The log-aesthetic curves include the logarithmic (equiangular) spiral, clothoid, and involute curves. Although most of them are expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them and they are expected to be utilized for practical use of industrial and graphical design. The discrete log-aesthetic filter based on the formulation of the log-aesthetic curve has successfully been introduced not to impose strong constraints on the designer's activity, to let him/her design freely and to embed the properties of the log-aesthetic curves for complicated ones with both increasing and decreasing curvature. In this paper, in order to define the log-aesthetic surface and develop surface filters based on its formulation, at first we reformulate the log-aesthetic curve with variational principle. Then we propose several new functionals to be minimized for free-form surfaces and define the log-aesthetic surface. Furthermore we propose new discrete surface filters based on the log-aesthetic surface formulation.

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The log-aesthetic curves include the logarithmic (equiangular) spiral, clothoid, and involute curves. Although most of them are expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them and they are expected to be utilized for practical use of industrial and graphical design. However, their input method proposed so far by use of three socalled control points can generate only an aesthetic curve segment with monotonic curvature variation and can not create a curve with the curvature-extremal point or the inflection point. Hence at first we propose a technique to input a log-aesthetic curve segment with an inflection point at its end. Then we propose a method to generate an log-aesthetic curve with G 2 continuity from a sequence of 2D points input with, for example, a liquid crystal pen tablet.

It is known that if the degree of the typical plane Bézier curve is increased infinitely, the curve will converge to the logarithmic (equiangular) spiral. The logarithmic spiral is one of log-aesthetic curves and they are formulated by : the slope of the logarithmic curvature graph. In this paper we define the non-stationary typical Bézier curve by making the transition matrix of the typical Bézier curve non-stationary and dependent on each side of the control polyline and defining the transition matrix in the Frenet frame. We propose a method that generates a curve such that if its degree is increased infinitely it will converge to a log-aesthetic curve with arbitrary  and : the slope of the logarithmic torsion graph in case of the space curve, by controlling the relationship between the rotation angle and the scaling factor. Furthermore we extend the non-stationarization for free-form surfaces and propose the non-stationary typical surface with the unit scaling factor.

In the field of industrial design, the aesthetic design is an important element to determine the quality of products 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 aesthetic curves, it is expected that the quality of the curve design improves drastically because we can use it as a standard to generate, evaluate, and deform the curves. The log-aesthetic curve was proposed to generate high-quality curves efficiently [6]. Harada et al. insist 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. Miura et al. defined 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 they derive general formulas of aesthetic curves that describe the relationship between their radiuses of curvature and lengths. Also they defined the self-affinity possessed by the curves satisfying the general equations of aesthetic curves. The proposed curve is called the log-aesthetic curve. Agari et al. [1] proposed a method to input compound-rhythm log-aesthetic curves by use of four control points. Yoshida and Saito [2] proposed a method to generate log-aesthetic curves using three control points by searching for one variable, although Agari’s method searches for two variables. In this paper we compare Agari’s method with Yoshida-Saito’s method, and check out whether both of them generate the same curve. Next, we propose an efficient method to input log-aesthetic curve segments with an inflection point. Furthermore we try to improve the quality and the efficiency of the aesthetic design by plugging an log-aesthetic curve module in a commercial geometric modeler “FullMoon”.