Geometric Modeling. Computer Graphics in Education
GraphiCon 2018
Techniques for modeling a high-quality B-spline curves by S-polygons in
a float format
Rushan Ziatdinov1, Valerijan Muftejev2, Rifkat Nabiyev2, Albert Mardanov2, Rustam Akhmetshin2
ziatdinov@kmu.ac.kr|muftejev@mail.ru|dizain55@yandex.ru|systems@list.ru|ahmetshin@bk.ru
1
Department of Industrial and Management Engineering, Keimyung University, Daegu, Republic of Korea;
2
Department of the Fundamentals of Mechanisms and Machines Design, Ufa State Aviation Technical University, Ufa,
Russian Federation
This article proposes a technique for the geometrically stable modeling of high-degree B-spline curves based on S-polygon in a float
format, which will allow the accurate positioning of the end points of curves and the direction of the tangent vectors. The method of
shape approximation is described with the purpose of providing geometrical proximity between the original and approximating curve.
The content of the notion of a "harmonious, regular form" of B-spline curve’s S-polygon in a float format is revealed as a factor in
achieving a high-quality of fit for the generated curve. The expediency of the shape modeling method based on S-polygon in a float
format at the end portions of the curve for quality control of curve modeling and editing is substantiated. The results of a comparative
test are presented, demonstrating the superlative efficacy of using the Mineur-Farin configuration for constructing constant and
monotone curvature curves based on an S-polygon in a float format. The findings presented in this article confirm that it is preferable
to employ the principle of "constructing a control polygon of a harmonious form (or the Mineur-Farin configuration) of a parametric
polynomial" to a B-spline curve’s S-polygon in a float format, and not to a B-polygon of the Bézier curve. Recommendations are given
for prospective studies in the field of applying the technique of constructing a high-quality B-spline curves to the approximation of logaesthetic curves, Ziatdinov’s superspirals, etc. The authors of the article developed a technique for constructing smooth connections of
B-spline curves with ensuring a high order of smoothness of the composite curve. The proposed techniques are implemented in the
FairCurveModeler program as a plug-in to engineering CAD systems.
Keywords: S-polygon, NURBS, fair curve, typical curve, transition curve.
"The Golden Age will include people who will learn to unite."
Konstantin Tsiolkovsky
1. Introduction
Mineur et al. [1] and Farin [2] proposed a method for
constructing Bézier curves with a monotone change in curvature,
based on the construction of the Bézier polygon (also known as
B-polygon) with a monotonic change in the control polygon legs
at fixed angles between the legs. The error in this method was
recently demonstrated [18], although this fact was known a
decade
ago
to
one
of
our
Japanese
colleagues, Norimasa Yoshida1.
In this paper, the aforementioned method adapts to the
construction of B-spline curves with monotone curvature and the
application of the S-polygon with the Mineur-Farin
configuration. A technique for modeling a curve with an Spolygon in a float format is proposed, which allows for the
precise positioning of the end points of the curve and the
direction of the tangents. More detailed equivalence parameters
for the two curves, such as the similarity of the m-level curvature
and the similarity of the m-level curvature of torsion, are
introduced. In addition, a method for approximating analytic
curves with the equivalence of curvature and curvature of torsion
to the mth level is proposed.
Modern CAD systems use a method of modeling class A
curves using S-polygons of high-degree NURBS curves. To
model closed curves, the S-polygon is used in a float format, and
the closed format of the S-polygon is used for non-closed curves.
Based on the configuration of the S-polygon in a float format, the
quality of the modeled curve is easily judged. The harmonious,
regular form of the S-polygon in a float format of the B-spline
curve allows us to ensure high quality in the generated curve
(hereinafter in this article we will clarify the concept of harmony
of the S-polygon shape). Using a closed S-polygon is dictated by
the need for precise positioning of the end points of the curve and
the directions of the tangent vectors at the end points. The end
portions of the closed S-polygon coincide with the initial portions
1
of the B-polygons of the end segments of the curve, which allows
precise positioning of the end points and directions of the tangent
vectors at these points. However, using an S-polygon in a closed
format causes instability in the curve modeling at the end
portions of the curve.
2. Techniques for modeling B-spline curves
The Cox-de Boor algorithm [4, 5] using the format of the
representation of the B-spline curve with an S-polygon in a float
format is numerically stable, since it uses a sequence of
operations dividing the segment into a given ratio. In addition,
the operations of calculating points and differential
characteristics of the curve are uniform on any segment.
Obviously, for the best approximation of the circular arc, the
segment of the B-spline curve of the vertex of the S-polygon in a
float format must be uniformly chosen on the circle. We
construct an S-polygon in a float format of a segment of a Bspline of the ninth degree along the vertices of a regular
dodecagon inscribed in a circle with a radius of R = 10. The point
of the largest deviation of the evolute of the curve from the center
(0, 0) of the circle is the point with the coordinates
(0.000000015321, 0.000000057206).
Transformations of the control polygon formats of B-spline
curves and Bézier splines are performed using BS and SB
algorithms [6-8]. In [9] NBS and NSB algorithms of
transformations for rational splines are proposed. SB (NSB)
algorithms for the transition from the format of the S-polygon Bspline to the format of the generalized B-polygon of the Bézier
spline are stable, since they contain only operations of dividing
the segments into a given ratio. BS (NBS) algorithms are less
stable, so they contain extrapolation operations of a segment in a
given ratio. The algorithms for converting S-polygon formats of
multi-segment splines partially contain the operations of BS
(NBS) and SB (NSB) algorithms.
Private communication.
324
September 24–27, 2018, Tomsk, Russia
The publication of the conference proceedings was supported by the RFBR, grant No. 18-07-20045\18
GraphiCon 2018
Геометрическое моделирование. Компьютерная графика в образовании
Fig. 1 shows the segment of the B-spline curve of the ninth
degree associated with the S-polygon in a float format and the
closed S-polygon coinciding with the B-polygon of the Bézier
curve of the ninth degree.
1295.780), (-8.274, -26.046), (-9.832, 5.520), and so on. The
error of circular arc approximation increases from 5.92221×10-8
to a value of 3.06354×10-1.
It is the stability of shaping with an S-polygon in a float
format at the end portions of the curve that makes this method
preferable.
The uncertainty of positioning the end points of the curve and
the values of the tangent vectors is eliminated by employing a
technique which consists of the following steps:
1.
Fig. 1. An S-polygon in a float format and a closed Spolygon of a segment of a B-spline curve of ninth degree.
The vertices of the closed S-polygon (i.e., the B-polygon) do
not lie on the circle. The legs of the S-polygon do not have the
same length (the lengths of the legs are 0.518838, 0.519949,
0.520746, 0.521227, 0.521387, ...). The angles between the legs
are also not equal: in radians they are 0.0653566, 0.0654361,
0.0654898, 0.0655169, ...). At the same time, it is this
configuration that provides a high-precision approximation of the
circular arc with an accuracy of 5.92221×10-8.
In modern CAD systems for modeling class A curves, Bsplines with S-polygons in closed format are used. It is
impossible to draw a closed S-polygon with the same
configuration as in Fig. 2 to approximate a circular arc with the
same level of accuracy. The difficulty of ensuring proper quality
at the end portions is emphasized in the guidelines for modeling
class A curves in CAD systems.
3. Technique of geometrically stable modeling of
high-degree B-spline curves using an S-polygon
in a float format
An S-polygon in a float format of a B-spline curve of degree
m can be used for quality control of modeling and for editing a
curve [10].
Differential characteristics of the spline are determined by
the divided differences of the S-polygon in a float format [5, 11].
A harmonious, regular form of an S-polygon in a float format of
a B-spline curve of degree m must satisfy the second order
isogeometry and regularity of an S-polygon of order m-1 [9].
Such a configuration of an S-polygon can be achieved by
providing the required law of variation of the discrete
approximation of the tangent vectors and curvature [9], as
computed from the central divided differences of the S-polygon
of the first and second order. More stringent requirements to the
harmonicity of the polyline configuration can be introduced if
one takes into account the condition that portions of a modeled
or approximated curve with a monotone change in curvature are
determined by segments of a polygonal line with a monotonous
variation of the discrete curvature, as, for example, segments of
a polygonal line with a Mineur-Farin configuration.
Modeling and editing of the high-degree B-spline curve at the
end portions of the curve by means of a closed S-polygon is
extremely unstable. Even simple rounding of the coordinates of
a closed S-polygon’s vertices can cause distortion of the shape.
For example, let the arc of a B-spline curve of the ninth degree
be given by a convex S-polygon in a float format (see Figure 1),
built on the vertices of a dodecagon inscribed in a circle of radius
R = 10. Rounding the coordinates of the second vertex of the
closed S-polygon (see Fig. 1) from the value (1.807034341228000, 8.748581800695999) to the value (-1.807,
8.748) causes a significant change in the shape of the curve and
an unpredictable change in the shape of the S-polygon in a float
format. The vertices of the original S-polygon (-7.071, -7.071),
(-9.659, -2.588), (-9.659, 2.588), etc. have changed to (-83.972,
2.
3.
4.
5.
6.
7.
Construct the S-polygon of a curve in a float format of
the proper shape.
Control the positions of the end points and the tangent
directions when formatting the closed S-polygon.
Define the position difference vector (the vector
determined by the difference of the actual end point
versus the desired end point) and the difference angle
of the direction (the angle between the actual tangent
direction and the required one).
Switch the current formatting to that of S-polygon in a
float format.
Eliminate misalignment in the position of the end
points, requiring translation of the m terminal vertices
of the S-polygon by the magnitude of the difference
vector in the direction of decreasing the error value.
Eliminate the difference in the direction of the tangent
vectors, requiring rotation of m end vertices of the Spolygon by the amount of the difference angle in the
direction of decreasing the difference value.
Edit the changed configuration of the S-polygon in a
float format until it reaches a harmonious, regular
shape. Repeat the position control on the closed Spolygon and edit the S-polygon in a float format until
a harmonious, regular form of the S-polygon in a float
format and the accuracy of the closed S-polygon
position are obtained.
4. Typical curves
In the works of Mineur et al. [1] and Farin [2] algorithms for
generation of the control points for the formation of typical
Bézier curves with a monotone change in curvature were
developed. In these algorithms, a B-polygon of the Bézier curve
is generated as the control polygon. The essence of the
algorithms lies in the construction of a B-polygon with a
monotonous change in the length of the leg, while maintaining a
constant value of the angle between the legs and the angle of
torsion. Such a configuration of a B-polygon is called the
Mineur-Farin configuration.
The principle of constructing a control polygon by a MineurFarin configuration is correct in itself, however, as will be shown
below, it is more appropriate to apply it to an S-polygon in a float
format of the B-spline curve, and not to the B-polygon of the
Bézier curve.
We will perform a comparative testing of these two different
approaches for the limiting case, namely, modeling the curve of
constant curvature. Obviously, the elongation coefficient of the
leg in the Mineur-Farin configuration must be equal to 1 for
generating the curve of constant curvature. It is also obvious that
the Bézier curve and polynomial splines do not accurately
approximate the geometry of the circular arc. However, a
comparison of the approximation accuracy can demonstrate the
advantage of using one method over another.
To generate the vertices of an S-polygon with a B-spline
curve, we use an equilateral polygon inscribed in a circle of
radius R = 10. For a degree of n = 3, a hexagon is used, and for
degrees of n = {5, 7, 9} a dodecagon is used.
24–27 сентября 2018, Томск, Россия
Работа опубликована при финансовой поддержке РФФИ, грант 18-07-20045\18
325
Geometric Modeling. Computer Graphics in Education
The construction of the vertices of a B-polygon is performed
as follows: The terminal vertices of the B-polygon which
coincide with the terminal vertices of the B-spline curve segment.
The tangent vectors at the endpoints are orthogonal to the radius
vectors from the center (0, 0). The intermediate vertices of the Bpolygon are defined in such a way as to obtain a configuration
with a constant leg length and a fixed angle between the legs.
Below is a table with the distance values of the evolute points
farthest from the zero center (0, 0), of the Bézier curve and the
B-spline curve.
Table 1. Comparison of approximation accuracy
Degree (n)
Bézier curve
B-spline curve
3
5
7
9
8.17283×10-1
8.06272×10-2
4.65855×10-2
3.19524×10-2
8.33333×10-1
1.11371×10-3
7.85284×10-6
5.92221×10-8
As can be seen from Table 1, an increase in the degree results
in an extremely slow increase in the approximation accuracy of
a circular arc by Bézier curve. Even for a degree of n = 9, the
Bézier curve does not provide acceptable accuracy (in modern
CAD systems, the NURBS degree does not exceed a tenth
degree). Concurrently, for n = 9, the B-spline curve achieves
accuracy acceptable for engineering calculations in CAD
systems. This test result shows the advantage of using the
Mineur-Farin method for constructing a curve of constant
curvature by means of an S-polygon in a float format. This
statement is also true in the case of modeling curves with a
monotone change in curvature. Consider the configuration of a
polygon of five points with a monotonous change in the length
of the leg with a scaling factor of two units and a fixed angle
between the legs equal to 90°. Let us define the Bézier curve of
the fifth degree on this configuration and on the same
configuration, as on the S-polygon in the float format, we define
the segment of the B-spline curve. Analysis of the evolute graphs
shows that the curvature of the Bézier curve is not monotonic,
and the B-spline curve is defined with a monotone change in the
curvature (Fig. 2).
GraphiCon 2018
To approximate the form, we propose to use additional,
highly detailed characteristics of the internal geometry of the
curve, proposed in [12]:
● Curvature of the first level is the curvature function itself
𝜅(𝑠).
● The torsion of the first level is the torsion function itself
𝜏(𝑠).
● The second level curvature is a function of curvature of the
curvature function 𝜅 (2) (𝑠) = (𝜅 ∘ 𝜅)(𝑠)
The second level curvature of the torsion is a function of
curvature of the torsion: 𝜅𝜏 (2) (𝑠) = (𝜅 ∘ 𝜏)(𝑠).
We define of the curvature of the nth level from the curvature
of a curve as follows
𝜅 (𝑛) (𝑠) = (𝜅 (𝑛−1) ∘ 𝜅)(𝑠),
and the curvature of the nth level from the torsion of the curve is
(𝑛−1)
(𝑛)
(𝑠) ∘ 𝜏) (𝑠).
𝜅𝜏 (𝑠) = (𝜅𝜏
Under shape preserving approximation of n-level of
equivalence we mean the equivalence of curvatures to the n-th
level and of the curvatures of torsion to the n-th level for the
original curve and the approximating curve. Equivalence of the
shape of the curves means that the number and order of inflection
and extrema points in the curves are the same. Thus, for highquality shape approximation, the curvature of the curvature and
the curvature of the torsion of the original and approximating
curve should be equivalent in shape, at least until the second
level. These arguments lead us to the following technique of
isogeometric shape approximation of an analytic curve by Bspline curve of high-degree n, where n>5. The vertices of an Spolygon in a float format of a B-spline curve are generated
directly on the analytic curve. With a proper distribution of points
and the distribution density on the analytic curve, it is possible to
ensure the equivalence of the shape of the analytic curve segment
and the B-spline curve not lower than the second level.
As an example, a shape approximation of a conical spiral by
a B-spline curve of the eighth degree is given (Fig. 3). The
conical spiral is determined by the parametric equation
𝑥(𝑠) = 2 + 𝑠 sin 𝑠
𝑦(𝑠) = 2 + 𝑠 cos 𝑠
𝑧(𝑠) = 𝑠
The vertices of the S-polygon are computed on an analytic curve
with the initial value of the parameter s = 0 and with a step h = 1
in the number of twenty points.
Fig. 2. Bézier curve (left) and B-spline curve (right) segments,
defined on the same control polygon.
5. Construction of a high-quality approximating
B-spline curve
Welters and Farin [3] proposed an approximation method
taking advantage of the geometric proximity of two curves (the
original curve and the approximating curve). In Welters and
Farin’s method, geometric proximity is estimated by examining
the relative equivalence of two curves with respect to the
parameters of curvature and torsion. Moreover, the authors
introduced the term shape approximation, and noted that
successful approximation guarantees the accuracy of the shape in
the sense that the geometry of the original curve is approximated
neatly [3].
Fig. 3. Approximation of the shape of a conical spiral.
The curvature graph as well as the graph of curvature of
torsion of the B-spline curve up to the second level are equivalent
in shape to the second level curvatures of the conical spiral.
This technique will also be useful for approximating the logaesthetic curves [13, 14] and Ziatdinov’s superspirals [15]. In
accordance with the multi-criteria approach to the evaluation of
326
September 24–27, 2018, Tomsk, Russia
The publication of the conference proceedings was supported by the RFBR, grant No. 18-07-20045\18
GraphiCon 2018
Геометрическое моделирование. Компьютерная графика в образовании
log-aesthetic curves [16], when importing log-aesthetic curves to
CAD systems in the form of NURBS templates, it is important to
preserve the characteristics of the internal geometry using
smoothness criteria.
6. Composite curves
Existing methods of modern CAD systems provide an order
of smoothness of no more than G4. With the float format of the
S-polygon, it is easy to construct composite log-aesthetic curves
by segments of B-spline curves of the same degree of n
approximating log-aesthetic curves not lower than the second
level of shape isogeometry. The S-polygons of the B-spline
curves are positioned and matched by additional vertices so that
the composite S-polygon has a harmonious, regular shape.
Hence, a composite B-spline curve will have an order of
smoothness of n-1.
7. Conclusion
This article demonstrates the techniques of applying Spolygons in a float format for modeling a high-quality B-spline
curve using smoothness criteria. In particular, the technique of
modeling the non-closed B-spline curve is proposed with the
provision of accurate positioning of the end points of the curve
and the directions of the tangent vectors at the end points. In
addition, a technique for modeling a B-spline curve with a
monotonic curvature through an S-polygon with a Mineur-Farin
configuration is proposed, as well as a technique for
approximating analytic curves with the equivalence of curvature
and curvature of torsion to a level m. A technique has been
developed for constructing smooth connections of B-spline
curves with a high order of smoothness of the composite curve.
The proposed techniques are implemented in the program
FairCurveModeler [17].
Acknowledgements
This manuscript was copy-edited by volunteers Mr. Duncan
Griffiths (Canada), Mr. Anthony Dowds (UK) and Mr. Marty
Smith (USA). Additionally, we thank Ms. Kimberley Hoffman
(USA), Ms. Linda Sherwood (USA) and Mr. Seth I. Rich (USA)
for useful remarks and suggestions. Without their kind
assistance, this publication would not be possible.
[8] Prautzsch, H., Boehm, W., Paluszny, M. Bézier and Bspline techniques. Springer, 2002, 190 p.
[9] Muftejev, V.G., Romanyuk, A.N., Mardanov, A.R.,
Farkhutdinov, I.M. Geometrically stable modeling of
NURBS curves and surfaces of arbitrary degrees.
Applied Geometry, 11, pp. 19-77.
[10] Muftejev, V.G., Mardanov, A.R., Romanyuk, A.N.,
Turta, V.G., Farkhutdinov, I.M. Program of
isogeometric modeling of high-quality curves. Webapplication for CAD systems. Proceedings of the
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[11] Golovanov, N.N. Geometric modeling. Moscow,
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[12] Ziatdinov, R., Muftejev, V.G. On a class of plane
curves with equal curvatures of the n-th level. In
Systems of design, technological preparation of
production and management of the life cycle of an
industrial product (CAD/CAM/PDM - 2017). Institute
of Control Problems, Russian Academy of Sciences.
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[13] Yoshida, N., Saito, T. Interactive aesthetic curve
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896-905.
[14] Miura, K.T. A general equation of log-aesthetic curves
and its self-affinity. Journal of the Japan Society for
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[15] Ziatdinov, R. Family of superspirals with completely
monotonic curvature given in terms of Gauss
hypergeometric function. Computer Aided Geometric
Design, 29 (7), 2012, pp. 510-518.
[16] Muftejev, V.G., Ziatdinov, R. Functionality and
aesthetics of curves in industrial design: a multicriteria approach to assessing the quality of shapes in
CAD
systems
of
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[17] FairCurveModeler app for AutoCAD v. 2019
https://apps.autodesk.com/ACD/en/Detail/Index?id=4
526969846340104233&appLang=en&os=Win32_64
&mode=preview&autostart=True&loginRequired=Tr
ue
[18] Wang, A., Zhao, G. Counter examples of “Class A
Bézier curves”. Computer Aided Geometric Design,
61, 2018, pp. 6-8.
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