Discretization of differential geometric structures of surfaces with singularities

2023 Fiscal Year Final Research Report

• Project/Area Number: 17K14197
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Hiroshima Institute of Technology
• Principal Investigator: Naokawa Kosuke 広島工業大学, 情報学部, 准教授 (60740826)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Keywords: 可展面 / 離散曲面 / メビウスの帯 / 特異点 / カスプ辺 / 交叉帽子 / 等長実現 / 折り紙

Outline of Final Research Achievements

The purpose of this research project is to clarify the differential geometrical properties of smooth surfaces with singularities, and to formulate and study the properties of their discrete objects. In particular, the following results are obtained;
(I)(1) constructions of discrete developable and geodesic Mobius strips with arbitrarily given knot types and twisting numbers, (2) the formulation and study of properties of singularities of cuspidal edge type and swallowtail type, appearing on discrete developable surfaces,
(II)(1) the problem of isometric realizations of a given Kossowski metric, (2) classifications of isomers of cuspidal edges up to congruence, and reserch on origami curved foldings, and (3) isometric realizations of cross cap singularities by formal power series.

Free Research Field

微分幾何学

Academic Significance and Societal Importance of the Research Achievements

近年，曲線や曲面の微分幾何学的な性質に着目し，その離散的対応物を定式化し研究する分野である「離散微分幾何学」の研究が盛んになりつつある．微分幾何学的に重要と考えられる性質に着目して離散化しているため，数値計算における単なる近似に比べ幾何構造を保つと考えられ，数学の枠を超えて建築やコンピュータグラフィックスを含む工学的，芸術的分野への応用も期待される．本研究では，可展面と特異点についての離散化を出発点として，離散曲面の研究と，その由来となる特異点をもつ曲面の微分幾何学的性質の研究の両面に対して，広く成果を得た．