Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions. (English) Zbl 1242.65037

Summary: Log-aesthetic curves (LACs) have recently been developed to meet the requirements of industrial design for visually pleasing shapes. LACs are defined in terms of definite integrals, and adaptive Gaussian quadrature can be used to obtain curve segments. To date, these integrals have only been evaluated analytically for restricted values (0,1,2) of the shape parameter \(\alpha \).
We present parametric equations expressed in terms of incomplete gamma functions, which allow us to find an exact analytic representation of a curve segment for any real value of \(\alpha \). The computation time for generating an LAC segment using the incomplete gamma functions is up to 13 times faster than using direct numerical integration. Our equations are generalizations of the well-known Cornu, Nielsen, and logarithmic spirals, and involutes of a circle.


65D17 Computer-aided design (modeling of curves and surfaces)
33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
33F05 Numerical approximation and evaluation of special functions
65D20 Computation of special functions and constants, construction of tables
Full Text: DOI


[1] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, (1965), Dover New York · Zbl 0515.33001
[2] Amore, P., Asymptotic and exact series representations for the incomplete gamma function, Europhysics letters, 71, 1, 1-7, (2005)
[3] Burchard, H.; Ayers, J.; Frey, W.; Sapidis, N., Designing fair curves and surfaces, (1994), SIAM Philadelphia, USA, pp. 3-28
[4] Dankwort, C.W.; Podehl, G., A new aesthetic design workflow: results from the European project FIORES, (), 16-30
[5] Dyakonov, V., Mathematica 5: programming and mathematical computations, (2008), DMK Press Moscow
[6] Farin, G., Class A Bézier curves, Computer aided geometric design, 23, 7, 573-581, (2006) · Zbl 1101.65016
[7] Farouki, R.T., Pythagorean-hodograph quintic transition curves of monotone curvature, Computer-aided design, 29, 9, 601-606, (1997)
[8] Frey, W.H.; Field, D.A., Designing Bézier conic segments with monotone curvature, Computer aided geometric design, 17, 6, 457-483, (2000) · Zbl 0945.68173
[9] Gradshtein, I.; Ryzhik, I., Tables of integrals, summations, series and derivatives, vol. 1, (1962), GIFML Moscow
[10] Harada, T.; Yoshimoto, F.; Moriyama, M., An aesthetic curve in the field of industrial design, (), 38-47
[11] Inoue, J.; Harada, T.; Hagihara, T., An algorithm for generating log-aesthetic curved surfaces and the development of a curved surfaces generation system using VR, (), 2513-2522
[12] Kronrod, A., Doklady akademii nauk SSSR, 154, 283-286, (1964)
[13] Laurie, D.P., Calculation of Gauss-kronrod quadrature rules, Mathematics of computation, 66, 219, 1133-1145, (1997) · Zbl 0870.65018
[14] Levien, R.; Séquin, C., Interpolating splines: which is the fairest of them all?, Computer-aided design and applications, 4, 91-102, (2009)
[15] Meek, D.; Walton, D., The use of cornu spirals in drawing planar curves of controlled curvature, Journal of computational and applied mathematics, 25, 1, 69-78, (1989) · Zbl 0662.65008
[16] Mehlum, E., Nonlinear splines, (), 173-207
[17] Mineur, Y.; Lichah, T.; Castelain, J.M.; Giaume, H., A shape controlled Fitting method for Bézier curves, Computer aided geometric design, 15, 9, 879-891, (1998) · Zbl 0910.68211
[18] Miura, K.; Sone, J.; Yamashita, A.; Kaneko, T., Derivation of a general formula of aesthetic curves, (), 166-171
[19] Miura, K.T., A general equation of aesthetic curves and its self-affinity, Computer-aided design and applications, 3, 1-4, 457-464, (2006)
[20] Olver, F.W.J., Asymptotics and special functions, (1997), A.K. Peters Wellesley, MA · Zbl 0303.41035
[21] Pogorelov, A., Differential geometry, (1974), Nauka Moscow, USSR
[22] Sapidis, N.S.; Frey, W.H., Controlling the curvature of a quadratic Bézier curve, Computer aided geometric design, 9, 2, 85-91, (1992) · Zbl 0756.65012
[23] Struik, D.J., Lectures on classical differential geometry, (1988), Dover New York · Zbl 0041.48603
[24] Temme, N.M., Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta function, Mathematics of computation, 29, 132, 1109-1114, (1975) · Zbl 0313.33002
[25] Temme, N.M., Special functions: an introduction to the classical functions of mathematical physics, (1996), John Wiley & Sons Inc. New York · Zbl 0863.33002
[26] Temme, N.M., Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters, Methods and applications of analysis, 3, 3, 335-344, (1996) · Zbl 0863.33002
[27] Wang, Y.; Zhao, B.; Zhang, L.; Xu, J.; Wang, K.; Wang, S., Designing fair curves using monotone curvature pieces, Computer aided geometric design, 21, 5, 515-527, (2004) · Zbl 1069.65538
[28] Whewell, W., Of the intrinsic equation of a curve, and its application, Cambridge philosophical transactions, 8, 659-671, (1849)
[29] Xu, L.; Mould, D., Magnetic curves: curvature-controlled aesthetic curves using magnetic fields, ()
[30] Yates, R., Intrinsic equations, (), 123-126
[31] Yoshida, N.; Fukuda, R.; Saito, T., Logarithmic curvature and torsion graphs, (), 434-443 · Zbl 1274.65054
[32] Yoshida, N.; Saito, T., Interactive aesthetic curve segments, The visual computer, 22, 9, 896-905, (2006)
[33] Yoshida, N.; Saito, T., Quasi-aesthetic curves in rational cubic Bézier forms, Computer-aided design and applications, 4, 9-10, 477-486, (2007)
[34] Yoshimoto, F.; Harada, T., Analysis of the characteristics of curves in natural and factory products, (), 276-281
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.