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ISSN 0361-7688, Programming and Computer Software, 2019, Vol. 45, No. 4, pp. 202–212. © Pleiades Publishing, Ltd., 2019.
Russian Text © The Author(s), 2019, published in Programmirovanie, 2019, Vol. 45, No. 4.
To the Development of Open Source Software
for the Reconstruction of CAD Models
S. E. Slyadneva,* and V. E. Turlapova,**
aLobachevsky State University of Nizhni Novgorod (UNN),
pr. Gagarina 23, Nizhni Novgorod, 603950 Russia
*e-mail: sergey.slyadnev@gmail.com
**e-mail: vadim.turlapov@itmm.unn.com
Receive d February 16 , 2019; revised March 19 , 2019; accepted March 19, 2019
Abstract—In this paper, we describe an open source software package aimed at solving reverse engineering
problems for CAD models defined in polygonal form. We briefly discuss the main principles behind the new
software, its architecture, and directions for its further development. The use of the software is illustrated by
examples of a turbine blade. In the first example, the turbine blade is reconstructed automatically from a
structured point cloud. Another example is the interactive reconstruction of the turbine blade from an
unstructured surface triangulation. In both the cases, we use a surface skinning strategy enhanced by a curve
fairing operator. We show that the modif ied skinning operator does not minimize the total bending energy of
the surface, but yields a smooth patch where input inaccuracies are compensated for. The reconstruction
result is a parametric model of the turbine blade where the design variables are the coordinates of the poles
for each profile curve. The proposed software architecture can be used for partial or complete parameteriza-
tion of reconstructed CAD models with the aim of their subsequent optimization.
DOI: 10.1134/S03617688190 4008X
1. INTRODUCTION
Reconstructing the exact digital model of an object
from an incomplete description of its geometric shape is
an important engineering problem [1]. These data are
generally obtained by digitizing the man-made object,
e.g., with 3D scanning. The result is then recon-
structed for further use in some engineering software.
This problem is solved by reverse engineering (reengi-
neering). The key stage of industrial reengineering is
modeling the geometric shape of the object as its
boundary representation (B-rep) [2], which is pres-
ently a de facto standard for computer-aided design
(CAD) systems. As noted in [1], the CAD model (in
B-rep) is a kind of “language” in which various design
and production automation tools communicate.
In engineering practice, reconstruction problems
have different origins. For instance, if a part was
designed without CAD, then its digital representation is
a priori unavailable. Sometimes the geometric model is
not available due to other reasons, e.g., the obsoles-
cence or loss of the original CAD system. In these cases,
reengineering is used to obtain a previously non-exis-
tent or lost digital model. The reconstructed geometry
can be used for subsequent CAD model editing, auto-
mated manufacturing on a CNC machine, solving
computational problems (strength analysis, thermal
analysis, aerodynamic analysis, etc.), and parametric
shape optimization. If parametric optimization is car-
ried out on a polygonal model, then the reengineering
problem can arise in the reverse transition from a
deformed mesh [3] to CAD representation.
Reverse engineering approaches are also employed
as the main tools for modeling objects that not yet exist
[4]. In the automotive industry, the physical model
made by a sculptor to demonstrate the outer contours
of a vehicle body is often digitized. Sometimes, geo-
metric modeling is initially carried out in polygonal
form; then, reengineering is employed to obtain an
exact B-rep [5]. In the latter case, reconstruction tech-
niques play basically the same role as traditional bot-
tom-up modeling techniques do, with the difference
that, instead of modeling from scratch, an auxiliary
polygonal “basis” is used. When solving reconstruc-
tion problems, in one way or another, the engineer
deals with some reference geometric representation in
the form of an unordered or structured point cloud or
a surface triangulation. The reference geometry guides
the modeling process and is used to match the target
shape of the object with its reconstructed shape.
Depending on the goal of reverse engineering, var-
ious criteria of success for the reconstruction process
can be considered. For instance, when creating a digi-
tal copy of a physical model, it is required to ensure the
minimum geometric discrepancy between the CAD
model and its polygonal representation. The automo-
PROGRAMMING AND COMPUTER SOFTWARE Vol. 45 No. 4 2019
TO THE DEVELOPMENT OF OPEN SOURCE SOFTWARE 203
tive industry also imposes strict requirements for the
aesthetic qualities of the product, which is why the
modeling process must use the so-called “class A” sur-
faces. On the other hand, when reconstructing the
digital model of a damaged part, or a part with signif-
icant wear, the high accuracy of reconstruction is not
required; moreover, it is undesirable in those regions
where the actual shape of the part significantly devi-
ates from the reference one. Minimizing only the geo-
metric discrepancy deprives the resulting model of
such key properties as axial symmetry, parallelism or
orthogonality of faces, etc. Thus, reverse engineering
is a multi-criteria process, in which the set and strict-
ness of quality criteria depend on a particular indus-
trial application.
To solve reverse engineering problems, specialized
software or plug-ins for traditional CAD systems are
employed. Difficulties encountered in the process of
reconstruction call for comprehensive scientific
research aimed at developing accurate, reliable, and
efficient reengineering algorithms. At the same time,
engineering practice suggests that, for the full-fledged
solution of even a narrow class of reconstruction prob-
lems, one algorithm is not enough. There is a need for
the consistent application of many geometric opera-
tors, as well as the implementation of the correspond-
ing object model for problem domain representation
and organization of computations. Unfortunately, the
available literature on the problem pays little attention
to the architecture of software packages meant for
reverse engineering. In our opinion, this is due to the
fact that almost all automated reengineering software
products are commercial ones, and their developers
are not interested in disclosing the details of their
implementation.
The main contribution of this work is the develop-
ment of an open software architecture aimed at solving
reverse engineering problems for CAD models defined
in polygonal (mesh) form. This paper provides general
information about the operation principles of the
developed software and outlines the directions for its
further development. Following the authors of [6], we
believe that the publication of open source systems
contributes to the independent reproduction of scien-
tific results. In addition, as mentioned in [7], the
implemented prototyping tools allow one to use the
proposed system to exchange ideas and transfer tech-
nologies from academic researchers to the industry.
The paper is organized as follows. Section 2
describes some existing methods for CAD model
reconstruction and outlines some criteria for their
classification. Section 3 introduces working hypothe-
ses for the proposed system and considers the main
steps of the interactive reconstruction procedure. Sec-
tion 4 describes principles underlying the developed
software. The application of the reconstruction system
to the parametric model of a turbine blade is illus-
trated in Section 5. In the Conclusions section, we
outline some directions for the further development of
the open source software.
2. STATE OF THE PROBLEM
In [8], it was noted that reverse engineering inves-
tigations often sidestep the question about software
packages that implement particular reconstruction
approaches. Meanwhile, it is the availability and
demand for the corresponding software (including
commercial one) that makes it possible to estimate
practical applicability of the published approaches and
algorithms, as well as their accuracy, reliability, and
efficiency. The review [8] compared some software
products in terms of the completeness of their reverse
engineering operators. However, all considered sys-
tems are commercial ones, which significantly com-
plicates their use by the scientific community. Thus,
the following problems arise. First, in most cases,
commercial systems do not disclose their bibliographic
databases that store the information about particular
reconstruction approaches implemented. Second, the
original investigations carried out by commercial com-
panies are rarely available through public means of sci-
entific communication because the preference is given
to issuing internal technical reports. In solidarity with
the author of [9], we believe that a truly scientific
approach to reverse engineering problems should be
based on the principles of open science, which implies
publication of the corresponding scientific papers, as
well as the source codes and input data for the reengi-
neering algorithms.
In [10], some CAD model reconstruction approaches
implemented in popular commercial software were
described. In particular, it was noted that interactive
methods that use manual segmentation of meshes are
still in demand in the industry, despite their complexity.
In addition, according to [10], interactive reconstruc-
tion with manual segmentation allows the user to
achieve the best quality of the resulting surfaces.
Of special interest are the approaches that allow one
to reconstruct a parametric family S(a) (where a ∈ RN is
the vector of design variables and N is the dimension
of the parameter space) rather than a single instance of
a geometric model. The parametric model is not only
a shape descriptor but also a model generator that
implements the so-called design intent for different
values of the vector a. Reengineering approaches that,
in addition to geometric artifacts, can reconstruct
engineering concepts (design intents) dominate, e.g.,
in turbine engineering [11, 12]. Reengineering meth-
ods for parametric models of mechanical parts are
developed. For instance, in [13], an efficient method
was proposed for synthesizing a CSG tree in the form
of a program that performs Boolean operations on a
predefined set of primitives recognized from polygo-
nal meshes. In [14], we proposed a method for volume
decomposition of boundary models. In [15], a set of
approaches to parametric model reconstruction based
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PROGRAMMING AND COMPUTER SOFTWARE Vol. 45 No. 4 2019
SLYADNEV, TURLAPOV
on the SolidWorks commercial system was described.
When considering systems for parametric model
reengineering from a software implementation per-
spective, of interest are not only geometric operators
but also linked data models.
Most of the reengineering methods published are
confined to the reconstruction of dumb models,
which do not have structural elements (features) and,
therefore, do not enable full-fledged editing. In par-
ticular, for models without history of their construc-
tion, the solution of parametric optimization problems
is complicated. In [16], two methods of parameteriza-
tion—partial and complete—for functional models
(which allow for aerodynamic flow) were distin-
guished. Infeasibility of model decomposition into
structural elements leaves room only for partial
parameterization.
Different software packages may use different geo-
metric representations. It is desirable that the recon-
structed model can be transferred to popular CAD
systems for further processing, e.g., subsequent editing
or manufacturing planning. Presently, the STEP for-
mat (ISO 10303), which specifies B-rep as a basic
model representation [2], is a lingua franca for CAD
systems. Non-uniform rational B-spline (NURBS) is
a universal form that enables modeling of both canon-
ical geometry and sculptured surfaces. However, the
reconstruction process can use other representations,
e.g., subdivision surfaces [17, 18] and T-splines [19],
provided that they are converted into the standard
form at the end of modeling. For example, popular
NURBS-compatible modeling tools that use subdivi-
sion surfaces are the Clayoo and Xirus plug-ins for the
Rhinoceros CAD system and the Power Surfacing
plug-in for SolidWorks.
Another aspect of reengineering systems is the
degree of automation. In [20], an automated approach
to the reconstruction of the B-rep composed of G1
smooth NURBS patches was proposed. However, the
automatically reconstructed topology of a model does
not reflect its design intent because it does not rely on
the engineering knowledge about its feature lines (if
they can be extracted). In [21], the popular commer-
cial system Tebis was described, in which the model
topology is defined by the user interactively (using the
rapid surface creation (RSC) technology). A similar
method is implemented in the PolyWorks system and,
e.g., in the Mesh2Surface plug-in for Rhinoceros. The
authors of [22] developed an interactive platform for
inverse modeling that enables reconstruction of free-
form objects, including kinematic surfaces. Interactive
systems are not intended to replace the human in the
process of reconstruction; rather, they are designed to
automate routine operations, which is in line with the
concept of intelligence amplification (IA) introduced
by Brooks in [23].
Open source software is widely employed to solve
various scientific and engineering problems. For exam-
ple, popular systems for 3D geometry processing are
OpenFlipper [24] and MeshLab [25]. Unfortunately,
these systems (which are used mostly to work with
polygonal models) do not enable full-fledged integra-
tion with geometric kernels that implement curvilinear
Fig. 1. Main steps of free-form reconstruction in the pro-
posed system.
Import STL
Prepare mesh
Evaluate mesh curvatures
Retopologize model
Fit surfaces
Evaluate deviations
Export NURBS
Outline curve network
Fig. 2. Curve network that forms the topology of the polyg-
onal model (vertices and edges are labeled).
e5
e6
e7
e4
e3
e2
e1
V1V2
V3
V4V5
V6
Fig. 3. Directed half-edges of the wireframe model.
ce1ce2
ce3
ce4
ce8
ce7ce6
ce5
PROGRAMMING AND COMPUTER SOFTWARE Vol. 45 No. 4 2019
TO THE DEVELOPMENT OF OPEN SOURCE SOFTWARE 205
B-rep. Our system is integrated with the OpenCascade
open source kernel, which provides multi-purpose
operators for surface and solid modeling, as well as
services for geometric data exchange (in the STEP and
IGES formats). General-purpose CAD operators
(skinning, extrusion, revolve operation, blending, etc.)
are key components of reconstruction systems as many
stages of reverse engineering are based on direct model-
ing [15]. In [26], a reengineering procedure that uses the
history of modeling operations and OpenCascade oper-
ators (to design features) was described. The growing
importance of open source software in solving design
problems was also noted in [27].
3. RECONSTRUCTION PROCEDURE
This section describes the interactive reconstruc-
tion procedure, which is the main procedure in the
proposed system. It should be noted, however, that
our software system can also be used for fully auto-
mated reconstruction of parametric models, which is
demonstrated in Section 5.
3.1. Main Stages
The reconstruction process begins with importing
the initial data in the form of a spatial triangulation
mesh. STL is a popular format for storing and transfer-
ring mesh data in CAD; it represents polygonal geom-
etry in the simplest form, without topology. Mesh data
often require additional processing, e.g., filling holes,
eliminating self-intersections, decimation, smooth-
ing, etc. Even though this processing is not directly
related to reconstruction, the corresponding geomet-
ric operators are generally implemented in reengineer-
ing systems, because the lack of these operators com-
plicates the reconstruction process. For manipulating
and displaying polygonal models, our system uses the
VTK library [28].
Model segmentation must take into account the
arrangement of feature lines. For this purpose, the
reengineering system evaluates the curvature of the
input mesh [29]. The curvature data are used to sup-
port the interactive specification of the topology by the
user. Once the model topology is defined, the system
constructs geometric primitives selected by the user.
In this case, various operators are employed, e.g.,
Coons and Gordon surface generation, ordered point
cloud interpolation, skinning, etc. At the final step,
the user evaluates the quality of reconstruction: if the
result is satisfactory, then the process is terminated;
otherwise, the user can return to the previous steps to
modify the topological model and associated parame-
ters. Figure 1 shows the main steps of free-form object
reconstruction. It should be noted that reengineering
is not a sequential process because, in order to achieve
a satisfactory result, the user needs to return to the pre-
vious steps.
When reconstructing a typical mechanical part, the
reengineering approach generally differs from the one
described above. In addition to the use of other seg-
mentation and reconstruction methods, the reengi-
neering procedure involves the recognition of the
mechanical features [30]. The latter is necessary to
yield an editable model that meets the design intent.
Our system implements feature recognition based on
the classical principle of analyzing attributed face
adjacency graphs [31].
3.2. Topological Model
Polygonal surface segmentation is one of the key
reengineering stages. As a working hypothesis, we
assume that the user performs interactive segmenta-
tion to define certain topology on the mesh model (see
Fig. 2). This approach was described in [32] as applied
to segmentation of free-form models.
There are some successful implementations of
interactive reconstruction based on curve networks in
commercial software [21]. The user partitions the
Fig. 4. Topology of the wireframe model defined by the
hierarchical oriented graph; the nodes p1 and p2 corre-
spond to two reconstructed patches shown in Fig. 2.
p1
ce1ce2ce3ce4ce8ce7ce6ce5
e5
e6
e7
e4
e3
e2
e1
V1V2V3V4V5V6
p2
Fig. 5. System Input and Output.
Reconstruction
scenario
Triangulation
Node Part Node
Fig. 6. Data node and the Presentation class that contains
a set of pipelines.
Data Node Presentation
Pipeline 1
Pipeline 2
Pipeline N
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PROGRAMMING AND COMPUTER SOFTWARE Vol. 45 No. 4 2019
SLYADNEV, TURLAPOV
model into patches based on certain practical consid-
erations and, possibly, a discrete curvature distribu-
tion map. In the simplest case, the system may restrict
the user to quadrangular curvilinear cells as they admit
covering with NURBS patches in their natural bound-
aries [33]. The latter allows one to do without trimmed
surfaces, which simplifies the data structures and
algorithms (e.g., smooth transition) used.
The curve network not only provides polygonal
data segmentation but also serves as a carrier for the
attributes that control the reconstruction process. In
particular, the edges of the model can store informa-
tion about the desired order of joint smoothness,
markers of contour lines, etc. In addition, the presence
of a curve network determines the method of model
editing, which consists in modifying support edges
and reconstructing the affected surfaces.
Each edge of the wireframe model has a natural
(geometric) orientation given by the parameterization
of the corresponding curve. To ensure the consistent
order of contour traversal, the patch of the wireframe
model includes occurrences of the edges with the ori-
entation sign, rather than the edges themselves (see
Fig. 3). Thus, the wireframe model is a directed hier-
archical graph (Fig. 4).
The structure of the wireframe model defines a
representation scheme that is equivalent to B-rep in
terms of the differentiation between “geometry” and
“topology” (see [34]). For full-fledged modeling with
the use of the curve network, the system must provide
the basic set of topology editing operators, including
edge and face splitting functions, boundary merging,
contour completion, etc. The work [35] devoted to an
interactive reengineering system reports some relevant
data on a sufficient set of topological operators for
editing subdivision surfaces. The same approaches are
also applicable in a NURBS-based representation
scheme.
3.3. Reconstruction Operators
In its current version, our system implements the
following geometric operators, not counting the algo-
rithms already available in the OpenCascade and VTK
libraries:
1. curve and surface interpolation (algorithms A9.1
and A9.4 from [36]);
2. surface skinning (as described in Chapter 10.3 of
the monograph [36]);
3. curve and surface fairing (based on the ideas dis-
cussed in [37]);
4. Coons patch construction in the form of
NURBS surfaces [33];
5. local editing of NURBS curves [38];
6. line segment projection onto triangulation for
curve network modeling;
7. envelope generation for point clouds on a plane [39].
3.4. Constructing the Boundary Model
Upon modeling the patches that fill the curvilinear
cells of the curve network, the geometrical and topo-
logical information in the structure of the target object
Fig. 7. Representation of the model edge by two curves (in
the modeling space and in the domain of the carrying sur-
face).
r(p) : R1 → R3
g(t) : R1 → R2
s(u, v) : R2 → R3
u
v
Fig. 8. Interaction between two geometric kernels.
OpenCascade
In-house NURBS library
JSON Direct conversion
Fig. 9. Code fragment for the first-degree B-surface in the
JSON format.
PROGRAMMING AND COMPUTER SOFTWARE Vol. 45 No. 4 2019
TO THE DEVELOPMENT OF OPEN SOURCE SOFTWARE 207
is completely defined. The synthesis of the final
boundary model is reduced to transforming the data
structures into the representation scheme of the
OpenCascade geometric kernel for subsequent trans-
lation into neutral formats (STEP or IGES).
4. ARCHITECTURE OF THE SYSTEM
Below, we describe the basic implementation princi-
ples behind the open source reconstruction platform.
4.1. Object Model
The central object of the system is the Part to be
reconstructed. The Part contains the resulting CAD
model in the OpenCascade B-rep. The system’s input
is a surface mesh that is uploaded to the Triangulation
node from the corresponding STL file (see Fig. 5).
The nodes of the data model and the relationships
among them are organized using the OCAF module of
OpenCascade [40]. This module implements a multi-
purpose hierarchical data storage that provides some
basic services, e.g., transaction support, undo/redo,
and CRUD (create, read, update, and delete) opera-
tions. Data access is based on the data access object
(DAO) design pattern.
Each data node is associated with an instance of the
Presentation class that contains one or several VTK
pipelines (Fig. 6).
In our system, the pipeline is an adaptation of the
VTK pipeline with the following reservations: (1) the
data source for our pipeline is the OCAF node that
stores the time of the last modification to enable effi-
cient redrawing and (2) it is assumed that one pipeline
is associated with one object of the scene.
4.2. Visualization
To work with CAD data, we implemented a visual-
ization subsystem, which translates the curvilinear
representation of the CAD model into a polygonal
shape, and the corresponding interactive services
based on the VTK library. In popular geometric ker-
nels (e.g., OpenCascade or Parasolid), the boundary
model includes errors caused by limited machine rep-
resentation of real numbers, approximation errors,
and geometric inaccuracies. In practice, these errors
are stored in the structure of the CAD model as geo-
metric tolerances locally associated with its boundary
elements [41]. Even though modern CAD systems
generally hide visible defects of the model, the very
presence of errors can cause incorrect operation of
modeling algorithms or problems with data transfer
from one engineering system to another. The visual-
ization subsystem of our platform is implemented in
such a way as to display the geometric and topological
defects to the user. For B-rep and mesh models,
defects, such as open and non-manifold edges, open
faces, dangling vertices, etc., are automatically recog-
nized and marked with color. The accentuation of
anomalies allows them to be eliminated as soon as at
the stage of modeling, which exempts one from the use
of healing operators [42] on the model.
High geometric tolerances are usually due to the
defects at the joints of model faces. These defects are
often caused by mismatched edges on adjacent con-
tours; moreover, mismatch can occur between the
parametric curve r in the modeling space and its own
pre-image g in the definition domain of the carrying
surface s(u, ) (see Fig. 7). Thus, the requirement of
synchronous parameterization arises, which is formal-
ized as follows:
(1)
where ε is the geometric tolerance, the parameter t
takes values on the interval [t0, t1] and ||⋅|| is the Euclid-
v
−<ε
|| ( ( )) ( )|| ,sgt rt
Fig. 10. Solid model of the turbine blade.
Fig. 11. Blade sections constructed by intersecting the tri-
angulation with the interactively positioned plane.
208
PROGRAMMING AND COMPUTER SOFTWARE Vol. 45 No. 4 2019
SLYADNEV, TURLAPOV
ean distance. Geometric modeling with guaranteed
observance of rule (1) becomes easier if the data struc-
ture of the model is organized on the 3D curve net-
work that defines the model topology (see Section
3.2). In this case, surface modeling does not use para-
metric curves g(t) up to the point at which the model
is converted into the data structures of the OpenCas-
cade kernel.
In addition, the system provides graph visualiza-
tion tools and tools for analyzing the differential prop-
erties of curves and surfaces. These tools were
described in detail in our previous work [43].
4.3. Geometric Kernel
As a basic geometric kernel, we employ the Open-
Cascade library [44]. The data structures of this library
are used to represent CAD models in the form suitable
for visualization, basic modeling, and exchange
through open translators (STEP and IGES). New
modeling operators are developed with the open
source NURBS library implemented in accordance
with the well-known monograph [36].
Data transfer between the two libraries is carried
out in the JavaScript object notation (JSON) format
or by using a specialized software module for curve
and surface editing (Fig. 8). The in-house library is
adapted to solving reverse engineering problems, while
OpenCascade acts as a general-purpose geometric
kernel. Similar architectural solutions were described,
e.g., in [45].
The JSON format (Fig. 9) is neutral for both the
libraries. It is used to implement module testing pro-
cedures.
4.4. Command Console
In its current version, the system provides a com-
mand console as a main user interaction tool. It sup-
ports more than one hundred commands for various
Fig. 12. Blade surface constructed by skinning after the
unification of the profile curves. The isolines correspond
to the knot values of B-splines. One can see typical con-
centration. The surface has more than 43 thousand poles.
Fig. 13. The surface constructed by skinning on a priori
compatible curves (with 3100 poles). The twisting effect is
due to the asynchronous parameterization of the profile
curves.
Fig. 14. Skinning with and without fairing (right and left
images, respectively) at λ = 1.
PROGRAMMING AND COMPUTER SOFTWARE Vol. 45 No. 4 2019
TO THE DEVELOPMENT OF OPEN SOURCE SOFTWARE 209
operations on the model. The corresponding graphical
interface will be implemented in future versions.
5. APPLICATIONS
5.1. Automatic Blade Reconstruction
One of the problems solved using the proposed sys-
tem is the reconstruction of the parametric model of a
turbine wheel from discrete streamlines [12]. Figure 10
shows the geometric model of a turbine blade, which,
at the next step, is replicated in the axial direction of
the rotor and is combined with the turbine hub model
by using the solid Boolean operator.
The turbine blade is an example of a functional sur-
face modeled using NURBS patches. At the same
time, to solve shape optimization problems, the geo-
metric model of the blade should include a parametric
structure, which is usually composed of its radial sec-
tions. Each section is parameterized in such a way as to
obtain a certain set of design variables that form the
search space for finding the optimal geometry. In [12],
design parameters were coordinates of section poles;
however, the same organization of computations is
suitable for other parameterization methods.
The key difference of the reengineering procedure
from direct bottom-up modeling is the need for the
elimination of noise in the input data. For this pur-
pose, a fairing technique [37] was implemented and
applied to the set of intermediate interpolation curves
used to obtain the final poles of the skinning surface.
5.2. Interactive Blade Reconstruction
The reengineering process described above is com-
pletely automatic due to the fact that the original point
cloud is structured and ordered. For instance, the
blade sections are initially represented in the cloud
structure by individual slices that belong to some sur-
faces of revolution whose generatrices are also recon-
structed automatically. If the reference geometry of
the blade is represented by triangulation, then auto-
matic reconstruction becomes complicated. Below,
we describe the interactive procedure for reconstruct-
ing the aerofoil geometry in such cases.
The procedure begins with the selection of sec-
tional planes to extract the points that form the profile
curves of the blade. The triangulation–plane intersec-
tion is carried out using the VTK library; then, the
resulting set of points is sorted in the sequence order by
the envelop construction algorithm [39] implemented
in our system (see Fig. 11).
The ordered points are fitted by the least squares
method to obtain B-curves with desired accuracy. For
further skinning, the section curves must be compati-
ble [46]; however, the previous reconstruction steps do
Fig. 15. Approximate bending energy versus fairing coeffi-
cient λ.
Bending energy
2e+5
1e+6
2e+6
5e+5
1e-8 1e-6 1e-4 0.01
Fairing coecient
Unexpected (?) minimum
1
Fig. 16. Maximum geometric deviation versus fairing coef-
ficient λ.
10
Deviation
5
2
1
1e-81e-7 1e-61e-5 1e-41e-3 0.01 0.1
Fairing coecient
No eect
1
Table 1. Maximum and minimum mean curvatures in the
neighborhood of a local minimum of the bending energy
(Fig. 15)
λMean curvature
(min)
Mean curvature
(max)
1e-4 –0.492 1.25
1e-3 –0.492 1.21
0.01 –0.493 1.21
0.1 –0.539 1.22
1–0.5571.22
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PROGRAMMING AND COMPUTER SOFTWARE Vol. 45 No. 4 2019
SLYADNEV, TURLAPOV
not guarantee that. In addition, bringing the sections
to a compatible form by unification of knots is unrea-
sonable in practice because the resulting surface
becomes too complex (Fig. 12).
As noted in [47], reducing the amount of data that
represent the blade geometry allows one not only to
speed up the reconstruction procedure but also to
obtain a smoother surface and improve memory con-
sumption. It should also be noted that the unreason-
ably high number and concentration of spline poles
complicates the parametric optimization process.
Moreover, the excessive complexity of the geometric
representation adversely affects the reliability of the
subsequently used modeling operators (e.g., surface
intersection). An easy way to reduce the complexity of
the resulting surface is to avoid the unification of the
profile curves by initially specifying them on a single
knot vector. For this purpose, the original curves are
sampled with a specified number of points and then
interpolated using centripetal parameterization [48].
The resulting surface has a much lower complexity;
however, shape defects remain (see Fig. 13).
To eliminate the blade twisting effect, each section
is presegmented. We use interactive segmentation;
however, this process can be automated [49].
The initial data for reconstruction usually have cer-
tain inaccuracy due to the imperfection of the sam-
pling method. In addition, user-defined feature lines
for blade segmentation may prove unintentionally tor-
tuous due to input errors or sparse point clouds. To
eliminate shape distortions, a skinning strategy
enhanced by a fairing operator is employed [12]. The
idea of the strategy is to change the spatial location of
the poles for the intermediate skinning curves to min-
imize the bending energy [50]. Adjusting the fairing
coefficient λ > 0 allows us to obtain a smoother surface
at the expense of increased geometric deviations (see
Fig. 14).
Thus, the interactive reconstruction procedure
consists of the following steps:
1. triangulation–plane intersection and sorting of
points by constructing their envelope based on K near-
est neighbors [39];
2. segmentation of each section to separate the
sides of the blade;
3. fitting of section points with a specified toler-
ance;
4. additional interpolation of the regular cloud
constructed by sampling profile curves [51] to define
them on a single knot vector;
5. merging the curves to form closed contours;
6. fairing-enhanced skinning [12].
To estimate the effect of the fairing coefficient on
the shape of the blade, we use an approximate value of
the bending energy E computed by the formula
(2)
Here, the operator (⋅)2 is a scalar square, and inte-
gration is performed over the entire domain of the sur-
face s(u, ).
Figure 15 shows the experimental dependence of
the bending energy on the fairing coefficient.
It can be seen that this dependence is not a mono-
tonically decreasing function. Indeed, the energy
functional (2) is not an objective to the procedure of
fairing the skinning directrices. According to [12],
fairing the curve c(u) guarantees the minimization of
functional (3) only; this functional aggregates the geo-
metric deviation c – c0 with the bending energy:
(3)
Let us make sure that the procedure of fairing-
enhanced skinning can increase the surface curvature
of the blade. Table 1 shows the increase in the absolute
value of the minimum mean curvature with the
increase in the fairing coefficient λ.
It should be noted that, in order to obtain a
smoother shape, it is reasonable to apply the fairing
operator also to the profile curves (in addition to the
skinning directrices).
Finally, let us analyze the growth in the geometric
deviation with the increase in the fairing coefficient.
Figure 16 shows the dependence of the maximum geo-
metric deviation (in the model’s units of measure-
ment). As expected, upon reaching a certain value of
λ, the distance between the initial triangulation and
the smoothed surface ceases to change (the skinning
directrices acquire the minimum possible bending
energy).
The measurements were carried out at the points
on the resulting surface s(u, ) with a step of 0.01 in the
parametric direction u and with a step of 0.005 in the
parametric direction (see Fig. 17).
∂∂∂
=++ ν
∂∂ν
∂∂ν
222
222
22
2
sss
Edud
u
u
v
=λ +−
22
0
{[''] [ ]}
c
Ecccdu
v
v
Fig. 17. Deviation map for the smoothed blade with
respect to the initial triangulation at λ = 0.001.
PROGRAMMING AND COMPUTER SOFTWARE Vol. 45 No. 4 2019
TO THE DEVELOPMENT OF OPEN SOURCE SOFTWARE 211
6. CONCLUSIONS
We believe that the further development of the pro-
posed system will result in the creation of a full-
fledged open source CAD platform for reverse engi-
neering. As of now, this system was used to implement
a number of algorithms integrated into commercial
software.
For real-time evaluation of geometric accuracy,
approaches that employ GPU-accelerated computing
are of interest [52]. Another gap in the current system
is the small set of surface fitting operators. From a
practical perspective, the strategy described in [53]
and the approach [54] that supplements it seem prom-
ising.
The source code of the system and its extended
documentation are available at www.analysissitus.org.
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Translated by Yu. Kornienko
