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Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function
Computer Aided Geometric Design 29 (2012) 510–518
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Family of superspirals with completely monotonic curvature given in 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
Article history: Available online 28 March 2012 Keywords: Spiral Log-aesthetic curve Superspiral Monotone curvature Fair curve Surface of revolution Superspiraloid
a b s t r a c t We present superspirals, a new and very general family of fair curves, whose radius of curvature is given in terms of a completely monotonic Gauss hypergeometric function. The superspirals are generalizations of log-aesthetic curves, as well as other curves whose radius of curvature is a particular case of a completely monotonic Gauss hypergeometric function. High-accuracy computation of a superspiral segment is performed by the Gauss– Kronrod integration method. The proposed curves, despite their complexity, are the candidates for generating G 2 , and G 3 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 that are 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 remains insufficiently defined (Levien and Séquin, 2009). The different mathematical definitions of fairness and aesthetic aspects of geometric modeling are briefly described by Sapidis (1994). The curves of monotone curvature were studied in recent works. Frey and Field (2000) analyzed the curvature distributions of segments of conic sections represented as rational quadratic Bézier curves in standard form. Farouki (1997) has used the Pythagorean-hodograph quintic curve as the monotone-curvature transition between a line and a circle. The monotone-curvature condition for rational quadratic B-spline curves is studied by Li et al. (2006). The use of Cornu spirals in drawing planar curves of controlled curvature was discussed by Meek and Walton (1989). The log-aesthetic curves (LACs), which are high-quality curves with linear logarithmic curvature graphs (Yoshida et al., 2010), have recently been developed to meet the requirements of industrial design for visually 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 Miura et al. (2012). A planar spiral called generalized log-aesthetic curve segment (GLAC) (Gobithaasan and Miura, 2011) has been proposed 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: [email protected], [email protected]. URL: http://www.ziatdinov-lab.com/.
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0167-8396/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cagd.2012.03.006
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three-dimensional case by Gobithaasan et al. (2012). According to the author of this work, a series of interesting works of Alexei 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 components of natural world objects: horns, seashells, bones, leaves, flowers, and tree trunks (Cook, 1903; Harary and Tal, 2011). In addition, they are used as a transition curves in rail-road and highway design (Walton and Meek, 1999, 2002; Habib and Sakai, 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 hypergeometric 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 still remain 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 G 2 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 of analytic functions, despite its representation using infinite series.
• Since superspirals have no inflection points in non-polynomial cases, it cannot be considered as a G 2 transition between a 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 hypergeometric functions and describe the constraints under which the radius of curvature function, defined in terms of the Gauss hypergeometric function, becomes completely monotonic and can be associated with fair curves. In Section 3, we propose the general equations of the superspirals and discuss their properties providing several examples on their shapes. In Section 5, we give some graphical examples of superspiraloids, which are actually the surfaces of revolution plotted in CAS Mathematica 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 C4 as 2 F 1 (a, b ; c ; z) =
∞ (a)n (b)n zn , (c )n n!
(1)
n =0
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 cut along [1, ∞) can be written as a contour integral, also known as the Barnes integral
(c ) 1 2 F 1 (a, b ; c ; z) = (a)(b) 2π i where
i ∞ −i ∞
(a + s)(b + s)(−s) (− z)s ds, (c + s)
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∞
t z−1 e −t dt
( z) = 0
is a gamma function (Abramowitz and Stegun, 1965; Lebedev, 1965). A complete table of analytic continuation formulas for the Gauss hypergeometric function, which allow its fast and accurate computation for arbitrary values of z and of the parameters a, b, c can be found in Becken and Schmelcher (2000). There are several specific values of the Gauss hypergeometric function in which we are interested: 2 F 1 (a, b ; c ; 0)
= 1, (c )(c − a − b) , 2 F 1 (a, b ; c ; 1) = (c − a)(c − b)
Re(c − a − b) > 0.
Besides the Gauss hypergeometric function, the function, which is called a confluent hypergeometric function (Kummer’s function), plays an important role in special functions theory:
Φ(a, b, z) =
∞ (n) n a z n =0
b(n)n!
= 1 F 1 (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, logarithmic, 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 analysis and its applications. For more exhaustive information on hypergeometric functions and their properties, the reader is referred 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 F 1 (a, b; c ; −θ) is completely monotonic for c > b > 0, a > 0 when 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) =
α = 1,
e λθ , 1
((α − 1)λ θ + 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) =
α = 1, Φ(α , α , λθ), 1 F ( , b ; b ; ( 1 − α )λθ), otherwise. 2 1 1 −α
It means that LACs, which include well-known spirals as Euler, logarithmic, and Nielsen’s spiral and involutes of a circle is the 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 and Séquin, 2009; Wang et al., 2004), and they, as well as class A Bézier curves (Farin, 2006) are very significant in computeraided design and aesthetic shape modeling (Dankwort and Podehl, 2000). We present the following new definition in this work. Definition 1. A superspiral is a planar curve with a completely monotone radius of curvature given in the form 2 F 1 (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
=
F (a, b; c ; −ψ) cos ψ 0 2 1
dψ
F (a, b; c ; −ψ) sin ψ 0 2 1
dψ
θ
ρ (ψ) =
,
(2)
where 0 θ < +∞. It is important to note that the integrals in Eq. (2) cannot be represented in analytical form except their representation in terms of infinite series, which will be discussed a little later, thus we will apply the adaptive Gauss–Kronrod integration
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(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):
d y (θ)
=
dx(θ)
d2 y (θ)
d y (θ) dx(θ)
=
dx2 (θ)
dθ
dθ
= tan θ,
1 + tan2 θ 2 F 1 (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 (Abramowitz and Stegun, 1965; Gradshtein and Ryzhik, 1962) we obtain
θ s=
θ
ρ (ψ) dψ = 0
2 F 1 (a, b ; c ; −ψ) dψ 0
=−
c−1 2 F 1 (a − 1, b − 1; c − 1; −θ) − 1 . (a − 1)(b − 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 F 1 (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 the origin 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 consider asymptotic approximations of Eq. (2). Taking into account the small-angle approximations,
cos ψ =
∞ (−1)n n =0
sin ψ =
(2n)!
ψ 2n ≈ 1 −
ψ2 2
(4)
,
∞ (−1)n ψ 2n+1 ≈ ψ, (2n + 1)!
(5)
n =0
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) 2a2 b2 − 10a2 b + 12a2 − 10ab2 + 2 c 2 − 5c + 6 2 F 1 (a − 3, b − 3; c − 3; −θ) + 2(a − 3)(b − 3)(c − 2)θ 2 F 1 (a − 2, b − 2; c − 2; −θ) + 50ab − 60a + 12b2 − 60b − 2c 2 + 10c + 60 +
1 2(a − 3)(a − 2)(a − 1)(b − 3)(b − 2)(b − 1)
× (c − 1) a2 b2 θ 2 − 2a2 b2 − 5a2 bθ 2 + 10a2 b + 6a2 θ 2 − 12a2 − 5ab2 θ 2 + 10ab2 + 25abθ 2 − 50ab − 30aθ 2 + 60a + 6b2 θ 2 − 12b2 − 30bθ 2 + 60b + 36θ 2 − 72 2 F 1 (a − 1, b − 1; c − 1; −θ) , (c 2 − 3c + 2) (a − 1) 2 F 1 (a − 2, b − 2; c − 2; −θ) (a − 2)(a − 1)(b − 2)(b − 1) − (a − 2) 2 F 1 (a − 1, b − 2; c − 2; −θ) − 1 .
y (θ) = −
The derived parametric equations do not contain integrals of special functions, and are actually simpler from a computation point 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), and operating with sums one can obtain1
1
The possibility to write in this form has been noted by one of reviewers.
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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 F 1 (0.1, 1; 2; −θ)−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 F 1 (1, 1; 2; −θ)−1 =
x(θ) =
θ ∞ 0 n =0
=
θ ∞ 0 n =0
=
n =0
θ ∞ 0 n =0
y (θ) =
∞
(a)n (b)n ψ n (−1)n 2n × ψ dψ = (c )n n! (2n)! ψn n!
n =0
i +2 j +1=n i , j ∈N0
, for θ 0.
i +2 j =n i , j ∈N0
θ ∞ 0 n =0
(6)
i +2 j =n i , j ∈N0
ψn
i +2 j +1=n i , j ∈N0
∞ n (a)i (b)i θ n +1 dψ = (−1)i (c )i (n + 1)! i n =0
− 1)−1 ,
(a)i (b)i (−1) j dψ (c )i i !(2 j )!
n =0
i +2 j =n i , j ∈N0
θ
log(1+θ )
∞ n n θ n +1 (a)i (b)i (a)i (b)i (−1)i (−1)i , dψ = (c )i (n + 1)! (c )i i i ∞
ψn n!
ψn
0 n =0
(a)n (b)n ψ n (−1)n × ψ 2n+1 dψ = (c )n n! (2n + 1)!
θ ∞ 0 n =0
θ ∞
9 10 θ((1 +θ) 10 9
(a)i (b)i (−1) j dψ (c )i i !(2 j + 1)!
i +2 j +1=n i , 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 shapes of superspirals with their curvatures are shown in Figs. 1–4, and an example of fillet modeling is shown in Fig. 5.
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Fig. 3. (a) An example of the superspiral with a = 2, b = 1, c = 2, and θ ∈ [−1, 10π ]. (b) Its curvature function, 2 F 1 (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 F 1 (2, 2; 2; −θ)−1 = (θ − 1)2 , for θ 0.
Fig. 5. (a) An example of fillet modeling: A G 1 transition superspiral between two straight lines is generated by the Yoshida–Saito method (Yoshida and Saito, 2006) and swept along the z-axis to obtain a transition surface between the two planes. (b) The same surface with generated zebra lines.
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Fig. 6. (a) The superspiraloid obtained by rotating about the y-axis with a = 1.1, b = 2, c = 2, 0 θ 32π . (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 rotating about the y-axis with a = 1.1, b = 2, c = 2, π2 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 about the x-axis with a = 1.1, b = 2, c = 2, 0 θ π2 , 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 following definition refers to these surfaces. Definition 2. A superspiraloid is a surface generated by rotating a two-dimensional superspiral curve segment about an axis.
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6. Conclusions and future work We have introduced analytic parametric equations for superspirals, whose radius of curvature is given by Gauss hypergeometric functions, which are completely monotonic under described conditions. Whereas previous authors deal with the specific curves having linear curvature graphs (Yoshida and Saito, 2006), the superspirals can cover a huge variety of fair curves. There are several directions for future work. It is possible to generalize the radius of curvature function and present it in terms of the generalized hypergeometric function, p F q (a1 , . . . , a p ; 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 introduced in Fox (1961), which is a generalization of the Meijer G-function. But, in such an approach, monotonicity conditions would be somehow much more complex or even not discovered. Proposed superspirals can logically be applied for generating nonlinear splines (Mehlum, 1974) using the Yoshida–Saito method for two-point G 1 Hermite interpolation (Yoshida and Saito, 2006), as well as for constructing non-linear spline with curvature continuity (which is actually G 2 multispiral; Ziatdinov et al., 2012a), the generating algorithm for which would be the scope of our next works. Finally, we will like to extend this approach 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 laboratory of whom I started my research on spirals. The authors appreciate the issues, remarks, and very important suggestions of the anonymous reviewers which helped to improve the quality of this paper. References Abramowitz, M., Stegun, I.A., 1965. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York. Baykal, O., Tari, E., Coskun, Z., Sahin, M., 1997. New transition curve joining two straight lines. Journal of Transportation Engineering 123 (5), 337–347. Becken, W., Schmelcher, P., 2000. The analytic continuation of the Gaussian hypergeometric function 2 F 1 (a, b; c ; z) for arbitrary parameters. Journal of Computational and Applied Mathematics 126 (1–2), 449–478. Chandrasekhar, S., 1998. The Mathematical Theory of Black Holes. Oxford Classic Texts in the Physical Sciences. Oxford University Press. Cook, T., 1903. Spirals in nature and art. Nature 68 (1761), 296. Dankwort, C.W., Podehl, G., 2000. A new aesthetic design workflow: results from the European project FIORES. In: CAD Tools and Algorithms for Product Design. Springer-Verlag, Berlin, Germany, pp. 16–30. Dimulyo, S., Habib, Z., Sakai, M., 2009. Fair cubic transition between two circles with one circle inside or tangent to the other. Numerical Algorithms 51, 461–476. Farin, G., 2006. Class A Bézier curves. Computer Aided Geometric Design 23 (7), 573–581. Farin, G., Hoschek, J., Kim, M., 2002. Handbook of Computer Aided Geometric Design. Elsevier. Farouki, R.T., 1997. Pythagorean-hodograph quintic transition curves of monotone curvature. Computer Aided Design 29 (9), 601–606. Fox, C., 1961. The G and H functions as symmetrical Fourier kernels. Transactions of the American Mathematical Society 98 (3), 395–429. Frey, W.H., Field, D.A., 2000. Designing Bézier conic segments with monotone curvature. Computer Aided Geometric Design 17 (6), 457–483. Gobithaasan, R., Miura, K., 2011. Aesthetic spiral for design. Sains Malaysiana 40 (11), 1301–1305. Gobithaasan, R.U., Yee, L.P., Miura, K.T., 2012. A generalized log aesthetic space curve. In: Proceedings of the 2012 Joint International Conference on HumanCentered Computer Environments, HCCE’12. ACM, New York, NY, USA, pp. 145–149. Gradshtein, I., Ryzhik, I., 1962. Tables of Integrals, Summations, Series and Derivatives, vol. 1, 4th edition. GIFML, Moscow. Habib, Z., Sakai, M., 2003. G2 cubic transition between two circles with shape control. International Journal of Computer Mathematics 80 (8), 959–967. Habib, Z., Sakai, M., 2004. Simplified and flexible spiral transitions for use in computer graphics and geometric modelling. In: Proceedings of the Third International Conference on Image and Graphics, pp. 426–429. Habib, Z., Sakai, M., 2005a. Family of G2 spiral transition between two circles. In: Advances in Geometric Design. John Wiley & Sons, Ltd., pp. 133–151. Habib, Z., Sakai, M., 2005b. G2 PH quintic spiral transition curves and their applications. Scientiae Mathematicae Japonicae 61 (2), 207–217. Habib, Z., Sakai, M., 2005c. Spiral transition curves and their applications. Scientiae Mathematicae Japonicae 61 (2), 195–206. Habib, Z., Sakai, M., 2006. An answer to an open problem on cubic spiral transition between two circles. In: Computer Algebra – Design of Algorithms, pp. 46–52. Habib, Z., Sakai, M., 2007. On PH quintic spirals joining two circles with one circle inside the other. Computer Aided Design 39 (2), 125–132. Habib, Z., Sakai, M., 2009. G2 cubic transition between two circles with shape control. Journal of Computational and Applied Mathematics 223 (1), 133–144. Harada, T., Yoshimoto, F., Moriyama, M., 1999. An aesthetic curve in the field of industrial design. In: IEEE Symposium on Visual Languages. Institute of Electrical Electronics Engineering, Tokyo, Japan, pp. 38–47. Harary, G., Tal, A., 2011. The natural 3D spiral. Computer Graphics Forum 30 (2), 237–246. Kronrod, A., 1964. Integration with control of accuracy. Doklady Akademii Nauk SSSR 154, 283–286 (in Russian). Kurnosenko, A., 2009. General properties of spiral plane curves. Journal of Mathematical Sciences 161, 405–418. Kurnosenko, A., 2010a. Applying inversion to construct planar, rational spirals that satisfy two-point G2 Hermite data. Computer Aided Geometric Design 27 (3), 262–280. Kurnosenko, A., 2010b. Two-point G2 Hermite interpolation with spirals by inversion of hyperbola. Computer Aided Geometric Design 27 (6), 474–481. Laurie, D.P., 1997. Calculation of Gauss–Kronrod quadrature rules. Mathematics of Computation 66 (219), 1133–1145. Lebedev, N., 1965. Special Functions and Their Applications. Prentice Hall, Inc., Englewood Cliffs, NJ. Revised English edition, translated and edited by Richard A. Silverman. Levien, R., Séquin, C., 2009. Interpolating splines: which is the fairest of them all? Computer Aided Design and Applications 4, 91–102. Li, Z., Ma, L., Meek, D., Tan, W., Mao, Z., Zhao, M., 2006. Curvature monotony condition for rational quadratic B-spline curves. In: Gavrilova, M., Gervasi, O., Kumar, V., Tan, C., Taniar, D., Lagana, A., Mun, Y., Choo, H. (Eds.), Computational Science and Its Applications – ICCSA 2006. In: Lecture Notes in Computer Science, vol. 3980. Springer-Verlag, Berlin, Heidelberg, pp. 1118–1126.
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Meek, D., Walton, D., 1989. The use of Cornu spirals in drawing planar curves of controlled curvature. Journal of Computational and Applied Mathematics 25 (1), 69–78. Mehlum, E., 1974. Nonlinear splines. In: Barnhill, R., Riesenfeld, R. (Eds.), Computer Aided Geometric Design. Academic Press, New York, pp. 173–207. Meijer, C.S., 1936. Über Whittakersche bezw. Besselsche Funktionen und deren Produkte. Nieuw Archief voor Wiskunde 18 (4), 10–39. Miller, K.S., Samko, S.G., 2001. Completely monotonic functions. Integral Transforms and Special Functions 12 (4), 389–402. Miura, K.T., 2006. A general equation of aesthetic curves and its self-affinity. Computer Aided Design and Applications 3 (1–4), 457–464. Miura, K., Sone, J., Yamashita, A., Kaneko, T., 2005. Derivation of a general formula of aesthetic curves. In: 8th International Conference on Humans and Computers (HC2005). Aizu-Wakamutsu, Japan, pp. 166–171. Miura, K.T., Usuki, S., Gobithaasan, R.U., 2012. Variational formulation of the log-aesthetic curve. In: Proceedings of the 2012 Joint International Conference on Human-Centered Computer Environments, HCCE’12. ACM, New York, NY, USA, pp. 215–219. Pogorelov, A., 1974. Differential Geometry. Nauka, Moscow, USSR. Sapidis, N., 1994. Designing Fair Curves and Surfaces: Shape Quality in Geometric Modeling and Computer-Aided Design. Society for Industrial and Applied Mathematics. Geometric design publications. Struik, D.J., 1988. Lectures on Classical Differential Geometry, 2nd edition. Dover, New York. Walton, D., Meek, D., 1999. Planar G2 transition between two circles with a fair cubic Bézier curve. Computer Aided Design 31 (14), 857–866. Walton, D.J., Meek, D.S., 2002. Planar G2 transition with a fair pythagorean hodograph quintic curve. Journal of Computational and Applied Mathematics 138 (1), 109–126. Walton, D.J., Meek, D.S., Ali, J.M., 2003. Planar G2 transition curves composed of cubic Bézier spiral segments. Journal of Computational and Applied Mathematics 157 (2), 453–476. Wang, Y., Zhao, B., Zhang, L., Xu, J., Wang, K., Wang, S., 2004. Designing fair curves using monotone curvature pieces. Computer Aided Geometric Design 21 (5), 515–527. Whewell, W., 1849. Of the intrinsic equation of a curve, and its application. Cambridge Philosophical Transactions 8, 659–671. Yamada, A., Shimada, K., Furuhata, T., Hou, K.-H., 1999. A discrete spring model for generating fair curves and surfaces. In: Seventh Pacific Conference on Computer Graphics and Applications, pp. 270–279. Yates, R., 1952. Intrinsic equations. In: A Handbook on Curves and Their Properties. J.W. Edwards, Ann Arbor, MI, pp. 123–126. Yoshida, M., 1997. Hypergeometric Functions, My Love. Vieweg Verlag, Leipzig. Yoshida, N., Saito, T., 2006. Interactive aesthetic curve segments. The Visual Computer 22 (9), 896–905. Yoshida, N., Fukuda, R., Saito, T., 2009. Log-aesthetic space curve segments. In: 2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling, SPM’09. ACM, New York, NY, USA, pp. 35–46. Yoshida, N., Fukuda, R., Saito, T., 2010. Logarithmic curvature and torsion graphs. In: Dahlen, M., Floater, M., Lyche, T., Merrien, J.-L., Morken, K., Schumaker, L. (Eds.), Mathematical Methods for Curves and Surfaces. In: Lecture Notes in Computer Science, vol. 5862. Springer-Verlag, Berlin, Heidelberg, pp. 434– 443. Ziatdinov, R., Yoshida, N., Kim, T., 2012a. Fitting G 2 multispiral transition curve joining two straight lines. Computer Aided Design 44 (6), 591–596. Ziatdinov, R., Yoshida, N., Kim, T., 2012b. Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions. Computer Aided Geometric Design 29 (2), 129–140.
Contents lists available at SciVerse ScienceDirect
Computer Aided Geometric Design www.elsevier.com/locate/cagd
Family of superspirals with completely monotonic curvature given in 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
Article history: Available online 28 March 2012 Keywords: Spiral Log-aesthetic curve Superspiral Monotone curvature Fair curve Surface of revolution Superspiraloid
a b s t r a c t We present superspirals, a new and very general family of fair curves, whose radius of curvature is given in terms of a completely monotonic Gauss hypergeometric function. The superspirals are generalizations of log-aesthetic curves, as well as other curves whose radius of curvature is a particular case of a completely monotonic Gauss hypergeometric function. High-accuracy computation of a superspiral segment is performed by the Gauss– Kronrod integration method. The proposed curves, despite their complexity, are the candidates for generating G 2 , and G 3 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 that are 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 remains insufficiently defined (Levien and Séquin, 2009). The different mathematical definitions of fairness and aesthetic aspects of geometric modeling are briefly described by Sapidis (1994). The curves of monotone curvature were studied in recent works. Frey and Field (2000) analyzed the curvature distributions of segments of conic sections represented as rational quadratic Bézier curves in standard form. Farouki (1997) has used the Pythagorean-hodograph quintic curve as the monotone-curvature transition between a line and a circle. The monotone-curvature condition for rational quadratic B-spline curves is studied by Li et al. (2006). The use of Cornu spirals in drawing planar curves of controlled curvature was discussed by Meek and Walton (1989). The log-aesthetic curves (LACs), which are high-quality curves with linear logarithmic curvature graphs (Yoshida et al., 2010), have recently been developed to meet the requirements of industrial design for visually 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 Miura et al. (2012). A planar spiral called generalized log-aesthetic curve segment (GLAC) (Gobithaasan and Miura, 2011) has been proposed 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: [email protected], [email protected]. 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
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three-dimensional case by Gobithaasan et al. (2012). According to the author of this work, a series of interesting works of Alexei 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 components of natural world objects: horns, seashells, bones, leaves, flowers, and tree trunks (Cook, 1903; Harary and Tal, 2011). In addition, they are used as a transition curves in rail-road and highway design (Walton and Meek, 1999, 2002; Habib and Sakai, 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 hypergeometric 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 still remain 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 G 2 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 of analytic functions, despite its representation using infinite series.
• Since superspirals have no inflection points in non-polynomial cases, it cannot be considered as a G 2 transition between a 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 hypergeometric functions and describe the constraints under which the radius of curvature function, defined in terms of the Gauss hypergeometric function, becomes completely monotonic and can be associated with fair curves. In Section 3, we propose the general equations of the superspirals and discuss their properties providing several examples on their shapes. In Section 5, we give some graphical examples of superspiraloids, which are actually the surfaces of revolution plotted in CAS Mathematica 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 C4 as 2 F 1 (a, b ; c ; z) =
∞ (a)n (b)n zn , (c )n n!
(1)
n =0
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 cut along [1, ∞) can be written as a contour integral, also known as the Barnes integral
(c ) 1 2 F 1 (a, b ; c ; z) = (a)(b) 2π i where
i ∞ −i ∞
(a + s)(b + s)(−s) (− z)s ds, (c + s)
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∞
t z−1 e −t dt
( z) = 0
is a gamma function (Abramowitz and Stegun, 1965; Lebedev, 1965). A complete table of analytic continuation formulas for the Gauss hypergeometric function, which allow its fast and accurate computation for arbitrary values of z and of the parameters a, b, c can be found in Becken and Schmelcher (2000). There are several specific values of the Gauss hypergeometric function in which we are interested: 2 F 1 (a, b ; c ; 0)
= 1, (c )(c − a − b) , 2 F 1 (a, b ; c ; 1) = (c − a)(c − b)
Re(c − a − b) > 0.
Besides the Gauss hypergeometric function, the function, which is called a confluent hypergeometric function (Kummer’s function), plays an important role in special functions theory:
Φ(a, b, z) =
∞ (n) n a z n =0
b(n)n!
= 1 F 1 (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, logarithmic, 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 analysis and its applications. For more exhaustive information on hypergeometric functions and their properties, the reader is referred 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 F 1 (a, b; c ; −θ) is completely monotonic for c > b > 0, a > 0 when 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) =
α = 1,
e λθ , 1
((α − 1)λ θ + 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) =
α = 1, Φ(α , α , λθ), 1 F ( , b ; b ; ( 1 − α )λθ), otherwise. 2 1 1 −α
It means that LACs, which include well-known spirals as Euler, logarithmic, and Nielsen’s spiral and involutes of a circle is the 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 and Séquin, 2009; Wang et al., 2004), and they, as well as class A Bézier curves (Farin, 2006) are very significant in computeraided design and aesthetic shape modeling (Dankwort and Podehl, 2000). We present the following new definition in this work. Definition 1. A superspiral is a planar curve with a completely monotone radius of curvature given in the form 2 F 1 (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
=
F (a, b; c ; −ψ) cos ψ 0 2 1
dψ
F (a, b; c ; −ψ) sin ψ 0 2 1
dψ
θ
ρ (ψ) =
,
(2)
where 0 θ < +∞. It is important to note that the integrals in Eq. (2) cannot be represented in analytical form except their representation in terms of infinite series, which will be discussed a little later, thus we will apply the adaptive Gauss–Kronrod integration
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(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):
d y (θ)
=
dx(θ)
d2 y (θ)
d y (θ) dx(θ)
=
dx2 (θ)
dθ
dθ
= tan θ,
1 + tan2 θ 2 F 1 (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 (Abramowitz and Stegun, 1965; Gradshtein and Ryzhik, 1962) we obtain
θ s=
θ
ρ (ψ) dψ = 0
2 F 1 (a, b ; c ; −ψ) dψ 0
=−
c−1 2 F 1 (a − 1, b − 1; c − 1; −θ) − 1 . (a − 1)(b − 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 F 1 (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 the origin 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 consider asymptotic approximations of Eq. (2). Taking into account the small-angle approximations,
cos ψ =
∞ (−1)n n =0
sin ψ =
(2n)!
ψ 2n ≈ 1 −
ψ2 2
(4)
,
∞ (−1)n ψ 2n+1 ≈ ψ, (2n + 1)!
(5)
n =0
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) 2a2 b2 − 10a2 b + 12a2 − 10ab2 + 2 c 2 − 5c + 6 2 F 1 (a − 3, b − 3; c − 3; −θ) + 2(a − 3)(b − 3)(c − 2)θ 2 F 1 (a − 2, b − 2; c − 2; −θ) + 50ab − 60a + 12b2 − 60b − 2c 2 + 10c + 60 +
1 2(a − 3)(a − 2)(a − 1)(b − 3)(b − 2)(b − 1)
× (c − 1) a2 b2 θ 2 − 2a2 b2 − 5a2 bθ 2 + 10a2 b + 6a2 θ 2 − 12a2 − 5ab2 θ 2 + 10ab2 + 25abθ 2 − 50ab − 30aθ 2 + 60a + 6b2 θ 2 − 12b2 − 30bθ 2 + 60b + 36θ 2 − 72 2 F 1 (a − 1, b − 1; c − 1; −θ) , (c 2 − 3c + 2) (a − 1) 2 F 1 (a − 2, b − 2; c − 2; −θ) (a − 2)(a − 1)(b − 2)(b − 1) − (a − 2) 2 F 1 (a − 1, b − 2; c − 2; −θ) − 1 .
y (θ) = −
The derived parametric equations do not contain integrals of special functions, and are actually simpler from a computation point 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), and operating with sums one can obtain1
1
The possibility to write in this form has been noted by one of reviewers.
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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 F 1 (0.1, 1; 2; −θ)−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 F 1 (1, 1; 2; −θ)−1 =
x(θ) =
θ ∞ 0 n =0
=
θ ∞ 0 n =0
=
n =0
θ ∞ 0 n =0
y (θ) =
∞
(a)n (b)n ψ n (−1)n 2n × ψ dψ = (c )n n! (2n)! ψn n!
n =0
i +2 j +1=n i , j ∈N0
, for θ 0.
i +2 j =n i , j ∈N0
θ ∞ 0 n =0
(6)
i +2 j =n i , j ∈N0
ψn
i +2 j +1=n i , j ∈N0
∞ n (a)i (b)i θ n +1 dψ = (−1)i (c )i (n + 1)! i n =0
− 1)−1 ,
(a)i (b)i (−1) j dψ (c )i i !(2 j )!
n =0
i +2 j =n i , j ∈N0
θ
log(1+θ )
∞ n n θ n +1 (a)i (b)i (a)i (b)i (−1)i (−1)i , dψ = (c )i (n + 1)! (c )i i i ∞
ψn n!
ψn
0 n =0
(a)n (b)n ψ n (−1)n × ψ 2n+1 dψ = (c )n n! (2n + 1)!
θ ∞ 0 n =0
θ ∞
9 10 θ((1 +θ) 10 9
(a)i (b)i (−1) j dψ (c )i i !(2 j + 1)!
i +2 j +1=n i , 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 shapes of superspirals with their curvatures are shown in Figs. 1–4, and an example of fillet modeling is shown in Fig. 5.
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Fig. 3. (a) An example of the superspiral with a = 2, b = 1, c = 2, and θ ∈ [−1, 10π ]. (b) Its curvature function, 2 F 1 (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 F 1 (2, 2; 2; −θ)−1 = (θ − 1)2 , for θ 0.
Fig. 5. (a) An example of fillet modeling: A G 1 transition superspiral between two straight lines is generated by the Yoshida–Saito method (Yoshida and Saito, 2006) and swept along the z-axis to obtain a transition surface between the two planes. (b) The same surface with generated zebra lines.
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Fig. 6. (a) The superspiraloid obtained by rotating about the y-axis with a = 1.1, b = 2, c = 2, 0 θ 32π . (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 rotating about the y-axis with a = 1.1, b = 2, c = 2, π2 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 about the x-axis with a = 1.1, b = 2, c = 2, 0 θ π2 , 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 following definition refers to these surfaces. Definition 2. A superspiraloid is a surface generated by rotating a two-dimensional superspiral curve segment about an axis.
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6. Conclusions and future work We have introduced analytic parametric equations for superspirals, whose radius of curvature is given by Gauss hypergeometric functions, which are completely monotonic under described conditions. Whereas previous authors deal with the specific curves having linear curvature graphs (Yoshida and Saito, 2006), the superspirals can cover a huge variety of fair curves. There are several directions for future work. It is possible to generalize the radius of curvature function and present it in terms of the generalized hypergeometric function, p F q (a1 , . . . , a p ; 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 introduced in Fox (1961), which is a generalization of the Meijer G-function. But, in such an approach, monotonicity conditions would be somehow much more complex or even not discovered. Proposed superspirals can logically be applied for generating nonlinear splines (Mehlum, 1974) using the Yoshida–Saito method for two-point G 1 Hermite interpolation (Yoshida and Saito, 2006), as well as for constructing non-linear spline with curvature continuity (which is actually G 2 multispiral; Ziatdinov et al., 2012a), the generating algorithm for which would be the scope of our next works. Finally, we will like to extend this approach 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 laboratory of whom I started my research on spirals. The authors appreciate the issues, remarks, and very important suggestions of the anonymous reviewers which helped to improve the quality of this paper. References Abramowitz, M., Stegun, I.A., 1965. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York. Baykal, O., Tari, E., Coskun, Z., Sahin, M., 1997. New transition curve joining two straight lines. Journal of Transportation Engineering 123 (5), 337–347. Becken, W., Schmelcher, P., 2000. The analytic continuation of the Gaussian hypergeometric function 2 F 1 (a, b; c ; z) for arbitrary parameters. Journal of Computational and Applied Mathematics 126 (1–2), 449–478. Chandrasekhar, S., 1998. The Mathematical Theory of Black Holes. Oxford Classic Texts in the Physical Sciences. Oxford University Press. Cook, T., 1903. Spirals in nature and art. Nature 68 (1761), 296. Dankwort, C.W., Podehl, G., 2000. A new aesthetic design workflow: results from the European project FIORES. In: CAD Tools and Algorithms for Product Design. Springer-Verlag, Berlin, Germany, pp. 16–30. Dimulyo, S., Habib, Z., Sakai, M., 2009. Fair cubic transition between two circles with one circle inside or tangent to the other. Numerical Algorithms 51, 461–476. Farin, G., 2006. Class A Bézier curves. Computer Aided Geometric Design 23 (7), 573–581. Farin, G., Hoschek, J., Kim, M., 2002. Handbook of Computer Aided Geometric Design. Elsevier. Farouki, R.T., 1997. Pythagorean-hodograph quintic transition curves of monotone curvature. Computer Aided Design 29 (9), 601–606. Fox, C., 1961. The G and H functions as symmetrical Fourier kernels. Transactions of the American Mathematical Society 98 (3), 395–429. Frey, W.H., Field, D.A., 2000. Designing Bézier conic segments with monotone curvature. Computer Aided Geometric Design 17 (6), 457–483. Gobithaasan, R., Miura, K., 2011. Aesthetic spiral for design. Sains Malaysiana 40 (11), 1301–1305. Gobithaasan, R.U., Yee, L.P., Miura, K.T., 2012. A generalized log aesthetic space curve. In: Proceedings of the 2012 Joint International Conference on HumanCentered Computer Environments, HCCE’12. ACM, New York, NY, USA, pp. 145–149. Gradshtein, I., Ryzhik, I., 1962. Tables of Integrals, Summations, Series and Derivatives, vol. 1, 4th edition. GIFML, Moscow. Habib, Z., Sakai, M., 2003. G2 cubic transition between two circles with shape control. International Journal of Computer Mathematics 80 (8), 959–967. Habib, Z., Sakai, M., 2004. Simplified and flexible spiral transitions for use in computer graphics and geometric modelling. In: Proceedings of the Third International Conference on Image and Graphics, pp. 426–429. Habib, Z., Sakai, M., 2005a. Family of G2 spiral transition between two circles. In: Advances in Geometric Design. John Wiley & Sons, Ltd., pp. 133–151. Habib, Z., Sakai, M., 2005b. G2 PH quintic spiral transition curves and their applications. Scientiae Mathematicae Japonicae 61 (2), 207–217. Habib, Z., Sakai, M., 2005c. Spiral transition curves and their applications. Scientiae Mathematicae Japonicae 61 (2), 195–206. Habib, Z., Sakai, M., 2006. An answer to an open problem on cubic spiral transition between two circles. In: Computer Algebra – Design of Algorithms, pp. 46–52. Habib, Z., Sakai, M., 2007. On PH quintic spirals joining two circles with one circle inside the other. Computer Aided Design 39 (2), 125–132. Habib, Z., Sakai, M., 2009. G2 cubic transition between two circles with shape control. Journal of Computational and Applied Mathematics 223 (1), 133–144. Harada, T., Yoshimoto, F., Moriyama, M., 1999. An aesthetic curve in the field of industrial design. In: IEEE Symposium on Visual Languages. Institute of Electrical Electronics Engineering, Tokyo, Japan, pp. 38–47. Harary, G., Tal, A., 2011. The natural 3D spiral. Computer Graphics Forum 30 (2), 237–246. Kronrod, A., 1964. Integration with control of accuracy. Doklady Akademii Nauk SSSR 154, 283–286 (in Russian). Kurnosenko, A., 2009. General properties of spiral plane curves. Journal of Mathematical Sciences 161, 405–418. Kurnosenko, A., 2010a. Applying inversion to construct planar, rational spirals that satisfy two-point G2 Hermite data. Computer Aided Geometric Design 27 (3), 262–280. Kurnosenko, A., 2010b. Two-point G2 Hermite interpolation with spirals by inversion of hyperbola. Computer Aided Geometric Design 27 (6), 474–481. Laurie, D.P., 1997. Calculation of Gauss–Kronrod quadrature rules. Mathematics of Computation 66 (219), 1133–1145. Lebedev, N., 1965. Special Functions and Their Applications. Prentice Hall, Inc., Englewood Cliffs, NJ. Revised English edition, translated and edited by Richard A. Silverman. Levien, R., Séquin, C., 2009. Interpolating splines: which is the fairest of them all? Computer Aided Design and Applications 4, 91–102. Li, Z., Ma, L., Meek, D., Tan, W., Mao, Z., Zhao, M., 2006. Curvature monotony condition for rational quadratic B-spline curves. In: Gavrilova, M., Gervasi, O., Kumar, V., Tan, C., Taniar, D., Lagana, A., Mun, Y., Choo, H. (Eds.), Computational Science and Its Applications – ICCSA 2006. In: Lecture Notes in Computer Science, vol. 3980. Springer-Verlag, Berlin, Heidelberg, pp. 1118–1126.
518
R. Ziatdinov / Computer Aided Geometric Design 29 (2012) 510–518
Meek, D., Walton, D., 1989. The use of Cornu spirals in drawing planar curves of controlled curvature. Journal of Computational and Applied Mathematics 25 (1), 69–78. Mehlum, E., 1974. Nonlinear splines. In: Barnhill, R., Riesenfeld, R. (Eds.), Computer Aided Geometric Design. Academic Press, New York, pp. 173–207. Meijer, C.S., 1936. Über Whittakersche bezw. Besselsche Funktionen und deren Produkte. Nieuw Archief voor Wiskunde 18 (4), 10–39. Miller, K.S., Samko, S.G., 2001. Completely monotonic functions. Integral Transforms and Special Functions 12 (4), 389–402. Miura, K.T., 2006. A general equation of aesthetic curves and its self-affinity. Computer Aided Design and Applications 3 (1–4), 457–464. Miura, K., Sone, J., Yamashita, A., Kaneko, T., 2005. Derivation of a general formula of aesthetic curves. In: 8th International Conference on Humans and Computers (HC2005). Aizu-Wakamutsu, Japan, pp. 166–171. Miura, K.T., Usuki, S., Gobithaasan, R.U., 2012. Variational formulation of the log-aesthetic curve. In: Proceedings of the 2012 Joint International Conference on Human-Centered Computer Environments, HCCE’12. ACM, New York, NY, USA, pp. 215–219. Pogorelov, A., 1974. Differential Geometry. Nauka, Moscow, USSR. Sapidis, N., 1994. Designing Fair Curves and Surfaces: Shape Quality in Geometric Modeling and Computer-Aided Design. Society for Industrial and Applied Mathematics. Geometric design publications. Struik, D.J., 1988. Lectures on Classical Differential Geometry, 2nd edition. Dover, New York. Walton, D., Meek, D., 1999. Planar G2 transition between two circles with a fair cubic Bézier curve. Computer Aided Design 31 (14), 857–866. Walton, D.J., Meek, D.S., 2002. Planar G2 transition with a fair pythagorean hodograph quintic curve. Journal of Computational and Applied Mathematics 138 (1), 109–126. Walton, D.J., Meek, D.S., Ali, J.M., 2003. Planar G2 transition curves composed of cubic Bézier spiral segments. Journal of Computational and Applied Mathematics 157 (2), 453–476. Wang, Y., Zhao, B., Zhang, L., Xu, J., Wang, K., Wang, S., 2004. Designing fair curves using monotone curvature pieces. Computer Aided Geometric Design 21 (5), 515–527. Whewell, W., 1849. Of the intrinsic equation of a curve, and its application. Cambridge Philosophical Transactions 8, 659–671. Yamada, A., Shimada, K., Furuhata, T., Hou, K.-H., 1999. A discrete spring model for generating fair curves and surfaces. In: Seventh Pacific Conference on Computer Graphics and Applications, pp. 270–279. Yates, R., 1952. Intrinsic equations. In: A Handbook on Curves and Their Properties. J.W. Edwards, Ann Arbor, MI, pp. 123–126. Yoshida, M., 1997. Hypergeometric Functions, My Love. Vieweg Verlag, Leipzig. Yoshida, N., Saito, T., 2006. Interactive aesthetic curve segments. The Visual Computer 22 (9), 896–905. Yoshida, N., Fukuda, R., Saito, T., 2009. Log-aesthetic space curve segments. In: 2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling, SPM’09. ACM, New York, NY, USA, pp. 35–46. Yoshida, N., Fukuda, R., Saito, T., 2010. Logarithmic curvature and torsion graphs. In: Dahlen, M., Floater, M., Lyche, T., Merrien, J.-L., Morken, K., Schumaker, L. (Eds.), Mathematical Methods for Curves and Surfaces. In: Lecture Notes in Computer Science, vol. 5862. Springer-Verlag, Berlin, Heidelberg, pp. 434– 443. Ziatdinov, R., Yoshida, N., Kim, T., 2012a. Fitting G 2 multispiral transition curve joining two straight lines. Computer Aided Design 44 (6), 591–596. Ziatdinov, R., Yoshida, N., Kim, T., 2012b. Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions. Computer Aided Geometric Design 29 (2), 129–140.