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Log-Aesthetic Curves in Industrial Design as Similarity Geometric Analogue of Euler’s Elastic Curves

Hold Date
2017-12-21 16:45〜2017-12-21 17:45
Lecture Room M W1-C-512, West Zone 1, Ito campus, Kyushu University
Object person
Kenji Kajiwara (Institute of Mathematics for Industry, Kyushu University)

The class of plane curves called the Euler’s elastic curves is one of the most important geometric objects and serves as a basic model in the elastic theory. It can be characterized (1) as a critical point of the elastic energy where it is given by the square of the Euclidean curvature, (2) as the stationary flow with respect to the isoperimetric deformation of plane curves in the Euclidean geometry governed by the modified KdV equation, which is one of the most typical integrable systems. In this talk, we consider a class of plane curves called the log-aesthetic curves (LAC) and their generalization which is used in the industrial design. We investigate those curves under the similarity geometry and characterize them as stationary integrable flow on plane curves which is governed by the Burgers equation. We introduce "fairing energy" and propose a variational formulation of those curves whose Euler-Lagrange equation yields the stationary Burgers equation. Our result suggests that the LAC and their generalization can be regarded as the similarity geometric analogue of the Euler’s elastic curves, which provides a new mathematical framework of those curves and would yield various generalizations. As an example, we propose a discrete analogue of LAC based on integrable deformation theory of the discrete curves in the similarity geometry, and discuss its discrete variational formulation by introducing a discrete analogue of the fairing energy.

* This talk will be given in English.