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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.

MSC:

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
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