PROF. DR. KENJI KAJIWARA

Kenji Kajiwara is a Professor the Institute of Mathematics for Industry  (IMI), Kyushu University, Japan since 2011. He has been taking the role of Director of the IMI since 2022. He was awarded a Ph.D. from The University of Tokyo in 1994, then served as a lecturer and an associate professor in the Department of Electrical Engineering at Doshisha University, Kyoto, Japan. He became an associate professor at the Faculty of Mathematics, Kyushu University, in 2001 and was promoted to professor in 2009. Upon the inauguration of the IMI, he served as a founding member. His mathematical expertise is integrable systems and (discrete) differential geometry. He founded the research activity group “Geometric Shape Generation” in JSIAM and is promoting its activity. Currently, he is leading the interdisciplinary research project “Evolving Design and Discrete Differential Geometry – towards Mathematics Aided Geometric Design,” funded by JST, with researchers in mathematics, architecture, and industrial design. He is contributing to the mathematical community in Japan by serving as a Vice President of Japan SIAM for 2020-2022 and as a Board Member since 2023, together with an associate member of the Science Council of Japan since 2023. At the international level, he has been serving as the Honorary Secretary of the Asia Pacific Consortium of Mathematics for Industry (APCMfI) since 2021 and also an Officer-at-Large of the International Council for Industrial and Applied Mathematics (ICIAM) since 2023.

GENERATION OF AESTHETIC SHAPES BY INTEGRABLE DIFFERENTIAL GEOMETRY

In this talk, we consider a class of plane curves called log-aesthetic curves (LAC) and their generalizations which have been developed in industrial design as the curves obtained by extracting the common properties among thousands of curves that car designers regard as aesthetic.  We consider these curves in the framework of similarity geometry and characterize them as invariant curves under the integrable deformation of plane curves governed by the Burgers equation. We propose a variational principle for these curves, leading to the stationary Burgers equation as the Euler-Lagrange equation.

We then extend the LAC to space curves by considering the integrable deformation of space curves under similarity geometry. The deformation is governed by the coupled system of the modified KdV equation satisfied by the similarity torsion and a linear equation satisfied by the curvature radius. The curves also allow the deformation governed by the coupled system of the sine-Gordon equation and associated linear equation. The space curves corresponding to the travelling wave solutions of those equations would give generalization of LAC to space curves.

We also consider the surface constructed by the family of curves obtained by the integrable deformation of such curves. A special class of surfaces corresponding to the constant similarity torsion yields quadratic surfaces (ellipsoid, one/two-sheeted hyperboloids and paraboloid) and their deformations, which may be regarded as a generalization of LAC to surface.

We discuss the construction of such curves and surfaces together with their mathematical properties, including integration scheme of the frame by symmetries, and present various examples of curves and surfaces.