Family of superspirals with ompletely monotoni urvature

given in terms of Gauss hypergeometri funtion

IComputer Aided Geometri Design 29(7), 510-518

Rushan Ziatdinov

a

a

Department of Computer and Instrutional Tehnologies, Fatih University, 34500 Buyukekmee,

Istanbul, Turkey

Abstrat

We present superspirals, a new and very general family of fair urves, whose radius of

urvature is given in terms of a ompletely monotoni Gauss hypergeometri funtion.

The superspirals are generalizations of log-aestheti urves, as well as other urves whose

radius of urvature is a partiular ase of a ompletely monotoni Gauss hypergeometri

funtion. High-a

uray omputation of a superspiral segment is performed by the Gauss-

Kronrod integration method. The proposed urves, despite their omplexity, are the

andidates for generating G2, and G3 non-linear superspiral splines.

Keywords: spiral, log-aestheti urve, superspiral, monotone urvature, fair urve,

surfae of revolution, superspiraloid

2010 MSC: 65D17, 68U07

1. Introdution

The present work was motivated by an opportunity of nding a very general analyti

way, in whih so-alled fair urves [32, 47 an be represented. The possibility to generate

fair urves and surfaes that are visually pleasing is signiant in omputer graphis,

omputer-aided design, and other geometri modeling appliations [42, 49.

A urve's fairness is usually assoiated with its monotonially varying urvature, even

though this onept still remains insuiently dened [32. The dierent mathematial

denitions of fairness and aestheti aspets of geometri modeling are briey desribed

in [42. The urves of monotone urvature were studied in reent works. Frey et al.

[12 analyzed the urvature distributions of segments of oni setions represented as

rational quadrati Bezier urves in standard form. Farouki [10 has used the Pythagorean-

Hodograph quinti urve as the monotone-urvature transition between a line and a irle.

The monotone urvature ondition for rational quadrati B-spline urves is studied by

Li et al.[33. The use of Cornu spirals in drawing planar urves of ontrolled urvature

was disussed in [34. The log-eastheti urves (LACs), whih are high-quality urves

IThis work is dediated to the 65th birthday of my Ph.D. supervisor Professor Yurii G. Ignatyev

(Lobahevsky Institute of Mathematis and Mehanis, Kazan Federal University, Russia)

Corresponding author, Tel.: +905310322493

Email address: rushanziatdinovyandex.ru, rushanziatdinovgmail.om (Rushan Ziatdinov)

URL: http://www.ziatdinov-lab.om/ (Rushan Ziatdinov)

with linear logarithmi urvature graphs [53, have reently been developed to meet the

requirements of industrial design for visually pleasing shapes [24, 38, 39, 54, 52, 56. LACs

were reformulated based on variational priniple, and their properties were analysed in

[40. A planar spiral alled generalized log-aestheti urve segment (GLAC) [13 has been

proposed using the urve synthesis proess with two types of formulation; -shift and-shift, and it was extended to three-dimensional ase in [14. A

ording to the author ofthis work, a series of interesting works of Alexei Kurnosenko [27, 28, 29 play an important

role in the researh on spirals.

Besides artiial objets, spirals, whih are the urves with the monotone urvature

funtion, are important omponents of natural world objets: horns, seashells, bones,

leaves, owers, and tree trunks [5, 25. In addition, they are used as a transition urves

in rail-road and highway design [44, 45, 18, 23, 17, 16, 19, 20, 21, 22, 21, 7, 2, 46, 55.

Main results

In this paper, we onsider a radius of urvature funtion of a planar urve in terms of

a very general Gauss hypergeometri funtion, whih is ompletely monotoni under some

onstraints. It allows us to enlose many well-known spirals, the family of log-aestheti

urves, and other types of urves with monotone urvature, the properties of whih an

be still remain unexplored beause of the urve's ompliated analyti expression in terms

of speial funtions.

Our work has the following features:

The proposed superspirals inlude a huge variety of fair urves with monotoni

urvatures;

The superspirals an be omputed with high a

uray using the adaptive Gauss-Kronrod method;

The superspirals might allow us to onstrut a two-point G2 Hermite interpolant,whih seems to be impossible to do by means of log-aestheti urves sine insuient

degrees of freedom.

and several deienies:

The proposed equations are integrals in terms of hypergeometri funtions and

annot be represented in terms of analyti funtions, despite its representation

using innite series;

Sine superspirals have no inetion points in non-polynomial ases, it annot be

onsidered as a G2 transition between a straight line and another urve;

For highly a

urate superspiraloid omputation, signiant time is neessary.

Organization

The rest of this paper is organized as follows. In Setion 2, we shortly disuss about

Gauss and onuent hypergeometri funtions and desribe the onstraints under whih

the radius of urvature funtion, dened in terms of the Gauss hypergeometri funtion,

2

beomes ompletely monotoni and an be assoiated with fair urves. In Setion 3, we

propose the general equations of the superspirals and disuss their properties providing

several examples on their shapes. In Setion 5, we give some graphial examples of

superspiraloids, whih are atually the surfaes of revolution plotted in CAS Mathematia

8. In Setion 6, we onlude our paper and suggest future work.

2. Preliminaries

In this setion, we give a short survey of the work related to the Gauss hypergeometri

funtion.

The Gauss hypergeometri funtion is an analytial funtion of a, b, c, z, whih isdened in C4 as

2F1(a, b; c; z) =

n=0

(a)n(b)n(c)n

zn

n!, (1)

where z is in the radius of onvergene of the series |z| < 1. This series is dened for anya C, b C, c C\{Z {0}}, and the Pohhammer symbol is given by

(x)n =

{1, n = 0x(x+ 1) (x+ n 1), n > 0.

In the general ase, where the parameters have arbitrary values, the analyti ontinuation

of F (a, b; c; z) into the plane ut along [1,) an be written as a ontour integral, alsoknown as the Barnes integral

2F1(a, b; c; z) =

(c)

(a)(b)

1

2pii

ii

(a+ s)(b+ s)(s)

(c+ s)(z)s ds,

where

(z) =

0

tz1et dt

is a gamma funtion [1, 31. A omplete table of analyti ontinuation formulas for

the Gauss hypergeometri funtion, whih allow its fast and a

urate omputation for

arbitrary values of z and of the parameters a, b, c an be found in [3. There are severalspei values of the Gauss hypergeometri funtion in whih we are interested:

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

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

(c a)(c b), Re(c a b) > 0.

Besides the Gauss hypergeometi funtion, the funtion, whih is alled a onuent

hypergeometri funtion (Kummer's funtion), plays an important role in speial fun-

tions theory:

3

(a, b, z) =

n=0

a(n)zn

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

where

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

is the rising fatorial.

The Gauss hypergeometri funtion is the generalization of many well-known fun-

tions suh as power, exponential, logarithmi, gamma, error, and inverse trigonometri

funtions, and ellipti, Fresnel, exponential integrals as well as Hermite, Laguerre, Cheby-

shev, and Jaobi polynomials. The funtions disussed above play an important role in

mathematial analysis and its appliations. For more exhaustive information on hyper-

geometri funtions and their properties, the reader is referred to [1, 31, 51.

3. The family of superspirals

It was shown in [37 that funtion 2F1(a, b; c;) is ompletely monotoni for c >b > 0, a > 0 when 0, and it an also be onsidered as an extension of the radiusof urvature funtion in terms of the tangent angle (it is also known as Cesaro equation

[50)

() =

{e, = 1

(( 1) + 1)1

1 , otherwise

of log-aestheti urves with the shape parameter [24, 38, 39, 54, 56 in the followingway

() =

{(, , ), = 1

2F1(

11

, b; b; (1 )), otherwise

.

It means that LACs, whih inlude well-known spirals as Euler, logarithmi, and Nielsen's

spiral and involutes of a irle is the subset of the set of urves with a ompletely monotone

urvature, given in terms of the Gauss hypergeometri funtion.

The urves with the monotonially varying urvature (radius of urvature) are often

being alled fair urves [32, 47, and they, as well as lass A Bezier urves [8 are very

signiant in omputer-aided design and aestheti shape modeling [6. We present the

following new denition in this work.

Denition 1. A superspiral is a planar urve with a ompletely monotone radius of

urvature given in the form () = 2F1(a, b; c;), where c > b > 0, a > 0. Its orre-sponding parametri equation in terms of the tangent angle is

S(a, b; c; ) =

(x()y()

)=

0

2F1(a, b; c;) cos d

0

2F1(a, b; c;) sin d

,

(2)

where 0 < +.

4

It is important to note that the integrals in Eq. (2) annot be represented in analytial

form exept their representation in terms of innite series, whih will be disussed a little

later, thus we will apply the adaptive Gauss-Kronrod integration [26, 30 for omputing

a urve segment with high a

uray, as it has been done in [54.

The rst and seond derivatives of a superspiral an be simply omputed from Eq. 2:

dy()

dx()=

dy()

d/dx()

d= tan ,

d2y()

dx2()=

1 + tan2

2F1(a, b; c;) cos .

The arlength of the parametri urve (2) an be obtained from well known in dif-

ferential geometry relationship [41, 43, ds/d = (), and after integration of the Gausshypergeometri funtion [1, 15 we obtain

s =

0

() d =

0

2F1(a, b; c;) d

= c 1

(a 1)(b 1)[2F1(a 1, b 1; c 1;) 1] . (3)

Eq. 3, whih relates the arlength with the tangent angle is often alled the Whewell

equation [48.

We are interested in the non-negative values of the tangent angle sine the restri-tions mentioned above, and this makes the properties of a superspiral to be as disussed

below.

(0) = 2F1(a, b; c; 0) = 1 for a, b, c, thus it an be simply seen from Eq. 2 that asuperspiral is always passing via the origin point, where it has = 0;

x-axis is a line tangent to a spiral at the origin;

For xed a, b, c, in nonpolynomial ases, a superspiral has no singularities;

Absene of upper or lower bounds for ;

Stritly monotone urvature.

4. Small angle approximation and representation in terms of innite series

For pratial purposes, it is also important to onsider the small values of the tangent

angle . Hene, we may onsider asymptoti approximations of Eq. 2. Taking intoa

ount the small-angle approximations,

cos =

n=0

(1)n

(2n)!2n 1

2

2, (4)

sin =n=0

(1)n

(2n+ 1)!2n+1 , (5)

5

and after integrating by parts in Eq. 2, we nally obtain

x() =1

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

[(c 1)(2a2b2 10a2b+ 12a2 10ab2 + 2(c2 5c+ 6)

2F1(a 3, b 3; c 3;) + 2(a 3)(b 3)(c 2)

2F1(a 2, b 2; c 2;)

+50ab 60a+ 12b2 60b 2c2 + 10c+ 60)]

+1

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

[(c 1)(a2b22 2a2b2 5a2b2 + 10a2b+ 6a22

12a2 5ab22 + 10ab2 + 25ab2 50ab 30a2+

60a+ 6b22 12b2 30b2 + 60b+ 362 72)

2F1(a 1, b 1; c 1;)],

y() = (c2 3c+ 2)

(a 2)(a 1)(b 2)(b 1)[(a 1)

2F1(a 2, b 2; c 2;)

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

The derived parametri equations do not ontain integrals of speial funtions, and are

atually simpler from a omputation point of view, despite the visual lumsiness.

There is another way to represent Eq. 2. Considering integrand funtions as innite

series using Eqs. 1, 4, 5, and operating with sums one an obtain

1

x() =

0

n=0

(a)n(b)n(c)n

n

n!

n=0

(1)n

(2n)!2n d

=

0

n=0

n

i+2j=ni,jN0

(a)i(b)i(c)i

(1)j

i!(2j)!

d

=

0

n=0

n

n!

i+2j=ni,jN0

(n

i

)(1)i

(a)i(b)i(c)i

d

=

n=0

n+1

(n+ 1)!

i+2j=ni,jN0

(n

i

)(1)i

(a)i(b)i(c)i

,

(6)

1

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

6

y() =

0

n=0

(a)n(b)n(c)n

n

n!

n=0

(1)n

(2n+ 1)!2n+1 d

=

0

n=0

n

i+2j+1=ni,jN0

(a)i(b)i(c)i

(1)j

i!(2j + 1)!

d

=

0

n=0

n

n!

i+2j+1=ni,jN0

(n

i

)(1)i

(a)i(b)i(c)i

d

=

n=0

n+1

(n + 1)!

i+2j+1=ni,jN0

(n

i

)(1)i

(a)i(b)i(c)i

.

(7)

5. Numerial examples

This setion gives numerial examples of the presented superspirals and superspiral

surfaes of revolution. Several shapes of superspirals with their urvatures are shown in

Figs. 1 - 4, and an example of llet modeling is shown in Fig. 5.

-0.5 0.5xHL

0.5

1.0

1.5

yHL

5 10 15 20 250

5

10

15

20

25

HL

(a) (b)

Figure 1: (a) An example of the superspiral with a = 0.1, b = 1, c = 2, and [1, 10pi] (b) Its urvature

funtion, 2F1(0.1, 1; 2;)1 = 109

((1 + )

9

10 1)1

, for 0.

7

-0.2 0.2 0.4 0.6xHL

0.2

0.4

0.6

0.8

1.0

yHL

5 10 15 20 250

5

10

15

20

25

HL

(a) (b)

Figure 2: (a) An example of the superspiral with a = 1, b = 1, c = 2, and [1, 10pi] (b) Its urvaturefuntion, 2F1(1, 1; 2;)

1 = log(1+) , for 0.

-0.2 0.0 0.2 0.4 0.6xHL

0.2

0.4

0.6

0.8

yHL

5 10 15 20 250

5

10

15

20

25

HL

(a) (b)

Figure 3: (a) An example of the superspiral with a = 2, b = 1, c = 2, and [1, 10pi] (b) Its urvaturefuntion, 2F1(2, 1; 2;)

1 = 1 + , for 0.

8

-0.2-0.1 0.0 0.1 0.2 0.3 0.4xHL

0.1

0.2

0.3

0.4

0.5

yHL

5 10 15 20 250

5

10

15

20

25

HL

(a) (b)

Figure 4: (a) An example of the superspiral with a = 2, b = 2, c = 2, and [1, 10pi] (b) Its urvaturefuntion, 2F1(2, 2; 2;)

1 = ( 1)2, for 0.

(a) (b)

Figure 5: (a) An example of llet modeling: A G1 transition superspiral between two straight lines is

generated by the Yoshida-Saito method [54 and swept along the z-axis to obtain a transition surfae

between the two planes. (b) The same surfae with generated zebra lines.

In Figs. 6-8, one may see the surfaes of revolution, whih are appliable to omputer

aided design of, for example, anal or pipe surfaes [9. The resulting surfae, therefore,

always has azimuthal symmetry. The following denition refers to these surfaes.

Denition 2. A superspiraloid is a surfae generated by rotating a two-dimensional su-

perspiral urve segment about an axis.

9

(a) (b)

Figure 6: (a) The superspiraloid obtained by rotating about the y-axis with a = 1.1, b = 2, c = 2, 0 3pi2 , (b) The superspiraloid obtained by rotating about the y-axis with a = 1.1, b = 2, c = 2, 0 pi.

(a) (b)

Figure 7: (a) The superspiraloid obtained by rotating about the y-axis with a = 1.1, b = 2, c = 2, 2pi 4pi, (b) The superspiraloid obtained by rotating about the y-axis with a = 1.1, b = 2, c = 2, pi2 5pi.

(a) (b)

Figure 8: (a) The superspiraloid obtained by rotating about the x-axis with a = 1.1, b = 2, c = 2, 0 2pi, (b) The superspiraloid obtained by rotating about the x-axis with a = 1.1, b = 2, c = 2, 0 pi2 ,whih is similar to the blak-hole model [4.

10

6. Conlusions and future work

We have introdued analyti parametri equations for superspirals, whose radius of

urvature is given by Gauss hypergeometri funtions, whih are ompletely monotoni

under desribed onditions. Whereas previous authors deal with the spei urves having

linear urvature graphs [54, the superspirals an over a huge variety of fair urves.

There are several diretions for future work. It is possible to generalize the radius

of urvature funtion and present it in terms of the generalized hypergeometri fun-

tion, pFq(a1, . . . , ap; b1, . . . , bq; z), or even the Meijer G-funtion [36, whih intends toinlude most of the known speial funtions as partiular ases, or as the Fox H-funtion

introdued in [11, whih is a generalization of the Meijer G-funtion. But, in suh

an approah, monotoniity onditions would be somehow muh more omplex or even

not disovered. Proposed superspirals an logially be applied for generating non-linear

splines [35 using the Yoshida-Saito method for two-point G1 Hermite interpolation [54,as well as for onstruting non-linear spline with urvature ontinuity (whih is atually

G2 multispiral [55), the generating algorithm for whih would be the sope of our nextworks. Finally, we will like to extend this approah to generate superspiral spae urve

segments and three-dimensional superspiral splines.

7. Aknowledgement

I would like to thank Prof. Norimasa Yoshida (Nihon University, Japan), Prof. Stefan

G. Samko (Universidade do Algarve, Spain) for useful omments and suggestions, and

Prof. Tae-wan Kim (Seoul National University, South Korea), in the laboratory of whom

I started my researh on spirals. The authors appreiate the issues, remarks, and very

important suggestions of the anonymous reviewers whih helped to improve the quality

of this paper.

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15

IntroductionPreliminariesThe family of superspiralsSmall angle approximation and representation in terms of infinite seriesNumerical examplesConclusions and future workAcknowledgement