# Superspirals

## Rushan Ziatdinov. (2012). Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function, Computer Aided Geometric Design 29(7), 510–518 [PDF].

## Abstract

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 G2, and G3 non-linear superspiral splines.

- 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 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).

## 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 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 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 G2 transition between a straight line and another curve.

For highly accurate superspiraloid computation, significant time is necessary.

## Please continue reading at ResearchGate

## 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 F1(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 G2 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.