Development and Extension of Discrete Integrable Geometry

2019 Fiscal Year Final Research Report

• Project/Area Number: 16H03941
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• 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)
• Co-Investigator(Kenkyū-buntansha): 増田 哲 青山学院大学, 理工学部, 教授 (00335457) ; 太田 泰広 神戸大学, 理学研究科, 教授 (10213745) ; 廣瀬 三平 芝浦工業大学, デザイン工学部, 准教授 (20743230) ; 井ノ口 順一 筑波大学, 数理物質系, 教授 (40309886)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 離散微分幾何 / 可積分系 / クライン幾何 / 曲面・曲線 / 離散正則函数 / ソリトン方程式 / パンルヴェ方程式 / 対数型美的曲線

Outline of Final Research Achievements

Discrete integrable differential geometry and its application have been studied, focusing on the integrable structure behind the discrete geometric objects. We have obtained the results on the discrete surfaces/curves and their deformation theory, discrete holomorphic functions, construction of discrete models of curves/surfaces, and stable and precise numerical method for the surfaces and interfaces. In particular, regarding the discrete surfaces/curves and their deformation theory, we formulated a good framework for the log-aesthetic curves developed in the area of the industrial design by using the Klein geometry and succeeded in generalization. Based on those results, we have proposed a project for JST CREST aiming at the development to the various areas of design, which has been successfully accepted.

Free Research Field

可積分系の理論

Academic Significance and Societal Importance of the Research Achievements

さまざまな離散的な曲面，曲線やその変形，またCGなどで用いられる複素正則函数の離散化を扱う理論的な基礎を確立し，その応用の一つとして設計諸分野で美しくアート性の高い形状の設計を容易にする美的形状の基本要素に関する数学的な理論を構築し，また土壌中の水浸透減少に関する高精度かつ高速な数値モデルを定式化した．