Development of Theory of Integrable Systems Describing Geometric Shapes

2023 Fiscal Year Final Research Report

• Project/Area Number: 21K03329
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Kyushu University
• Principal Investigator: Kajiwara Kenji 九州大学, マス・フォア・インダストリ研究所, 教授 (40268115)
• Project Period (FY): 2021-04-01 – 2024-03-31
• Keywords: 幾何学的形状生成 / 微分幾何 / 離散微分幾何 / 可積分系の理論 / 建築設計 / 工業意匠設計

Outline of Final Research Achievements

Log-aesthetic curves(LAC) are considered in similarity geometry, characterized as shape-invariant curves with respect to integrable deformations of plane curves and formulated by the variational principle, and their integrable discretization and space curve and surface versions are proposed. The integrable structure is investigated in detail. From the viewpoint of the self-affinity of plane curves, we identified quadratic curves as another family of aesthetic curves, and suggested equi-affine geometry and projective geometry as theoretical frameworks. We also constructed a method for generating Michell-Prager type truss structures of architecture by discrete holomorphic functions. We constructed an algorithm for approximating a given (discrete) plane curve by a LAC. We constructed explicit formulas for shape-invariant curves in Euclidean geometry for integrable deformations of spatial (discrete) curves using the theta functions.

Translated with www.DeepL.com/Translator (free version)

Free Research Field

幾何学的形状生成，離散微分幾何，可積分系

Academic Significance and Societal Importance of the Research Achievements

工業意匠設計や建築に動機を得て，クライン幾何における曲線論，曲面論を活用した幾何学的形状生成と，そこに現れる可積分構造を詳細に調べることで，（離散）微分幾何，可積分系の理論に対する新しい発展の方向性を与えた．同時に，工業意匠設計分野や建築分野に対して数学を活用した新しい形状設計手法をもたらした．このように，産業や社会に関わる問題をドライビングフォースとして，数学の新研究領域を開拓し，元の分野と数学双方を活性化する「マス・フォア・インダストリ」の一つの典型的な例を提示できたことに意義がある．