Computer Aided Geometric Design 29 (2012) 510–518

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Computer Aided Geometric Design

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Family of superspirals with completely monotonic curvature givenin terms of Gauss hypergeometric function ✩

Rushan Ziatdinov ∗

Department of Computer and Instructional Technologies, Fatih University, 34500 Büyükçekmece, Istanbul, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:Available online 28 March 2012

Keywords:SpiralLog-aesthetic curveSuperspiralMonotone curvatureFair curveSurface of revolutionSuperspiraloid

We present superspirals, a new and very general family of fair curves, whose radius ofcurvature is given in terms of a completely monotonic Gauss hypergeometric function.The superspirals are generalizations of log-aesthetic curves, as well as other curves whoseradius of curvature is a particular case of a completely monotonic Gauss hypergeometricfunction. High-accuracy computation of a superspiral segment is performed by the Gauss–Kronrod integration method. The proposed curves, despite their complexity, are thecandidates for generating G2, and G3 non-linear superspiral splines.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

The present work was motivated by an opportunity of finding a very general analytic way, in which so-called fair curves(Levien and Séquin, 2009; Wang et al., 2004) can be represented. The possibility to generate fair curves and surfaces thatare visually pleasing is significant in computer graphics, computer-aided design, and other geometric modeling applications(Sapidis, 1994; Yamada et al., 1999).

A curve’s fairness is usually associated with its monotonically varying curvature, even though this concept still re-mains insufficiently defined (Levien and Séquin, 2009). The different mathematical definitions of fairness and aestheticaspects of geometric modeling are briefly described by Sapidis (1994). The curves of monotone curvature were studiedin recent works. Frey and Field (2000) analyzed the curvature distributions of segments of conic sections represented asrational quadratic Bézier curves in standard form. Farouki (1997) has used the Pythagorean-hodograph quintic curve asthe monotone-curvature transition between a line and a circle. The monotone-curvature condition for rational quadraticB-spline curves is studied by Li et al. (2006). The use of Cornu spirals in drawing planar curves of controlled curvature wasdiscussed by Meek and Walton (1989). The log-aesthetic curves (LACs), which are high-quality curves with linear logarith-mic curvature graphs (Yoshida et al., 2010), have recently been developed to meet the requirements of industrial design forvisually pleasing shapes (Harada et al., 1999; Miura et al., 2005; Miura, 2006; Yoshida and Saito, 2006; Yoshida et al., 2009;Ziatdinov et al., 2012b). LACs were reformulated based on variational principle, and their properties were analyzed by Miuraet al. (2012). A planar spiral called generalized log-aesthetic curve segment (GLAC) (Gobithaasan and Miura, 2011) has beenproposed using the curve synthesis process with two types of formulation: ρ-shift and κ-shift, and it was extended to

✩ This work is dedicated to the 65th birthday of my Ph.D. supervisor Professor Yurii G. Ignatyev (Lobachevsky Institute of Mathematics and Mechanics,Kazan Federal University, Russia).

* Tel.: +90 5310322493.E-mail addresses: rushanziatdinov@yandex.ru, rushanziatdinov@gmail.com.URL: http://www.ziatdinov-lab.com/.

0167-8396/$ – see front matter © 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cagd.2012.03.006

R. Ziatdinov / Computer Aided Geometric Design 29 (2012) 510–518 511

three-dimensional case by Gobithaasan et al. (2012). According to the author of this work, a series of interesting works ofAlexei Kurnosenko (2009, 2010a, 2010b) play an important role in the research on spirals.

Besides artificial objects, spirals, which are the curves with the monotone-curvature function, are important componentsof natural world objects: horns, seashells, bones, leaves, flowers, and tree trunks (Cook, 1903; Harary and Tal, 2011). Inaddition, they are used as a transition curves in rail-road and highway design (Walton and Meek, 1999, 2002; Habib andSakai, 2003, 2004, 2005a, 2005b, 2005c, 2006, 2007, 2009; Dimulyo et al., 2009; Baykal et al., 1997; Walton et al., 2003;Ziatdinov et al., 2012a).

1.1. Main results

In this paper, we consider a radius of curvature function of a planar curve in terms of a very general Gauss hyperge-ometric function, which is completely monotonic under some constraints. It allows us to enclose many well-known spirals,the family of log-aesthetic curves, and other types of curves with monotone curvature, the properties of which can be stillremain unexplored because of the curve’s complicated analytic expression in terms of special functions.

Our work has the following features:

• The proposed superspirals include a huge variety of fair curves with monotonic curvatures.• The superspirals can be computed with high accuracy using the adaptive Gauss–Kronrod method.• The superspirals might allow us to construct a two-point G2 Hermite interpolant, which seems to be impossible to do

by means of log-aesthetic curves since insufficient degrees of freedom;

and several deficiencies:

• The proposed equations are integrals in terms of hypergeometric functions and cannot be represented in terms ofanalytic functions, despite its representation using infinite series.

• Since superspirals have no inflection points in non-polynomial cases, it cannot be considered as a G2 transition betweena straight line and another curve.

• For highly accurate superspiraloid computation, significant time is necessary.

1.2. Organization

The rest of this paper is organized as follows. In Section 2, we shortly discuss about Gauss and confluent hypergeo-metric functions and describe the constraints under which the radius of curvature function, defined in terms of the Gausshypergeometric function, becomes completely monotonic and can be associated with fair curves. In Section 3, we proposethe general equations of the superspirals and discuss their properties providing several examples on their shapes. In Sec-tion 5, we give some graphical examples of superspiraloids, which are actually the surfaces of revolution plotted in CASMathematica 8. In Section 6, we conclude our paper and suggest future work.

2. Preliminaries

In this section, we give a short survey of the work related to the Gauss hypergeometric function.The Gauss hypergeometric function is an analytical function of a,b, c, z, which is defined in C

4 as

2 F1(a,b; c; z) =∞∑

n=0

(a)n(b)n

(c)n

zn

n! , (1)

where z is in the radius of convergence of the series |z| < 1. This series is defined for any a ∈ C, b ∈ C, c ∈ C \ {Z− ∪ {0}},and the Pochhammer symbol is given by

(x)n ={

1, n = 0,

x(x + 1) · · · (x + n − 1), n > 0.

In the general case, where the parameters have arbitrary values, the analytic continuation of F (a,b; c; z) into the plane cutalong [1,∞) can be written as a contour integral, also known as the Barnes integral

2 F1(a,b; c; z) = �(c)

�(a)�(b)

1

2π i

i∞∫−i∞

�(a + s)�(b + s)�(−s)

�(c + s)(−z)s ds,

where

512 R. Ziatdinov / Computer Aided Geometric Design 29 (2012) 510–518

�(z) =∞∫

0

tz−1e−t dt

is a gamma function (Abramowitz and Stegun, 1965; Lebedev, 1965). A complete table of analytic continuation formulasfor the Gauss hypergeometric function, which allow its fast and accurate computation for arbitrary values of z and of theparameters a,b, c can be found in Becken and Schmelcher (2000). There are several specific values of the Gauss hypergeo-metric function in which we are interested:

2 F1(a,b; c;0) = 1,

2 F1(a,b; c;1) = �(c)�(c − a − b)

�(c − a)�(c − b), Re(c − a − b) > 0.

Besides the Gauss hypergeometric function, the function, which is called a confluent hypergeometric function (Kummer’sfunction), plays an important role in special functions theory:

Φ(a,b, z) =∞∑

n=0

a(n)zn

b(n)n! = 1 F1(a;b; z),

where

a(n) = a(a + 1)(a + 2) · · · (a + n − 1)

is the rising factorial.The Gauss hypergeometric function is the generalization of many well-known functions such as power, exponential, log-

arithmic, gamma, error, and inverse trigonometric functions, and elliptic, Fresnel, exponential integrals as well as Hermite,Laguerre, Chebyshev, and Jacobi polynomials. The functions discussed above play an important role in mathematical anal-ysis and its applications. For more exhaustive information on hypergeometric functions and their properties, the reader isreferred to Abramowitz and Stegun (1965), Lebedev (1965), Yoshida (1997).

3. The family of superspirals

It was shown by Miller and Samko (2001) that function 2 F1(a,b; c;−θ) is completely monotonic for c > b > 0, a > 0when θ � 0, and it can also be considered as an extension of the radius of curvature function in terms of the tangent angle(it is also known as Cesàro equation; Yates, 1952)

ρ(θ) ={

eλθ , α = 1,

((α − 1)λ θ + 1)1

α−1 , otherwise

of log-aesthetic curves with the shape parameter α (Harada et al., 1999; Miura et al., 2005; Miura, 2006; Yoshida and Saito,2006; Ziatdinov et al., 2012b) in the following way

ρ(θ) ={

Φ(α,α,λθ), α = 1,

2 F1(1

1−α ,b;b; (1 − α)λθ), otherwise.

It means that LACs, which include well-known spirals as Euler, logarithmic, and Nielsen’s spiral and involutes of a circle isthe subset of the set of curves with a completely monotone curvature, given in terms of the Gauss hypergeometric function.

The curves with the monotonically varying curvature (radius of curvature) are often being called fair curves (Levien andSéquin, 2009; Wang et al., 2004), and they, as well as class A Bézier curves (Farin, 2006) are very significant in computer-aided design and aesthetic shape modeling (Dankwort and Podehl, 2000). We present the following new definition in thiswork.

Definition 1. A superspiral is a planar curve with a completely monotone radius of curvature given in the form ρ(ψ) =2 F1(a,b; c;−ψ), where c > b > 0, a > 0. Its corresponding parametric equation in terms of the tangent angle is

S(a,b; c; θ) =(

x(θ)

y(θ)

)=

( ∫ θ

0 2 F1(a,b; c;−ψ) cos ψ dψ∫ θ

0 2 F1(a,b; c;−ψ) sin ψ dψ

), (2)

where 0 � θ < +∞.

It is important to note that the integrals in Eq. (2) cannot be represented in analytical form except their representationin terms of infinite series, which will be discussed a little later, thus we will apply the adaptive Gauss–Kronrod integration

R. Ziatdinov / Computer Aided Geometric Design 29 (2012) 510–518 513

(Kronrod, 1964; Laurie, 1997) for computing a curve segment with high accuracy, as it has been done by Yoshida and Saito(2006).

The first and second derivatives of a superspiral can be simply computed from Eq. (2):

dy(θ)

dx(θ)= dy(θ)

dθ

/dx(θ)

dθ= tan θ,

d2 y(θ)

dx2(θ)= 1 + tan2 θ

2 F1(a,b; c;−θ) cos θ.

The arclength of the parametric curve (2) can be obtained from well known in differential geometry relationship(Pogorelov, 1974; Struik, 1988), ds/dθ = ρ(θ), and after integration of the Gauss hypergeometric function (Abramowitzand Stegun, 1965; Gradshtein and Ryzhik, 1962) we obtain

s =θ∫

0

ρ(ψ)dψ =θ∫

0

2 F1(a,b; c;−ψ)dψ = − c − 1

(a − 1)(b − 1)

[2 F1(a − 1,b − 1; c − 1;−θ) − 1

]. (3)

Eq. (3), which relates the arclength with the tangent angle is often called the Whewell equation (Whewell, 1849).We are interested in the non-negative values of the tangent angle θ since the restrictions mentioned above, and this

makes the properties of a superspiral to be as discussed below.

• ρ(0) = 2 F1(a,b; c;0) = 1 for ∀a,b, c, thus it can be simply seen from Eq. (2) that a superspiral is always passing via theorigin point, where it has θ = 0.

• x-axis is a line tangent to a spiral at the origin.• For fixed a,b, c, in non-polynomial cases, a superspiral has no singularities.• Absence of upper or lower bounds for θ .• Strictly monotone curvature.

4. Small-angle approximation and representation in terms of infinite series

For practical purposes, it is also important to consider the small values of the tangent angle θ . Hence, we may considerasymptotic approximations of Eq. (2). Taking into account the small-angle approximations,

cosψ =∞∑

n=0

(−1)n

(2n)! ψ2n ≈ 1 − ψ2

2, (4)

sin ψ =∞∑

n=0

(−1)n

(2n + 1)!ψ2n+1 ≈ ψ, (5)

and after integrating by parts in Eq. (2), we finally obtain

x(θ) = 1

2(a − 3)(a − 2)(a − 1)(b − 3)(b − 2)(b − 1)

× [(c − 1)

(2a2b2 − 10a2b + 12a2 − 10ab2 + 2

(c2 − 5c + 6

)2 F1(a − 3,b − 3; c − 3;−θ)

+ 2(a − 3)(b − 3)(c − 2)θ 2 F1(a − 2,b − 2; c − 2;−θ) + 50ab − 60a + 12b2 − 60b − 2c2 + 10c + 60)]

+ 1

2(a − 3)(a − 2)(a − 1)(b − 3)(b − 2)(b − 1)

× [(c − 1)

(a2b2θ2 − 2a2b2 − 5a2bθ2 + 10a2b + 6a2θ2 − 12a2 − 5ab2θ2 + 10ab2 + 25abθ2 − 50ab

− 30aθ2 + 60a + 6b2θ2 − 12b2 − 30bθ2 + 60b + 36θ2 − 72)

2 F1(a − 1,b − 1; c − 1;−θ)],

y(θ) = − (c2 − 3c + 2)

(a − 2)(a − 1)(b − 2)(b − 1)

[(a − 1) 2 F1(a − 2,b − 2; c − 2;−θ)

− (a − 2) 2 F1(a − 1,b − 2; c − 2;−θ) − 1].

The derived parametric equations do not contain integrals of special functions, and are actually simpler from a computationpoint of view, despite the visual clumsiness.

There is another way to represent Eq. (2). Considering integrand functions as infinite series using Eqs. (1), (4), (5), andoperating with sums one can obtain1

1 The possibility to write in this form has been noted by one of reviewers.

514 R. Ziatdinov / Computer Aided Geometric Design 29 (2012) 510–518

Fig. 1. (a) An example of the superspiral with a = 0.1, b = 1, c = 2, and θ ∈ [−1,10π ]. (b) Its curvature function, 2 F1(0.1,1;2;−θ)−1 = 109 θ((1+θ)

910 −1)−1,

for θ � 0.

Fig. 2. (a) An example of the superspiral with a = 1, b = 1, c = 2, and θ ∈ [−1,10π ]. (b) Its curvature function, 2 F1(1,1;2;−θ)−1 = θlog(1+θ)

, for θ � 0.

x(θ) =θ∫

0

∞∑n=0

(a)n(b)n

(c)n

ψn

n! ×∞∑

n=0

(−1)n

(2n)! ψ2n dψ =θ∫

0

∞∑n=0

(ψn

∑i+2 j=ni, j∈N0

(a)i(b)i

(c)i

(−1) j

i!(2 j)!)

dψ

=θ∫

0

∞∑n=0

(ψn

n!∑

i+2 j=ni, j∈N0

(n

i

)(−1)i (a)i(b)i

(c)i

)dψ =

∞∑n=0

(θn+1

(n + 1)!∑

i+2 j=ni, j∈N0

(n

i

)(−1)i (a)i(b)i

(c)i

), (6)

y(θ) =θ∫

0

∞∑n=0

(a)n(b)n

(c)n

ψn

n! ×∞∑

n=0

(−1)n

(2n + 1)!ψ2n+1 dψ =

θ∫0

∞∑n=0

(ψn

∑i+2 j+1=n

i, j∈N0

(a)i(b)i

(c)i

(−1) j

i!(2 j + 1)!)

dψ

=θ∫

0

∞∑n=0

(ψn

n!∑

i+2 j+1=ni, j∈N0

(n

i

)(−1)i (a)i(b)i

(c)i

)dψ =

∞∑n=0

(θn+1

(n + 1)!∑

i+2 j+1=ni, j∈N0

(n

i

)(−1)i (a)i(b)i

(c)i

). (7)

5. Numerical examples

This section gives numerical examples of the presented superspirals and superspiral surfaces of revolution. Several shapesof superspirals with their curvatures are shown in Figs. 1–4, and an example of fillet modeling is shown in Fig. 5.

R. Ziatdinov / Computer Aided Geometric Design 29 (2012) 510–518 515

Fig. 3. (a) An example of the superspiral with a = 2, b = 1, c = 2, and θ ∈ [−1,10π ]. (b) Its curvature function, 2 F1(2,1;2;−θ)−1 = 1 + θ , for θ � 0.

Fig. 4. (a) An example of the superspiral with a = 2, b = 2, c = 2, and θ ∈ [−1,10π ]. (b) Its curvature function, 2 F1(2,2;2;−θ)−1 = (θ − 1)2, for θ � 0.

Fig. 5. (a) An example of fillet modeling: A G1 transition superspiral between two straight lines is generated by the Yoshida–Saito method (Yoshida andSaito, 2006) and swept along the z-axis to obtain a transition surface between the two planes. (b) The same surface with generated zebra lines.

516 R. Ziatdinov / Computer Aided Geometric Design 29 (2012) 510–518

Fig. 6. (a) The superspiraloid obtained by rotating about the y-axis with a = 1.1, b = 2, c = 2, 0 � θ � 3π2 . (b) The superspiraloid obtained by rotating

about the y-axis with a = 1.1, b = 2, c = 2, 0 � θ � π .

Fig. 7. (a) The superspiraloid obtained by rotating about the y-axis with a = 1.1, b = 2, c = 2, 2π � θ � 4π . (b) The superspiraloid obtained by rotatingabout the y-axis with a = 1.1, b = 2, c = 2, π

2 � θ � 5π .

Fig. 8. (a) The superspiraloid obtained by rotating about the x-axis with a = 1.1, b = 2, c = 2, 0 � θ � 2π . (b) The superspiraloid obtained by rotating aboutthe x-axis with a = 1.1, b = 2, c = 2, 0 � θ � π

2 , which is similar to the black-hole model (Chandrasekhar, 1998).

In Figs. 6–8, one may see the surfaces of revolution, which are applicable to computer-aided design of, for example,canal or pipe surfaces (Farin et al., 2002). The resulting surface, therefore, always has azimuthal symmetry. The followingdefinition refers to these surfaces.

Definition 2. A superspiraloid is a surface generated by rotating a two-dimensional superspiral curve segment about an axis.

R. Ziatdinov / Computer Aided Geometric Design 29 (2012) 510–518 517

6. Conclusions and future work

We have introduced analytic parametric equations for superspirals, whose radius of curvature is given by Gauss hyper-geometric functions, which are completely monotonic under described conditions. Whereas previous authors deal with thespecific curves having linear curvature graphs (Yoshida and Saito, 2006), the superspirals can cover a huge variety of faircurves.

There are several directions for future work. It is possible to generalize the radius of curvature function and presentit in terms of the generalized hypergeometric function, p Fq(a1, . . . ,ap;b1, . . . ,bq; z), or even the Meijer G-function (Meijer,1936), which intends to include most of the known special functions as particular cases, or as the Fox H-function introducedin Fox (1961), which is a generalization of the Meijer G-function. But, in such an approach, monotonicity conditions wouldbe somehow much more complex or even not discovered. Proposed superspirals can logically be applied for generating non-linear splines (Mehlum, 1974) using the Yoshida–Saito method for two-point G1 Hermite interpolation (Yoshida and Saito,2006), as well as for constructing non-linear spline with curvature continuity (which is actually G2 multispiral; Ziatdinov etal., 2012a), the generating algorithm for which would be the scope of our next works. Finally, we will like to extend thisapproach to generate superspiral space curve segments and three-dimensional superspiral splines.

Acknowledgements

I would like to thank Prof. Norimasa Yoshida (Nihon University, Japan), Prof. Stefan G. Samko (Universidade do Algarve,Spain) for useful comments and suggestions, and Prof. Tae-wan Kim (Seoul National University, South Korea), in the labora-tory of whom I started my research on spirals. The authors appreciate the issues, remarks, and very important suggestionsof the anonymous reviewers which helped to improve the quality of this paper.

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