**“There
are no barriers to human thought.”**

Sergei Korolev

There is an extensive class of products
with functional curves and surfaces that determine their essential design
characteristics. These are, for example, external contours of aircraft,
vessels, working surfaces of soil-cultivating units, cam profiles, and axial
lines of the road route, airfoils, turbine blades, and compressors.

In [1-2], basic requirements to the quality
of functional curves are constructed by motion analysis of a material point
along the trajectory of a spatial curve. These requirements, as generalized in
[3], relate to the smoothness criteria and are invariant with respect to the
specific operation of a particular product. They are as follows:

1.
The order of smoothness not lower than four.

2.
Absence or the minimal number of curvature
extremes.

3.
Small values of variation for the
curvature and the rate of its change.

4.
The small value of the potential energy of the
curve.

5.
Aesthetic analysis from the standpoint of the
laws of technical aesthetics.

Curves of high quality that satisfy the
above smoothness criteria are called class F curves (Functional) [3-5], in
contrast to the term class A (Aussenhaut, from its German origin) curves, which was borrowed by Gerald
Farin from the SYRKO CAD system description of Daimler-Benz AG for the modeling
of external surfaces of the car body [6].

Everything is designed with humans in mind;
a product is tailored to the needs of its consumer. Products are created by
means of material and spiritual production. They fill the person’s living
space—a place where he reveals the product’s content and gives it an
appropriate assessment. However, man evaluates the finished product by its necessity,
in the aggregate of all its qualities. The usefulness of the product is primarily
obtained through the visual communication channel. In this sense, the perceived
shape of the product expresses, in its characteristics, the main qualitative
indicators. As a rule, a harmonious form is an adequate reflection of the main
technical, operational, and aesthetic qualities of an industrial product. In
turn, the aesthetic qualities of the shape, embodied in design tools, play an important
role in the product’s sales. "Commercial success of any product depends
largely on its design" [23]. Furthermore, a large part of product design
can be attributed to its qualitative surface geometry. High aesthetic qualities
of the shape contribute to the evaluation of the car by the buyer, and also
become significant for many other industrial design objects. In the context of
a car’s industrial design, its basic aesthetic aspect is understood as the
ability of a shape based on the high quality of the geometric relationships of
the individual shape modeling areas among themselves, positively influencing
the psycho-emotional sphere of its user, and forming the advertising aesthetic
image necessary for the purchase of goods. Its content depends on the given
compositional structure of the shape, various aspects of which are oriented
towards providing a positive emotional perception of the geometry of the
product surface by a person in different subject-spatial conditions with any
kind of lighting. In this case, it is implied that, regardless of the nature
and conditions of illumination (e.g., scattered, direct incident light,
daytime, night, etc.), the steadily smooth motion of the light flare on the
surface of the car's shape should be an indicator of the ideal smoothness of
the connection of individual shape modeling areas of the product. The high
culture of smoothness of the transition between individual surfaces creates a
visual purity of the shape, which, at the level of the visual communication channel,
is transformed into an aesthetic quality.

It is important to emphasize that the
aesthetic properties of the shape of the design product, with the ideal
conjugations of the shape modeling surfaces, remain constant in any
subject-spatial situations, do not collapse depending on the perceived conditions
and experience of the product’s aesthetic image. This is due to the fact the
emotional-psychological state of the consumer is transformed in a different
object-spatial environment.

Thus, aesthetics become an important
component of the product. Clearly, smoother curves and surfaces are more
aesthetic [25]. Actually, the term a smooth curve carries an aesthetic
component. In this article, the basic criteria for smoothness are considered as
aesthetic criteria in the evaluation of the quality of shapes.

In work [3], the curves, the aesthetic
qualities of which are an important component of the consumer properties of the
product, are defined as aesthetic functional curves. To assess the quality of
aesthetic functional curves, a multi-criteria approach is proposed. The authors
of [3] believe that, in the evaluation of a product’s design, priority should
be given to its fulfillment of the smoothness criteria. Expert assessment from
the standpoint of the laws of technical aesthetics is valid only after
assessment for smoothness or in the absence of the possibility of such an
analysis.

To visualize class F curves and surfaces,
the FairCurveModeler software package was developed, implemented in AutoCAD
[7], Fusion360 [8], nanoCAD, COMPAS [9], Mathematica [4] applications, and the Web
FairCurveModeler cloud product [10]. Translating a program to other CAD systems
using their API presents a laborious task. Also, the functional expansion of
the FairCurveModeler software package in all its implementations also proves laborious.

In [11], an idea is proposed about the
possibility of using a universal software platform that is invariant with
respect to the specifics of certain CAD systems. As such a software platform, the
use of a combination of Microsoft Excel VBA, the FairCurveModeler software
complex, and computer algebra systems is proposed. The software platform is
designed to perform the following tasks:

•
*For visualization of high-quality curves and
surfaces for any CAD system*. In this instance,
there is no need to transfer the functionality of FairCurveModeler to other CAD
using their API. It is enough to write tools for exchanging geometric models
between Microsoft Excel and CAD or use standard exchange via a DXF file.

•
*For development of specialized applications
(applied CAD) using the functionality of FairCurveModeler*. To do this, a set of open source programs that implement the
interface between Microsoft Excel and FairCurveModeler is provided.

A necessary component of the software platform
and, in general, CAD, are also systems of computational mathematics. The
necessity and importance of the mathematical apparatus for CAD were well
understood by Samuel Geisberg and Mike Payne, who introduced the Mathcad
package into PTC products [12]. Computer algebra systems, together with the
FairCurveModeler software, are used in the following ways:

•
Calculation of the Hermite data, necessary for the
approximation of the analytic curve by NURBS curve and input in the CAD of the
analytical curve in the form of the NURBS template [13]. Hermite data is
represented in the form of a table of coordinates of the points of the support
polyline, the vectors of the first derivatives, the positive curvature values,
the lengths of the segments of the curve between the support points and the
unit curvature vectors

•
Deep and comprehensive analysis of the NURBS
curves constructed using the smoothness criteria

•
Analysis of the NURBS curve for compliance with
the engineering target (for example, for analyzing the geometric parameters of
the cam profile for matching the specified macro parameters and the dynamic
characteristics of the cam mechanism).

Thus, the software platform is an
integrated system integrating Microsoft Excel VBA, FairCurveModeler, and
computer algebra systems.

**Description of the Software Platform**

A universal software platform for
visualizing class F curves and surfaces and developing applied CAD products
with functional curves and surfaces comprises the following components:

•
FairCurveModeler, a program complex for
visualization of class F curves

•
Microsoft Excel VBA

•
Computer algebra systems, Mathcad, and
Mathematica

The method for constructing a high-quality
curve is comprised of the following steps:

•
A virtual curve (*v-curve*) of class C5 is
constructed on two types of Hermite data of the form by a support polyline or
tangent polyline.

•
On the *v-curve*, a Hermite data is formed
in the form of a table of coordinates of points, tangent vectors, and curvature
values.

•
NURBzS curve (rational Bezier spline curve) or
NURBS curve (non-uniform rational B-spline of high even degree *m*, where *m*
= {6, 8, 10}) is constructed by using Hermite data isogeometrically while preserving
the quality of the *v-curve*.

The theoretical foundations of the program
complex are described in [14-17]. The software package consists of two parts:
the FairCurve.exe COM component and the interface part.

Computer algebra systems Mathcad and
Mathematica are necessary for the analysis of NURBS curves, data preparation
for the approximation of analytical curves in FairCurveModeler, and for solving
computational problems in applied CAD-systems.

**Integration of the Software Platform
with Fusion360**

The concept of using a universal software
platform allows you to transfer the functionality of the FairCurveModeler
software complex to any CAD system without special labor. For example, this
concept is easily implemented for the perspective CAD Fusion360, an ambitious
project of Autodesk. The functionality of Fusion360 provides all stages of
product design. The Fusion360 is essentially an automated engineer workstation.
Therefore, if we supplement the Fusion360 with the visualization functions of
the class F curves, the Fusion360 will become a self-sufficient high-class CAD
system capable of modeling functional and aesthetic curves as well as product
surfaces, up to the outer contours of supercars.

The FairCurveModeler software complex is implemented
in Fusion360 in [8]. The platform is used to extend the functionality of the
application "FairCurveModeler app Fusion360" and for the possibility
of developing an application CAD system in an integrated system software
platform withFusion360. To ensure integration with the API (Application
Programming Interface) Fusion360 developed a plugin that provides the exchange
of geometric objects with Microsoft Excel. As part of the software platform,
the Microsoft Excel FCModeler
+ Fusion360.xlsm book is presented, which provides the
transfer of NURBS models to Fusion360.

The technique of working with the software
platform for visualizing class F curves in CAD system Fusion360 is described
below.

**1. ****Construction of a ***v-curve* on a support polyline

a) Build a polyline in Fusion360.

b) Call the plugin using the plug_in_FairCurveModeler
button in the ADD-INS tab of the Model workspace.

c) Turn on the switch to Points in the FAIRCURVEMODELER dialog of the application in the Exchange Options area.

d) Select the vertices of the polyline in succession.

e) Click the Exchange button in the FAIRCURVEMODELER dialog box.

f) The text of the object model will appear in the Exchange Box text
box. Place the cursor in the Exchange
Box text box, select all the text (use the End Shift,
and Home keys) and copy to the clipboard (Ctrl and Insert).

g) Go to the Microsoft Excel FCModeler.xlsm workbook on the Polyline page. Paste the contents of the clipboard into the sheet area,
starting with the cell Set_XYZ (C12).

h) In the Topology area, select the unclosed checkbox (a broken line is not
closed). In the Type GD area, select Basic (polyline).

i) To create the *v-curve* on the polyline, click Create Curve. The
program will build the curve of the curvature curve function on the Graphics
page (Figure 1) and form the NURBzS curve model on the NURBS page.

Fig.
1. The graph of its curvature curve on the Graphics page.

There are two ways to enter the NURBS model
into the Fusion360 project:

•
Inputting of the NURBS model into the project by
a plug-in;

•
Inputting the NURBS model in the project via a DXF-file.

**Entering the NURBS model into the project by a plug-in**

a) To transfer the
curve to Fusion360, go to the Excel workbook FCModel + Fusion360.xlsm (the apps
Fusion360 folder) on the Fusion360 page. Click on the from NURBS to Fusion360 button (Figure 2). The program will copy the NURBS model to the
clipboard.

Fig.
2. Microsoft Excel FCModel
+ Fusion360.xlsm, page Fusion360.

b) Go to Fusion360.

c) Insert the text in the Exchange
Box.

d) The application displays the message Press OK to create Object.

e) Click OK in the message window. Click OK in the FAIRCURVEMODELER dialog box.

The Fusion360 APIs provide the means to input curve models in the Fusion360
NURBS format transient (intermediate). However, the API Fusion360 lacks a method of
directly entering a curve from the transient format into the project. To
enter a curve in the project, the curve in the plugin is interpolated, and an
internal representation spline is generated at the interpolated points.

f) The application will display the message “Standard curve successfully
created on points of a fair curve, number of points = ... “

g) Click OK. The curve is constructed.

h) Check the quality of the curve. Select the curve. Select Toggle Curvature Display from the context menu.

i) The program will construct a curvature graph over the curve.

Importing NURBS models in the transient
format makes it possible to compute any parameter, but does not allow you to
include in the project an exact model of the constructed curve. It is only possible
to introduce the model into the project indirectly by approximating the
internal representation with a spline. The main drawback of the spline of the
internal representation is that it does not preserve the values of
the boundary parameters of the original curve.

To construct non-closed curves with
preservation of the quality of the *v-curve*, the following technique is
proposed.

**Entering NURBS Model in the Project via
DXF-File**

a) After plotting the curve in the
Microsoft Excel workbook FCModel.xlsm, go to the NURBS page.

b) Click the ViewCVT button. The program interpolates the curve and constructs the
curvature graphs. The program also generates a DXF file for the NURBS curve.
The DXF file is saved in the folder

C://FairCurveModeler_TEMP/temp/with
the name 'r_out_dxf.dxf'/ .

c) Next, go to Fusion360.

d) Call the plug_in_FairCurveModeler application.

e) Set the switch to Curve
from DXF.

f) Clean the Change
Box.

g) The program will display the message Press OK to create Object.

h) Click OK in the message window. Click OK in the FAIRCURVEMODELER
dialog box.

i) The program will generate a curve from the DXF file.

j) Because of the difference in the units of measure in Fusion360 and in the
DXF file, the curve will be 10 times smaller than the original.

k) Select > Modify
> Scale.

l) Select the curve in the object tree.

m) Set the Zoom settings.

Scale Type = Uniform

Scale Factor = 10

n) The curve passes through the vertices of the original polygonal line. Importing
a NURBS model through a DXF file preserves the geometry of the curve.

**2. Construction of a v-curve on a
Tangent Polyline**

a) Build an arbitrary polyline in
Fusion360.

b) Run the plug_in_FairCurveModeler plug-in from Add-Ins.

c) Turn the switch to Points in the FAIRCURVEMODELER dialog of the
application.

d) Select the vertices of the polyline in succession.

e) Click the Exchange button in the FAIRCURVEMODELER dialog box.

f) The object model will appear in the Exchange Box text box. Place the cursor
in the Exchange Box text box, select all the text (use the End, Shift, and
Home keys) and copy to the clipboard (Ctrl + Insert).

g) Go to the Microsoft Excel FCModeler.xlsm workbook on the Polyline page.

h) Paste the contents of the clipboard into the sheet area, beginning with the
cell Set_XYZ (C12). In the Topology area, turn on the Closed switch (polyline closed). In the Type GD area, enable Tangent (polyline tangent).

i) To create a v-curve on a polyline, click on Create Curve.

j) The program will build a curve and form the NURBzS curve model on the NURBS
page.

k) To transfer the curve to Fusion360, go to the Excel Workbook FCModel + Fusion360.xlsm (the apps
Fusion360 folder) on the Fusion360 page. Click the from NURBS to Fusion360 button. The program will copy the NURBS model to the clipboard.

l) Go to Fusion360.

m) Enter the text in the Exchange
Box.

n) The application displays the message Press OK to create Object.

o) Click OK in the message window. Click OK in the FAIRCURVEMODELER
dialog box.

p) NURBS from the transient format is converted into a spline of the internal representation.

q) The application will display the message “Standard curve successfully
created on points of a fair curve, number of points = ...”

r) Click OK in the message. The curve is constructed (Figure 3).

Fig.
3. The NURBS model approximated by the spline of the internal representation.

If the curve is closed, then when
converting to an internal spline, the program generates an open curve. Close it
by selecting the curve, then in the context menu select Open / Close Spline Curve. Check the quality of the curve. Select the curve and select Toggle Curvature Display from the context menu. Closed curves are converted while
maintaining the quality of the v-curve (Figure 4).

Fig.
4. Curve closure and quality control by plotting the curvature graph.

Obviously, class *F* curves are
aesthetic unto themselves. However, along with just beautiful curves, there are
"top models,” log-aesthetic curves. Estimation of log-aesthetic curves is
based on the mathematical characteristics of the forms revealed in real-world
objects (e.g., butterfly wings) [18-19]. Aesthetic evaluation of curves is
carried out by the laws of technical aesthetics and should reveal the aesthetic
appropriateness of the curve in conjunction with the ability to meet the
requirements of rationality [24]. In view of compositional order and structural
coherence, the log-aesthetic curve has also expressive qualities. Hence,
smoothness as a geometric property of the curve forms its expressive basis and
visual purity. In terms of the emotional and psychological influences,
smoothness contributes to the psychological comfort of a person, since it is
associated with calmness, constancy, lack of aggression, as opposed to a broken
line that gives rise to turbulent associations. To model aesthetic forms, the
so-called log-aesthetic curves [20] are proposed, which have a linear curvature
graph in the logarithmic scale. Several known spirals, including the clothoid,
are particular cases of this class of curves. In [21], the broadest class of
curves with a monotonic curvature function, superspirals, was proposed. The
equations of these curves are expressed in terms of Gaussian hypergeometric
functions and are numerically integrated by adaptive methods such as the Gauss-Kronrod
method. The unique formula of the superspiral allows changing the three shape
parameters *(a, b, c)* to get any of the known spirals and any spiral with
a monotone change in curvature. In particular, for values of the parameters *a*
= 0.5, *b* = 1,* c* = 1, the superspiral is a clothoid.

For importing analytical curves into CAD
system, it is advisable to approximate them with NURBS curves. Thus, the NURBS
template of the analytical curve is created. In the proposed software platform,
Mathematica is used to calculate the most complicated superspiral formula. In
Mathematica, a dynamic procedure for approximating and visualizing the superspiral
was developed (Fig. 5). In a dynamic procedure, a Hermite data is formed in the
form of a table of coordinates of the points of the support polyline, vectors
of the first derivatives, positive curvature values, lengths of segments of the
curve between the support points, and unit curvature vectors. The dynamic
procedure uses the FairCurve.exe component directly and performs not only the
preparation of the Hermite data for the approximation of the analytic curve but
also its approximation by NURBS and estimate of the accuracy of the
approximation. With a proper choice of the approximation parameters and the use
of the Golden Mean Technique (clipping of the end sections with the form
perturbation), one can achieve a high accuracy of approximation.

Fig. 5 shows the superspiral, representing
the clothoid for *a* = 0.5, *b* = 1, *c* = 1. The clothoid is
approximated by the *B*-spline curve of the eighth degree with the approximation
parameters *Number of Points* = 16, the initial value of the parameter *s*_{0}
= -1, the first increment of the parameter *h*_{s0} = 0.1, and the
last increment step is *h*_{sk} = 1. To eliminate the shape
perturbation on the final section of the *B*-spline curve, the last three
segments are removed.

**Fig.
5.** Superspiral representing the clothoid.
Approximation of clothoid by *B*-spline.

The dynamic procedure provides for the high
quality and accuracy of NURBS-template construction. The *B*-spline curve
visually coincides with the original clothoid. The curvatures of the *B*-spline
curve and the original curve visually coincide as well. The program also gives
a numerical estimate of the maximum deviations of the NURBS template from the
original: max = max [0.00140453], min = min [-0.00417795]. The program
generates the NURBS-model of the curve in Microsoft Excel format and exports it
to an external file.

Next, to transfer the NURBS model from an external
file to Fusion360, you must perform the following actions:

a)
Open this file in Microsoft Excel, select the
region with the model and copy it to the clipboard. Go to the FCModeler.xlsm book on the NURBS page. Paste the text from the clipboard, starting
at cell A2.

b)
Delete the last three spline segments. In the Extract
segments area, set the parameters Start segment = 0, Number of segments = 12. Click the Extract
segments button. Next, interpolate the curve to form
the DXF file (ViewCvt button).

c)
Go to Fusion360.

d)
Call the application plug_in_FairCurveModeler in Excel.

e)
Set the switch to Curve from DXF.

f)
Clean the Change Box.

g)
The program will display the message Press OK to create Object. Click OK in the message box. Click OK in the FAIRCURVEMODELER
dialog box.

h)
The program will generate a curve from the DXF
file.

i)
Due to the difference in the units of
measurement in Fusion360 and in the DXF file, the curve will be 10 times
smaller than the original.

j)
Select Modify-Scale. Select a curve in the
object tree. Set the following scaling parameters:

Scale Type = Uniform

Scale Factor = 10.

k)
Superspiral will be displayed in the project.
Check the quality of the superspiral by plotting the curve of its curvature
(Fig. 6).

**Fig. 6**
. Superspiral and its curvature
function in the Fusion360 project.

Despite the self-sufficiency of Fusion360,
it is recommended to use AutoCAD in addition to Fusion360 [22]. The authors of
this article suggest that AutoCAD should be used to fix high-quality curves in
three-dimensional models in order to avoid fitting the curves with a spline of
the internal representation and preserve the high quality of class* F*
curve. It is also recommended to use the application 'FairCurveModeler app
AutoCAD' [7], which has the most complete functionality and allows modeling
curves and surfaces. The application has an advanced toolkit for editing curves
and surfaces for various types of data.

The software platform
is complete with FairCurveModeler applications for CAD (Fusion360, AutoCAD,
nanoCAD and COMPAS 3D). Before you start, you must transfer the activation code
Code_Activation.txt from the application
folder to the software platform folder FairCurveModeler
app Excel VBA.

1.
The program complex FairCurveModeler for
visualization of high-quality curves and surfaces using criteria of smoothness
is developed.

2.
A universal software platform for visualizing
class F curves and developing specialized applications for any CAD system based
on the use of Microsoft Excel VBA, FairCurveModeler, and computer algebra
systems is proposed.

3.
The integration of the software platform with
Fusion360 is demonstrated. The limitations of the Fusion360 API are revealed
when the NURBS model is included in the project. The technique of transferring
the exact NURBS model from the software platform to Fusion360 via a DXF file is
demonstrated.

4.
Application of the software platform for
visualization of functional and log-aesthetic curves in integration with
Fusion360 is demonstrated.

We should like to thank Rebecca Ramnauth of
the Department of Computer Science at Long Island University and the Ravendesk
Team, USA, who has generously given her valuable time to substantively edit and
review this paper. Her care, competence, and conscientiousness are much
appreciated. Additionally, we thank Ms. Elizabeth Jorgensen (USA) for useful
remarks and suggestions.

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