Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions

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)(0,1,2) of the shape parameter α.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 α. 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.

Graphical abstractFigure optionsDownload full-size imageDownload high-quality image (290 K)Download as PowerPoint slideHighlights► The equations of log-aesthetic curves are derived using incomplete gamma functions. ► LACs are generalizations of several well-known spirals. ► The computation time for generating an LAC segment became up to 13 times faster. ► Analytic equations allow to compute an LAC segment accurately.