Variational Formulation of the Log-Aesthetic Surface and Development of Discrete Surface Filters

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.