудов;

- раскрывает искусство как процесс очи-щения души, катарсические возмож-ности искусства в воздействии на этическую сто-рону жизни человека;

- предвосхищает будущность синтеза искусств и наук, возникновение мон-тажного художественно-технического стиля мышле-ния, взаимосвязи чертежа, рисунка, проекта, диалектическая основа которых послужит зарождению нового типа художника – дизай-нера – исследователя;

- позволяет соотнести учет связи глобаль-ности проблем диалектического единства искусств и наук с их социально-духовными аспектами, что является очень важной пред-посылкой для теоретической и практической разработки интегративных научно-учебных


33

Fig. 26. Анатомические зарисовки. 1508 – 1510 г.г.


пособий, способствующие реализации ин-теллектуальных и духовных (нравственных, художественных, эстетических) способно-стей учащейся молодежи в сохранении и развитии духовно-материальных, природ-ных ценностей вселенной для человеческо-го существования;

- углубляет понимание социального значе-ния этических основ искусства и науки, этики образования, предполагающее выдвижение и решение тех этических проблем, которые по сути составят позитивную перспективу воспитания, эстетическое отношение к жиз-ни человека и природы, действующую этику будущих поколений. Какой она будет?... .

Главное – не утратить представление о Че-

ловеке и его Душе… .Литература1. Фролов И.Т. О человеке и гуманизме: Ра-

боты разных лет. – М.: Политиздат, 1989.2. Леонардо да Винчи. Суждения о науке и

искусстве. – СПб.: Азбука, 2001.3. Ардаширова Э.Т. Интеграция музыкаль-

ного искусства с естественно-математиче-скими и гуманитарными науками в педвузе. – М.: «Гуманитарный издательский центр Владос», 2001.

4. Баткин Л.М. Леонардо да Винчи и осо-бенности ренессанского творческого мыш-ления. М., 1990.

5. Санти Бруно. Леонардо да Винчи. Пер. с итальянского С.И. Козлова. М., 1997.


34

Fig. 27. Анатомические зарисовки. 1508 – 1510 г.г.

Fig. 28. Анатомический анализ движений плеча и шеи.

Ок. 1509 – 1510 г.г.


1735

Fig. 29. Наброски фигур для фрески «Тайная вечеря».

Ок. 1480 г.

Fig.30. Наброски цветов. Ок. 1481 - 1483 г.г.

Перо и чернила на тонированной бумаге.

Fig. 31.Набросок композиции фрески «Тайная вечеря».

Ок. 1495 г.


36

CONTEMPORARY PRINCIPLES OF 3D-MODELLING IN INDUSTRIAL DESIGN EDUCATION

Albina F. BASHAROVA,Konstantin S. IVSHINUdmurt State University



37

AbstractThis article will consider the principles of

modern 3D modelling in industrial design, aiming to establish the relationship between the process of thinking through an object’s structure



38

and its computer modelling. 3D modelling in industrial design is classified by individual elements. The features of modelling A-class, B-class, C-class surfaces are determined. Four principles of digital modelling (traditional, inverted, generative and interactive) are identified and described.

KeywordsModelling, industrial design, traditional,

inverted, generative, interactive, principles.1. IntroductionComputer 3D modelling is the principal means

employed in contemporary design to implement a designer’s conception of an object’s shape; it is also an effective tool throughout the whole creative process, from sketching to layout drawing.

Computer 3D modelling is now classified by model structure: frame, polygon, surface, solid, finite element, generative modelling [1-4]. Each modelling type has its own place in the industrial design process. However, this classification fails to properly consider 3D modelling application principles and their interaction in product design. Industrial design is in constant need of shape renovation, as well as the rethinking of principles of objects and space interaction and organization. Reduction of the time required for product modelling also remains a highly topical problem. Applying cross-disciplinary principles to design education contributes to the achievement of the solutions to these issues.

The wider objective of the article is to describe

interdisciplinary connections in research on contemporary modelling principles aimed at design sustainability, searching for new object shapes and analysing innovatory modelling types. Implementing these principles in design education will enable students to grasp modelling as an integral process, rather than a separate discipline: from sketching to prototyping and the resulting production. The principles under study are arranged by complexity: Bachelor’s and Master’s level, and consequent acquirement of contemporary modelling skills.

Modern shareware 3D-scanners and budget-priced 3D-printers make the below-listed principles implementable not only in expensive design companies, but by the students themselves, both at home and in their universities. The principles were put into practice by the authors within the academic process at the Udmurt State University, Russia.

Grasping the below-listed principles of computer 3D modelling enables students to adequately apply modelling efficiently for particular projects and search for innovations to implement designers’ ideas, promoting new thinking and creative approaches to modelling as a design method.

2. Principles of computer 3d modellingModelling principles research is carried out

by design students at various levels in the education system. Bachelor’s degree students focus on traditional and inverted modelling principles under the following scheme: 1st

Fig. 32. Traditional principle of computer 3D modelling.



39

year – 2D and 3D frame modelling; 2nd year – basics of surface and solid modelling; 3rd year – finite element modelling; 4th year – integration of modelling types. At Master’s degree level, two contemporary modelling principles are taken into close consideration: generative and interactive (described below). The above-mentioned modelling types are usually integrated to be put into practice, thus building a particular principle of object shaping.

After analysis of the way modelling types are commonly employed and integrated both by students and small design studios, four main modelling principles have been couched.

2.1. Traditional principleThe traditional principle consists of four

sequential stages (see Figure 32): sketch modelling (or polygonal and mesh modelling), surface modelling (A/B/C-class), solid modelling, and prototyping.

The traditional modelling principle is taught during all years of Bachelor courses. Acquiring competence in applying this principle enables a designer to visualize conceptions in a three-dimensional computer environment, study object shaping methods, and easily set up quick object prototyping. Various model shapes can thus be created from drawings and digital sketching in a relatively short time. It is highly valuable for modelling conceptual art objects or conceptual cars, as well as for conceptual design solutions for products of plain, average and advanced shape complexity.

The sketch (polygonal or mesh) modelling stage implies creation of two-dimensional (drawings, drafts, profiles, sketches) or three-dimensional (digital sketching polygon model) graphics of an object. Graphical data become the basis for further surface modelling. An adequate surface model serves as a ground for further development of a solid model. Solid modelling is based on the resulting surface model as follows: a designer sets thickness, then adds structure and building units. CAM and CAE software systems enable design via both solid and surface models, depending on reporting type and particular system requirements. The resulting solid model can be 3D- or CNC-printed (other printing facilities can also be employed). The inverted modelling principle is based on similar stages; however, these stages are connected differently.

2.2. Inverted principleThe inverted principle is laid out in five

consecutive stages (see Figure 33): modelling, 3D scanning, surface modelling (А/В-class), solid modelling, prototyping.

The inverted modelling principle resembles the traditional one, but supposes creation of a surface model resulting from 3D-scanning of a handmade product draft, or a prototype. This principle is becoming more common among students and design studios due to the availability of budget-priced home-use 3D scanners and downloadable ‘David’ [5] shareware.

Fig. 33. Inverted principle of computer 3D modelling.



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Using 3D scanning data, it becomes possible to model precise object shapes. Designers can also include appropriate element data. Object shaping is determined by the prototype, i.е., by its source data. This approach is valuable for modelling objects at the final design stage and for object shape restyling, as well as being applicable to the creation of objects with partially unified elements. Compared to traditional modelling, this principle proves more complicated, due to the necessity of creating or obtaining a handmade prototype and taking extra time to 3D-scan the project; still, strict compliance of the object parameters with the output product is highly achievable.

Surface modelling plays the most significant role in both the traditional and inverted modelling principles. During their studies of these modelling types students commonly face the problem of A-class surface modelling. This stage implies major object shaping without specifying material thickness. Solid models are much harder to handle than other modelling types and surface modelling is commonly used at creating 3D polysurface. Thus, the inverted principle places the surface modelling stage before the surface solid stage.

The creation of A-class surfaces using various software applications remains a highly topical issue in current research publications. The

Fig. 34. А, В, and С-class surfaces.



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principles of generating surfaces are common for all existing applications. However, close study of the issue reveals neither a common term to derive an A-class surface, nor any assessment criteria, plotting methods, or classification. Therefore, students tend to have a lot of questions on A-class surfaces. How can you tell the class? Are there any other surface classes?

Based on a study of current scientific background and ‘Autodesk Alias Studio’ and ‘ICEMSurf’ software for A-class surface modelling, we have subdivided multiple surfaces into three classes by visual quality: А, В, and С (see Figure 34). The classes are specified by

quantitative parameters: presence or lack of particular adjacency types (G0, G1, G2, G3), surface integrity (lack of accidental breaches), etc. Qualitative parameters, e.g., material specifications, are also taken into consideration (glossy/opaque texture).

An A-class surface is a multiple surface constructed on the ground of higher order continuity inheritance G2, G3 for smooth blending and G0 for edge modelling. When visualized, the model appears invisibly adjacent with gradient highlights all across the surface. A-class surfaces are applicable to modelling objects with glossy texture, complex-shape moulding, and high-quality product design.



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A B-class surface is a multiple surface entirely constructed by continuity types not above G1 (G1, G0). The visualization of the model appears with invisible adjacency, but still acquires certain highlight creases. The model is applicable to design of products with opaque and semi-opaque surfaces of all types, like vacuum cleaners, hair dryers, mobile phones, etc.

A C-class surface is entirely constructed with the inheritance not above G0. This surface class is visualized revealing partially visible adjacency and creased highlights. Applicable to least-significant model sections, it is commonly used for modelling of various spare parts, such as engines, gear boxes, etc. The model is widely used in part modelling, when design conception is of lower importance; it is commonly integrated with B and A-classes, enabling creation of peculiarly aggressive object shapes. Employing a C-class surface means sticking to engineering modelling.

Complex glossy texture modelling, e.g., in car body shaping, employs all continuity types G0, G1, G2. Each type application depends on both the designer’s objectives and the exact method used to divide the complex surface into simple ones. The main task of a designer in complex A-class surface modelling is to split the model into simple surfaces and strive for the visual effect of a solid surface, i.e., to make a highlight glide

gradually across the multiple surface sections and to vary its direction and curvature, fully guided by the conception rather than modelling issues.

An A-class surface is a multiple surface generated by two or more adjacent surfaces with G0, G1 and G2 inheritance to ensure conscious operation and variation of highlights for the purposes of object presentation improvement and further end-to-end modelling. The most common means of highlight operation is manual surface modifying, which requires considerable extra time.

Modelling objects that it makes sense to create via A-class surface plotting are those with compound object structure and of complex convex-concave shape, like, for instance, car or aeroplane bodies, complex casings (e.g., of a gadget, helmet, etc.), and certain art objects and household appliances where thorough product shaping and high-quality surface generation (e.g., high-quality products) are required. Otherwise, A-class surface modelling proves economically impractical due to the required time expenditure.

2.3 Generative principleThe generative principle is represented in

three stages (see Figure 35): information modelling, geometry modelling, and prototyping. Information and geometry modelling is not a sequential but an interpenetrating process.

Fig. 35. Generative principle of computer 3D-modelling.



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Generative (parametric) modelling is becoming more and more popular among architects and designers. This principle is widely applicable to parametric and generative architecture, interaction design, and exhibition design, when the object system varies via time and space parameters, or complex multiple non-recurring structure of the system is involved. Output: complex-shaped structure dissolvable into sectors, patterns, fractals, etc. Application of the interactive principle is currently experimental in industrial design. Industrial designers rarely focus on generative modelling packages, paying attention mainly to the two former principles. The task of this article is to underline the significance of the generative and interactive modelling principles in encouraging students’ creative and innovative approach to models they generate.

The philosophy of generative architecture and design (parametricism) [6] is based on study and elaboration of object system algorithms, applicability of fractal geometry principles, and visualization of physical, biological and mathematical phenomena. Thus, minimal surfaces recently derived via mathematical formulas are currently being adapted by designers thanks to generative modelling software. Minimal-surface-based objects economize material to the utmost, alongside being rigid and aesthetically impressive [7]. Gaudi’s membrane structures and surfaces also inspire constructions and images of proponents of parametricism – for instance, the Gaudi Stool designed by Bram Geenen [8].

The generative principle is based on generative modelling as a synthesis of information and geometry modelling. At the information modelling stage an object is conceptualized and infographically projected. The developed conception is implemented via generative modelling (‘Rhinoceros’ software with ‘Grasshopper’ plug-in). An object information model is a system (‘definition’) consisting of different parameters, components and connections; parameters contain data, components contain actions. This information model further enables a designer to modify a

model at various design stages and indicate particular model parameters as variables in compliance with different applications and external environments, etc.; the designer may also create object systems on the ground of complex non-recurring shaping principles (fractal geometry, mathematical sequences, and physical phenomena of membranes). An elaborated information model can be generated as a linear, surface, or solid model at the third modelling stage with further prototyping. Contemporary generative modelling software is capable of developing schemes for unfolding a 3D object as a flat pattern, unfolding flat compound cross-sections, and numbering model elements. These properties greatly improve the process of visualizing a real object via generative modelling.

Putting this principle into practice revealed that adjusting a generative model to different production methods, or 3D-printing requirements, turns out to be faster and more effective due to the ability to modify online a lot of properties (such as material thickness, or section quantity).

Perfectly new shapes and constructions most often arise from innovative production technologies. However, innovative modelling technologies can also result in completely new shaping principles. As an example, the work of Zaha Hadid [9] can be mentioned: along with active use of the existing production technologies, she has developed unique shaping based on little-implemented fractal modelling.

For instance, the project of a city bench was elaborated on the ground of the generative modelling principle (see Figures 36, 37). The 3D model possesses the complex of variable parameters to adjust the object to environmental and exploitational needs. Here, the variable parameters are the following: bench length, sitting options, geometrical arrangement (straight, arched, or waved), thickness of plywood (depending on the product), number of segments, and even number of sections, as the basis of the object shape. All the parameter data are contained in the generative model,



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which can be varied with a geometry model on a designer’s or producer’s web-site, thus building interaction between a consumer and a designer.

An example of potential consumption can be given as follows: a consumer visits an online shopping site in order to buy a bench for his garden, where he is limited in terms of space. He finds a bench of a shape to his liking, but still its length does not fit. He presses ‘Modify’ and submits the following data: number of people – 3, length – 1.5 m, colour – optional. Then he presses ‘Pay and produce’. The individual model is then uploaded to ‘FabLab’ [11] and manufactured by smart machines, with certain processes of final assembly remaining with the consumer or an assembly crew (one

more novelty in modern production). In this case, a model is a whole system of objects, some adjustable transformer, able to modify its shape in accordance with a consumer’s needs or environment. Projecting one model, a designer can achieve a scope of products; hence, the cost and time are reduced. This idea is applicable for cushioned and case furniture, light fittings, park and garden furniture, garden fixtures (arbours, pavilions), and so on.

New design structures therefore promote new structures of production and consumption. They reduce the time required for complex multiple object and structure modelling, as well as allowing the modification of all object parameters at the final modelling stage, when the geometry model outcome has already been

Fig. 36. Object information and geometry model of city bench concept [10].



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generated. This enables extra 3D shaping research, adjustment of system parameters to suit the environment, simultaneous creation of scaled objects with similar features but different proportions, or other properties.

Examples of the generative modelling principle include: Zaha Hadid [12], ‘The Layer Table and Chair’ by Dyvikdesign [13], ‘Branchpoint’ project from Russia [14].

2.4. Interactive principleThe interactive principle is represented

in three interconnected stages (see Figure 38): generative modelling, ‘Firefly’ decoding, ‘Arduino’/‘Freeduino’ interactive prototyping.

The interactive principle is based on integration of the generative model and a shared interactive prototyping platform via information model

decoding (‘Firefly’ plug-in). Such platforms allow the creation of digital sensor systems (light sensor, aspect sensor, etc.) for environmental interaction, servo controllers, motors, and so on [15]. This modelling principle enables the creation of generative models able to interact with the environment and people through the sensors employed (see Figure 41). Creation of interactively variable kinetic prototypes in compliance with computer generative model variation is also available.

Problem revealed: currently, to generate a prototype students need to have basic skills in wiring electric schemes for sensors to be connected with the electronic prototyping platform ‘Arduino’. Still, this problem is successfully solved by the availability of a great

Fig. 37. Prototyping using Dimension SST 1200 ES.

Fig. 38. Interactive principle of computer 3D modelling.



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number of online video-lessons on the Internet, as well as a fairly simple application principle developed by the Arduino company.

Students’ experiments with such systems can result in brand new objects equipped with innovative functions (a great example is a washing machine that tweets its owner when the laundry is done). The influence of such objects can be analysed by the students with the help of the interactive principle. Application sphere: establishment of synergies between an object/a 3D model and the environment, research into people’s reaction to objects and their shape, imitation of people in how an object reacts, imitation of communication.

For instance, the ‘Itwig’ project focuses the owner’s attention on the plant and creates an invisible emotional connection between plant and man (see Figures 39, 40).

The ensemble of the project represents a tree-twig, a bush-twig, and a twig set not far from trees, bushes or small plants. Just drive the Itwig object into the ground next to the plant, plug it in and admire!

The main idea which differentiates this project from other sensory analogues of ‘smart gardens’ is that it turns ordinary gardening into communication with nature. ‘Itwig’ needs no water or fertilizer, but gives signals and blinks when a man approaches, or sends e-mails. Sometimes it might be used like an illumination art installation (without plants). The twig in the pot helps beginners and avid botanists alike to take care of plants. The tree-twig and the bush-twig can also be used both as décor, operating in night-highlighting mode or for New Year’s illuminations – light-emitting diodes look quite dull during day-time and become

Fig. 39. Itwig’ concept of smart garden [10].



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brighter at night. With the help of sensors inset into the object, ‘Itwig’ can determine seasons and weather conditions, and create a special atmosphere.

The objects consist of a hard plastic frame, a metal rod, a three-colour LED, sensors, a control module, an accumulator and solar batteries. They have Wi-Fi and are automatically programmed according to species of plants and climatic zone. The consumer only has to choose a plant species on a computer or mobile phone.

Other projects that could be listed as examples of interactive modelling principle employment include: ‘Vertical Lake’ by the Australian design studio ‘Urban Art Projects’ [16], and ‘Kinetic Pavilion’ by the designers Yannick Bontinckx and Elise Vanden Elsacker [17].

3. ConclusionHaving revealed and systemized four

modelling principles, we can draw the following conclusions. Generative and interactive modelling principles have fewer stages, require less time and enable complicated object shape modelling, as compared to traditional and inverted principles. The generative modelling principle allows separate processing of geometry and information, and the latter enables the modification of object shape parameters at any modelling stage. However, A-class modelling is currently available only for the two former principles.

Traditional and inverted principles are propaedeutic; they ensure basic knowledge of a shape and shape modelling and develop the spatial three-dimensional thinking required for shaping. These principles also prepare students for practice and provide a basis for studying the generative and interactive principles at Master’s level. Employment of the two latter principles enables students to work with a large amount of modelling data, implementing both theoretical and practical tasks of industrial design.

Hence, the revealed differences serve as significant decision-making criteria for particular principle application and insight into computer 3D modelling.

Any curve or surface can be derived via mathematical formulas, thus being decoded

into pure information; however, traditional and inverted modelling principles do not require a designer to manually decode geometry into information, because this part of the work remains with the software. The basis of the two former principles (modelling ground) is geometry modelling, while the latter principles are based on information modelling. This means, for instance, that when using the generative principle a student does not, for the most part, deal with curves and surfaces as they are; he or she deals with their component systems containing information on each step or action (bend, shift, spatial attitude), as well as


Fig. 40. Scenarios of interaction, information modelling.

Fig. 41. IInteractive prototyping using Arduino Uno.


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sets of input and output data (parameters). This way, the process becomes more flexible, but still more intricate. Hence, the recommendation has been outlined: Bachelor’s degree students focus on traditional and inversion modelling principles under the following scheme: 1st year – frame modelling; 2nd year – basics of surface and solid modelling; 3rd year – finite element modelling; 4th year – modelling types integration. At Master’s degree level, two contemporary modelling principles are taken into close consideration: generative and interactive, as described above. This approach enables Master’s degree students to keep up to speed with trends in modelling theory, as well as to apply the described principles to multivariate tasks research, i.e., when removal of certain parameters to ensure further variation of source data is required (applicability scope is very wide, e.g., ergonomics in creating generative model of an automatically adjustable chair).

References[1] Russo M. Polygonal Modeling: Basic

and Advanced Techniques, 2005, Wordware Publishing Inc..

2] Lombard M. Solid Works, 2011, Wiley Publishing Inc.

[3] Krüger R. Three Dimensional Finite Element Analysis of Multidirectional Composite

DCB, SLB and ENF Specimens, ISD-Report No. 94/2, University of Stuttgart, 1994.

[4] Khabazi M., Algorithmic modeling with Grasshopper, 2009 [online]. Available at: http://proquest.safaribooksonline.com/1587050773 [accessed 01/03/2012].

[5] DAVID Vision Systems GmbH. David 3.x user manual pages [online]. Available at: http://www.david-laserscanner.com/wiki/david3_user_manual/overview [accessed 01/03/2012].

[6] Shumacher P. Parametricism as Style - Parametricist Manifesto, 2008 [online]. Available at: http://www.patrikschumacher.com/Texts/Parametricism%20as%20Style.htm [accessed 01/03/2012].

[7] Brakke K. Triply Periodic Minimal Surfaces, 2000 [online]. Available at:

http://www.susqu.edu/brakke/evolver/

examples/periodic/periodic.html [accessed 01/03/2012].

[8] Geenen B. Gaudi Stool [online]. Available at: http://www.studiogeenen.com [accessed 01/03/2012].

[9] Zaha Hadid Architects, [online] Available at: http://www.zaha-hadid.com [accessed 01/03/2012].

[10] Basharova A. F., Generative Bench project, 2011 [online]. Available at: http://cargocollective.com/bishenka [accessed 01/03/2012].

[11] Fablab from Torino [online] Available at: http://www.fablabtorino.org [accessed 01/03/2012].

[12] Dyvik J., Dyvikdesign [online]. Available at: http://www.dyvikdesign.com [accessed 01/03/2012].

[13] Branchpoint project, 2010 [online]. Available at: http://branchpoint.ru [accessed 01/03/2012].

[14] Johnson J. K., Payne A. Firefly Primer – version 1.006, 2011 [online]. Available at: http://www.fireflyexperiments.com [accessed 01/03/2012].

[15] Arduino. Open-source electronics prototyping platform [online]. Available at: http://www.arduino.cc [accessed 01/03/2012].

[16] Zalek Y. Another Wave in the Wall: Vertical Lake Building Façade, 2010

[online]. Available at: http://weburbanist.com/2010/07/23/another-wave-in-the-wall-vertical-lake-building-facade/?ref=search [accessed 01/03/2012].

[17] Elsacker E., Bontinckx Y. Kinetic pavilion, 2011 [online]. Available at: http://www.kineticpavilion.com [accessed 01/03/2012].


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mathematical design & technical aesthetics 1(1)

Design

bournemouth university

fatih university

free fatherland

russiadepartment of

computer animation

beloved country

industrial design education

rifkat nabiyev