=Paper= {{Paper |id=Vol-2744/paper68 |storemode=property |title=Multi-Criteria Assessment of Shape Quality in CAD Systems of the Future |pdfUrl=https://ceur-ws.org/Vol-2744/paper68.pdf |volume=Vol-2744 |authors=Valerijan Muftejev,Rushan Ziatdinov,Rifkat Nabiyev }} ==Multi-Criteria Assessment of Shape Quality in CAD Systems of the Future== https://ceur-ws.org/Vol-2744/paper68.pdf
    Multi-Criteria Assessment of Shape Quality in CAD
                   Systems of the Future*

Valerijan Muftejev 1,2 [0000-0003-4352-3381], Rushan Ziatdinov3 [0000-0002-3822-4275], and
                          Rifkat Nabiyev4[0000-0002-0920-6780]
 1 Department of the Fundamentals of Mechanisms and Machines Design, Ufa State Aviation

                       Technical University, Ufa, Russian Federation
                               muftejev@mail.ru
                              http://www.spliner.ru
     2 C3D Labs, Altufevskoe Shosse 1, Office 112, 127106 Moscow, Russian Federation


      3 Department of Industrial Engineering, Keimyung University, Daegu, South Korea

            ziatdinov@kmu.ac.kr, ziatdinov.rushan@gmail.com
                      http://www.ziatdinov-lab.com

    4 Department of Management and Service in Technical Systems, Ufa State Petroleum

                      Technological University, Ufa, Russian Federation
                                 dizain55@yandex.ru

       Abstract. Unlike many other works, where authors are usually focused on one
       or two quality criteria, the current manuscript, which is a generalization of the
       article [35] published in Russian, offers a multi-criteria approach to the
       assessment of the shape quality of curves that constitute component parts of the
       surfaces used for the computer modelling of object shapes in various types of
       design. Based on the analysis of point particle motion along a curved path,
       requirements for the quality of functional curves are proposed: a high order of
       smoothness, a minimum number of curvature extrema, minimization of the
       maximum value of curvature and its variation rate, minimization of the potential
       energy of the curve, and aesthetic analysis from the standpoint of the laws of
       technical aesthetics. The authors do not set themselves the task of giving a simple
       and precise mathematical definition of such curves. On the contrary, this category
       can include various curves that meet certain quality criteria, the refinement and
       addition of which is possible in the near future. Engineering practice shows that
       quality criteria can change over time, which does not diminish the need to
       develop multi-criteria methods for assessing the quality of geometric shapes.
       Technical issues faced during edge rounding in 3D models that affect the quality
       of industrial design product shape have been reviewed as an example of the
       imperfection of existing CAD systems.

       Keywords: High-quality Curve, Class F Curve, Class A Curve, G2 Continuity,
       Shape Modelling, Shape Quality, Technical Aesthetics, CAD.


Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).
* Corresponding author. Tel.: (+82)-053-580-5286
2 V. Muftejev, R. Ziatdinov, R. Nabiyev

                                          ‘There is no such thing as an unsolvable problem.’
                                                                              Sergei Korolev


1        Introduction

In engineering design, plane and spatial curves that specify certain functional
characteristics of an object are known as functional curves [1]. Among the functional
curves, a subclass of engineering curves may be distinguished; such curves prescribe
some design characteristic of an object in a single optimum way. These kinds of curves,
for instance, include the Archimedean spiral used for shaping the profile of gear teeth,
as well as the brachistochrone ‒ the fastest descent curve for transporting items [2]. A
catenary used for dome and hanging structure surface design, as well as the clothoid,
which is used to design smooth roadway transitions [3], can also serve as examples of
engineering curves.
    Engineering curves are widely used for solving various tasks and issues faced in
different branches of technology and industry. Below are some examples:
    1.    The wing profile of an aeroplane creates lift; therefore, when designing a
          profile curve it is necessary to maximize the lift while minimizing the drag.
    2.    A road ensures comfortable and safe vehicle driving at a given speed, which is
          why maximum road smoothness should be achieved within given limits and
          restrictions.
    3.    A cam profile defines the movement of a pusher with a valve to ensure a
          necessary gas distribution pattern; therefore, its design should ensure smooth,
          impactless valve movement.
    4.    The external surface of a vehicle body and the curved architectural shapes of a
          building can also be functional surfaces if one views aesthetics and beauty as
          a design property of a product that determines its usability.
Plane free-form functional curves can be locally convex (with a curvature function of
constant sign) and may feature points of inflection (areas with a curvature function of
variable sign). Furthermore, functional curves may be spatial and may, therefore, have
torsion. Those interested in plane curves are advised to read a well-known reference
book by Savelov [4].
    In their previous works, the authors have defined basic and supplemental quality
requirements for functional curves using smoothness criteria applicable to technical
objects [1], [5-7]. Several studies have been dedicated to the development of modelling
methods for aesthetic curves and their quality assessment from the standpoint of the
laws of technical aesthetics [8-9]. In the present manuscript, these results have been
clarified, expanded and systematized. Functional and aesthetic curves have been viewed
from a unified standpoint; common quality assessment criteria have been suggested.
Methods of modelling the curves meeting these requirements have been reviewed. Key
points of the methods crucial to authors’ priorities have been described in detail in the
manuscript.
               Multi-Criteria Assessment of Shape Quality in CAD Systems of the Future… 3


2      Quality of Geometric Shapes

Regardless of the specifics of the items designed, one can derive universal requirements
for the quality of geometric shapes arising from free-form functional curves. This
section offers a general list of the quality requirements for functional curve shapes that
are invariant as regards the specifics of an item design. Additionally, readers who are
interested in high-quality shapes are recommended to try the FairCurveModeler app,
which can be accessed online at http://fair-nurbs.ru/FairCurveModeler3D.aspx.
2.1    Order of Smoothness Not Less Than 4
Smoothness is a property of a function or a geometric figure (a curve, a surface, etc.)
indicating that this function can be differentiated or that each point of the given figure
has surroundings that can be defined using differentiable functions. Different types of
design use splines of different orders of smoothness. For example, when designing road
routes, clothoid splines are used and smoothness of at least the 2nd order is ensured.
For profiling the camshaft cam of high-speed engines, smoothness of at least the 3rd
order is required; therefore, the profile design begins with drawing a smooth graph of
the 3rd order derivative [10]. To ensure continuity of the torsion function when
modelling spatial curves, the curve must have 3rd order smoothness. A spatial curve
with smooth torsion should have 4th order smoothness, which follows from the analysis
of the spatial curvilinear trajectory of the point particle [11].
    By analogy with the concept of jerk (a quick, sharp, sudden movement) for plane
curves, meaning a sharp change in the rate of change of curvature, we can introduce the
concept of jerk for a sharp change in the rate of change of torsion. Only spline curves
of the 5th degree or higher (at least 4th order of smoothness) provide a smooth change
in torsion and can be used to model functional curves. A surface can be drawn from a
network of plane curves. However, the jets of a medium (air along the wing, water along
the propeller blade, soil along the plough blade) do not flow around an object, in the
general case, along planar curves; they flow around its surface along spatial trajectories
with torsion. If the jet trajectory does not have smoothness 3, then the discontinuities of
the derivatives of the 3rd order will inevitably cause discontinuities in the torque
function. Pulsating moments of forces (sharp pulsations at smoothness order 2 and
smoother pulsations at order 3) acting on spatial jets of the medium cause flow
pulsation, which, inevitably, increases the dynamic resistance of the surface to the
movement of the medium flow. Therefore, planar or spatial curves in the curve network
must also have a smoothness order of 4 or higher. In addition, the formula for defining
a surface on a network of curves must provide an order of 4 or higher for any
isoparametric curve of a surface.

2.2    Absence or Minimum Quantity of Curvature Extrema
The smoothness of the line also depends on the shape of the graph of the change in
curvature along the length of the motion line. According to the basic dynamics equation
[11], oscillations of the curvature function will cause the pulsation of centrifugal forces
acting on the point particle. Therefore, the section of the motion line must have a
minimum number of extrema of the curvature or a minimum number of vertices of the
curve. For instance, the presence of redundant extrema of the curvature in the shape of
4 V. Muftejev, R. Ziatdinov, R. Nabiyev

the designed item may result in the following deviations:
      1.   It can cause undue runout of the pusher that ultimately leads to premature
           mechanism wear.
      2.   It can cause soil build-up on a plough section with curvature concentration
           at the soil movement trajectory, which leads to increased resistance of the
           plough and ultimately increases the energy intensity of the ploughing process
           [12].
      3.   Their presence on the aerodynamic profile can lead to excessive pulsation of
           the medium flowing around the profile, which increases the drag on the
           profile and can cause a flow stall, as well as an increase in the pressure force
           on the profile.
      4.   It can cause the need for excessive braking and acceleration, which would
           ultimately increase the energy required for the movement of a vehicle [13].
      5.   Their presence on the curves of vehicle body part surfaces and architectural
           forms can result in distorting mirror effects [14].
      6.   They may cause incorrect visual perception of computer graphics and CAD
           objects [15].
2.3    Small Values of Curvature Variation and Its Variation Rate
In some applications, a requirement is introduced to minimize the variation in the
curvature. For example, such limitation to the minimum value of the curvature radius
(max curvature) is introduced naturally during a road design, where the minimum bend
radius is limited based on the allowed vehicle speed [16-17].
   An important quality attribute of a curve is the rate of variation in its curvature.
When designing a road route, this attribute defines the rate of centrifugal force increase
impacting a vehicle at bends in the road, and it is easily controlled through applying the
segments of the clothoid with a linear curvature function variation [16-17].

2.4    Small Value of the Potential Energy of the Curve
The curve with a minimum value of potential energy is called an elastica [18]:
                               𝑙1
                    𝐸𝑀𝐸𝐶 = ∫ 𝜅 2 (𝑠) 𝑑𝑠 → 𝑚𝑖𝑛                           (1)
                              𝑙0
It is an axis line of a deformed elastic bar between two fixed endpoints. The quality of
elasticas has been proven by the centuries-old shipbuilding experience. Elastic bars
(physical splines) have been used in the profile lofting of transverse frame ribs, buttocks
and water lines in the design and construction of marine vessels and, later on, in the
production of automobiles and aircraft.
     Mathematically accurate modelling of the contour of a curved physical spline is used
in the KURGLA curve modelling program for the AUTOKON ship design system [14],
[19]. In one of the KURGLA algorithms, a virtual physical spline is approximated by
clothoid segments. According to [20], the curvature between the fixed points of a
physical spline varies linearly as is the case with a clothoid.
               Multi-Criteria Assessment of Shape Quality in CAD Systems of the Future… 5


    The curve smoothness is believed to be directly related to the potential energy of
such a curve. The need to choose a functional curve with a small potential energy value
is justified by the following assumption. When an object with a functional surface
moves at a high speed, the medium flowing around the object behaves like an elastic
body, and less pressure will be required to deform the elastic medium along streamlines
with less potential energy. When a point particle moves along a concave curved path,
with friction taken into account the work spent on moving it will be less with a lower
value of the potential energy of the moving path [1]. This situation is also true for the
point particle movement along a curvilinear plane trajectory given the friction.
    The development of scientific visualization methods opens up new possibilities for
the mathematical modelling of geometric shapes† . There is an opportunity to study
polynomial and nonlinear splines by means of a computational experiment and, as a
result, obtain high-quality visualizations with high resolution. In such visualizations,
points are determined by pixels, and calculating an area with a resolution of 100 × 100
pixels can take several minutes. In [21], visualizations were obtained for the potential
energy function of a quadratic Bézier curve with a monotonic curvature function.

2.5    Aesthetic Analysis from the Standpoint of the Laws of Technical
       Aesthetics, Based on Eleven Criteria
What is a Beautiful Shape?

The beauty of the shape of an industrial product is a measure of quality perfection
expressed in visually perceived characteristics (geometry, colour, style, etc.) of the
shape, resulting from objectively acting conditions: function, design, properties of
materials, compliance with human factors (anthropometric, ergonomic, aesthetic, etc.)
and formed by means of design shaping (proportioning, compositional balance, tectonic
pattern, volumetric spatial organization, colour harmony, etc.).
    The expressed beauty of the shape of an industrial product effectively embodies the
content (function, purpose) and causes a positive emotional and psychological reaction
in a person.

Necessity of Aesthetic Analysis

Design practice carried out in the field of high-tech industrial production through the
mathematical modelling of industrial products and the evaluation at the production site
of manufactured industrial samples is needed to ensure maximum efficiency, economy
and performance of the product throughout its entire life cycle. However, the
performance of an industrial sample is not limited to technical characteristics only. The
product functions in all the variety of its relations with a person who reacts to objective
stimuli and evaluates them not only on the basis of rational judgments and conclusions,
but also in terms of the emotional–sensual attitude to the world. In that sense, a future
design solution should include not only a rational but also an aesthetical feasibility


†     ‘Visualization Methods in the Mathematical Modeling of Interpolating Curves with
Monotonic Curvature Function.’ YouTube, uploaded by Geometric Analysis, December 22,
2016, https://www.youtube.com/watch?v=xIUFKVageu8
6 V. Muftejev, R. Ziatdinov, R. Nabiyev

model at the pre-design analysis stage already. Such a dialectical unity is transformed
into a harmoniously integrated image that gives rise to a motive: an incentive for the
emotional perception of the formal qualities of a shape to be a factor in the desire to
reveal the useful qualities of the product, which in general will determine the value
judgment about it.
    From the standpoint of technical aesthetics, the achievement of the unity of rational
and emotional aspects in the image of a product is determined by the objective laws of
shaping. In terms of their objectification, it is important to identify the characteristics
of the primary elements of a shape, the content of which in many respects sets the
qualitative properties of a product design solution.
    The quality assessment of a curve, including from the standpoint of the laws of
technical aesthetics, must be carried out according to the proposed objective method for
assessing smoothness. A designer who is not restricted by the need to search for an
engineering curve can model free-form curves using given Hermite data.

Aesthetic Analysis of Quadratic Bézier Curves

A planar Bézier curve‡ was used as such a primary element in [8-9]. Its geometric
properties were analysed and evaluated from the standpoint of their aesthetic feasibility
together with the ability to meet the efficiency requirements. The principle of ‘structural
unity of a shape’ was used as a basis for scrutiny of the formative, plastic and expressive
properties of the curves. The shape-forming features of the geometry of the curves were
evaluated according to the following eleven criteria: conciseness-integrity,
expressiveness, proportional consistency, compositional balance, structural
organization, imagery, efficiency, dynamism, scale, plasticity and harmony (more
detailed information on these criteria can be found in [38-39]).
    Art–design analysis of available samples of Bézier curves revealed regularities of
shaping at the level of geometric features of the curve based on the fundamental
principles of the volumetric and spatial organization of the shape [9]. It is also
worthwhile to note that the data objectification was performed by employing a
questionnaire. Its goal was to differentiate curve structure assessment by professionals
creating product samples and their design, as well as the emotional and sensual response
of ordinary consumers to the features of the Bézier curves offered to them for appraisal.
Art and design analysis does not exhaust all aspects of the issue under consideration
and offers the prospect of further development.
    The study in [9] contains a detailed aesthetic analysis of 24 segments of Bézier
curves of the second order, 8 of which had a monotonic curvature function. Each of the
above eleven criteria was assessed according to a seven-point scale from -3 to 3
(maximum degree, medium degree, minimum degree, no criterion, minimum deviation,
medium deviation, maximum deviation). The analysis has shown that in four segments
of the Bézier curves with a monotonic curvature function the rounded average value of
fairness (RAVF) for all criteria is 0; in three segments it is 1, i.e. the criteria are of
minimum degree, and in one curve segment the criteria breach has been identified.
    To support the authors’ conclusions [9], a questionnaire was administered in one of

‡
      ‘Bernstein Polynomials and Bernstein-Bézier Curves.’ YouTube, uploaded by Rushan
Ziatdinov, June 30, 2015, https://www.youtube.com/watch?v=AL0vcsLlYp4
               Multi-Criteria Assessment of Shape Quality in CAD Systems of the Future… 7


the leading schools in Istanbul, Turkey to 240 teenagers from 14 to 17 years of age to
investigate the ‘aesthetic feasibility’ of different segments of Bézier curves, and its
results completely matched those of the authors’ ones. The choice of the age group was
based on consideration of teenagers’ psychological peculiarities in forming an
emotional picture of the world, with the characteristic absence of psychological
dependence of children’s consciousness on professional dogmas, norms and
instructions inherent in the adult audience. In this sense, this age group makes it possible
to give answers to the questionnaires based more on intuition and sensual perception
than on rational judgments and inferences, which is necessary to objectivize the results
of the questionnaire.
    Bézier curves having a monotonic curvature function (class A Bézier curves) are
often considered aesthetic (fair) curves [43], although their aesthetic analysis has never
been performed. A detailed aesthetic analysis carried out in [9] showed that this
statement is erroneous.
    The authors of the current work believe that assessment using the criteria of
smoothness is a priority. An expert assessment from the standpoint of the laws of
technical aesthetics is valid only after an assessment of smoothness or in the absence of
the possibility of such an analysis.

Natural Beauty of Spiral Curves

There is another approach to assessing the aesthetics of a curve that is based on the
mathematical characteristics of shapes found in real-world objects (e.g. the outlines of
butterfly wings) [22-23]. To generate beautiful (aesthetic) shapes, the so-called log-
aesthetic curves§ – which have a linear graph of curvature in a logarithmic scale – are
suggested [24-26]. Many well-known spirals [42], including a clothoid, are special
cases of this class of curves. The most generalized class of curves with a monotonic
curvature function, called superspirals, was introduced in [27] and studied via similarity
geometry in recent works [40-41]. Equations of these curves are expressed through
Gaussian hypergeometric functions and are numerically integrated by adaptive
integration methods such as the Gauss–Kronrod method.


3      Class F Curves

3.1    Modelling Methods
Thus, building a very smooth trajectory of the motion requires a minimum number of
reference points of the generated spline motion trajectory and a high level of smoothness
of at least the 4th order, smooth torsion of the spatial curve, restriction of the maximum
value of curvature and the variation rate of curvature, and minimization of the potential
energy function. Functional curves satisfying these requirements are called class F
curves**,†† [29], [36]. The authors do not set themselves the task of giving a simple and

§ ‘Interactive Aesthetic Curve Segments.’ Personal webpage of Norimasa Yoshida, 2006,

http://www.yoshida-lab.net/aesthetic/pg2006iacs.wmv
** Authors should not confuse this term with so-called F-curves proposed by Ferguson [37].
††     ‘Methods of high-quality surface modeling.’ YouTube, uploaded by Rushan Ziatdinov,
8 V. Muftejev, R. Ziatdinov, R. Nabiyev

precise mathematical definition of such curves. On the contrary, this category can
include various curves that meet certain quality criteria, the refinement and addition of
which is possible in the near future. Engineering practice shows that quality criteria can
change over time, which does not diminish the need to develop multi-criteria methods
for assessing the quality of geometric shapes.
    In the Russian language, a curve model is called the determinant [29], which
consists of the geometric part plus the algorithm for generating curve points or the
procedure for constructing an approximating spline. The geometric part of the
determinant can be considered the geometric determinant of the curve. The most
common and natural forms of the geometric determinant are the sets of points (the type
of the polyline vertices) or the set of tangent lines (namely, the form of the tangent
polyline). Also, the so-called control spline polygons of NURBS curves are used in the
applied geometry. Different types of geometric determinants have their own advantages
and disadvantages. The incidence line enables accurate positioning of the curve, the
tangent line uniquely and accurately sets the shape of the modelled curve, and the
NURBS S-polygon of the high-degree curve enables local change of the shape of the
curve, guaranteeing high-quality spatial curves according to the criteria for smooth
curvature and torsion.
    A general curve modelling algorithm includes the following steps [5]:
      1. Sketching the curve. Preliminary information about a curve may be specified
           as a) a curve gauge and its digital representation, b) multiple points captured
           from a full-scale replica using a measurement device, c) a line drawn by an
           engineer on paper or on a screen and recorded as a digital set of points, d) a
           digital set of points the curve should cross, or e) a fixed analytical curve.
      2.   Plotting a geometric determinant of a curve defining the geometric structure
           of the curve on the sketch.
      3.   Isogeometric approximation of the geometric determinant through
           developing an analytical (or piecewise-analytical) curve of a given class or
           plotting the results of an algorithm for generating curve points based on the
           given parameters of the geometric determinant.
      4.   Transition to another type of curve determinant by equivalent transformation
           or by isogeometric approximation of the curve determinant, editing it using
           the parameters of the new geometric determinant.
      5.   Transition to another type of curve determinant by equivalent transformation
           or by isogeometric approximation of the curve determinant to solve metric
           and positional tasks in CAD systems. In this case, the new curve determinant
           is called a curve pattern.
3.2    Absence or Minimum Number of Curvature Extrema
General requirements for curve modelling methods are formulated in [6], [14] and [29-
30]. These requirements include dimensional stability or isogeometry, invariance under
affine and projective transformations, high quality according to the criteria of
smoothness and aesthetics, flexibility, instrumental diversity and a possibility of using
analytical curves.

August 22, 2018, https://www.youtube.com/watch?v=YuNTTIz7K70 [in Russian].
                Multi-Criteria Assessment of Shape Quality in CAD Systems of the Future… 9


4       Comparative Assessment of Curves

An unbiased comparison of curves requires that they be based on the same Hermite data
and subjected to comparative analysis against the smoothness criteria. When comparing
two curves constructed using the same Hermite data, the number of curvature extrema
is checked, and the curve with the larger number is rejected. Then the order of
smoothness is compared, and the curve with less smoothness is rejected. Further on, the
curves are compared using the value of potential energy. The last stage of this
assessment can be an aesthetic analysis from the standpoint of the laws of technical
aesthetics.


5       Analysis of the Functional Capabilities of CAD Systems in the
        Context of Quality Assessment for the Surfaces of Industrial
        Design Products

Today, computer-aided design systems available on the market provide the industrial
designer with tools to create digital prototypes of products at the required level of
accuracy. Principally, they are generated through solid modelling. However, it is known
that this method does not to fully solve the issue of creating the complex geometry of a
form. Therefore, for this purpose, surface modelling is used. In this regard, many CAD
systems feature special modules focused on creating products with complex surface
geometry. However, for the productive use of this toolkit, an industrial designer needs
to have a sufficiently deep understanding of the theoretical aspects and patterns of the
technology used and needs to spend much time searching for solutions to purely
technical problems while using a software product. Otherwise, the existing product
design concept cannot be fully developed by the creator in the software environment,
which negatively affects the entire design process. A particularly urgent problem is the
rounding of the edges of the 3D model in the process of modelling a design product. In
many CAD systems, the issue of automatic and high-quality edge rounding has not yet
been resolved. But its solution would save designers from the CAD mathematical
apparatus and allow them to focus on the process of finding the optimal form of a
product designed. That is, to proceed to the tasks of their immediate area of expertise.
Let us look at a few examples that visually illustrate the issue of rounding edges in CAD
systems such as Rhinoceros ‡‡ , Altair Inspire Studio §§ , ANSYS SpaceClaim *** and
Autodesk Inventor Professional†††. These programs were given the task of rounding, in
automatic mode, all the edges of a solid body consisting of two mutually perpendicular
parallelepipeds making contact on their faces (Fig. 1).




‡‡ https://www.rhino3d.com/
§§ https://solidthinking.com/product/inspire-studio/
*** http://www.spaceclaim.com/en/default.aspx
††† https://www.autodesk.com/products/inventor/overview
10 V. Muftejev, R. Ziatdinov, R. Nabiyev




                        Fig. 1. A solid body used for edge rounding.




               Fig. 2. Automatic edge rounding result in ANSYS SpaceClaim.

    The edges have been rounded with the continuity of curvature G2. According to [33],
‘G is known to connect profiled curved surfaces with the curvature continuity to the
  2

boundary surfaces. With this connection type, one curve transfers to the other and the
end point of the former coincides with the starting point of the latter one. Besides,
tangent angles and radii at these points coincide.’ The programs under consideration
formally coped with the task of forming secondary surfaces, mating with primary
surfaces with continuity G2 (Figs. 2, 3, 4), except for Rhinoceros. This application failed
the task of rounding edges in the area of their intersection (Fig. 5). A peculiarity of the
edge rounding in Autodesk Inventor Professional 2020 was the creation of a set of extra
surfaces, making the topology of the object more complex (Fig. 4).
      Multi-Criteria Assessment of Shape Quality in CAD Systems of the Future… 11




           Fig. 3. Automatic edge rounding result in Inspire Studio.




Fig. 4. Automatic edge rounding result in Autodesk Inventor Professional 2020.




           Fig. 5. Automatic edge rounding result in Rhinoceros 6.
12 V. Muftejev, R. Ziatdinov, R. Nabiyev




   Fig. 6. Analysis of the surface of the designed object using zebra lines in Inspire Studio.

    It is known that an industrial design product should embody useful beauty in its
form, the characteristics of which express the functional expediency of the product.
From a technical aspect, one of the conditions for creating such a product is the visual
purity of its shape, expressed in the uniform movement of light flare over its surface.
This is especially true for products in which class A and F surfaces are used. In a
software environment, the behaviour of light flare can be predicted by surface analysis
using several methods. Here zebra lines have been used. Zebra lines enabled
identification of the smoothness criterion breach between two surfaces in almost all of
the examples reviewed: object analysis after edge rounding in Inspire Studio, ANSYS
SpaceClaim and Rhinoceros 6 identified sharp bends in the areas of contact between the
rounded surfaces and the original surfaces (Figs. 6, 7, 8).




            Fig. 7. Analysis of the surface of the designed object using zebra lines
                           in Autodesk Inventor Professional 2020.




            Fig. 8. Analysis of the surface of the designed object using zebra lines
                                    in ANSYS SpaceClaim.
               Multi-Criteria Assessment of Shape Quality in CAD Systems of the Future… 13




             Fig. 9. Manually creating and editing edge rounding in Rhinoceros 6.




    Fig. 10. Analysis of the surface of the designed object using zebra lines in Rhinoceros 6.

    Thus, the considered software products could not cope with the creation of the
rounded edges in the automatic mode, smoothly transferring one surface into another.
Autodesk Inventor Professional 2020 created surfaces that met the criteria for
smoothness, but at the same time created many unnecessary surfaces that complicated
the topology of the object. A high-quality result, in which the zebra lines did not reveal
excessively high curvature in the locations of mating surfaces, became possible with the
manual edge rounding process (Fig. 10). Rhinoceros 6 has the best toolkit for this
purpose.
    The above analysis emphasizes the urgent need to improve the software kernel in
order to automate many routine operations related to the modelling of industrial design
products, one of which has been examined above.


6       Conclusion

This manuscript suggests a multi-criteria approach to assessing the quality of the shapes
of functional curves that form surfaces, the quality of which substantially determines
14 V. Muftejev, R. Ziatdinov, R. Nabiyev

the functional characteristics of designed objects. The aesthetic functional curves are
proposed to include the aesthetic curves that form the basis for shaping industrial design
products and determining their consumer properties.
    Based on the analysis of the motion of a point particle along a curved path,
requirements for the quality of functional curves for the unstressed smooth motion of a
point particle have been developed. A general list of quality requirements for the
functional curves is defined (high order of smoothness, minimum number of extrema
of the curvature, minimization of the maximum value of the curvature, minimization of
the variation rate of the curvature, minimization of the potential energy of the curve).
Additional requirements for aesthetic functional curves are defined from the standpoint
of the laws of technical aesthetics. The curves satisfying the requirements for the
functional curves are defined as class F curves.
    To compare the quality of various CAD systems, an analysis of the rounded surfaces
of the 3D model obtained in different computer-aided design systems according to the
smoothness criterion G2 has been carried out. The purpose of a visual demonstration by
the method of comparative analysis of the low quality of fillets in various CAD systems
was to show the imperfection of the mathematical apparatus of the geometric kernel in
the software that was used.


7      Acknowledgements
We would like to thank the reviewers for their thoughtful comments and efforts towards
improving our manuscript.

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