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General Equations of Aesthetic Curves and Their Self-anity

The curve is the most basic design element to determine shapes and silhouettes of industrial products and works for shape designers and it is inevitable for them to make it aesthetic and attractive to improve the total quality of the shape design. If we can find an equation of the aesthetic curves, it is expected that the quality of the curve design improves drastically because we can use them as standards to generate, evaluate, and deform the curves. Harada et al. insisted that natural aesthetic curves like birds’ eggs and butterflies’ wings as well as artificial ones like Japanese swords and key lines of automobiles have such a property that their logarithmic curvature histograms (LCHs) can be approximated by straight lines and there is a strong correlation between the slopes of the lines and the impressions of the curves. In this paper，we define the LCH analytically with the aim of approximating it by a straight line and propose new expressions to represent an aesthetic curve whose LCH is given exactly by a straight line. Furthermore, we derive general equations of aesthetic curves that describe the relationship between their radii of curvature and lengths. Furthermore, we define the self-anity possessed by the curves satisfying the general equations of aesthetic curves.