Study of surfaces from the perspective of singularity theory of mappings and its applications

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14312
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Nagasaki University
• Principal Investigator: Kabata Yutaro 長崎大学, 情報データ科学部, 助教 (40830097)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 特異点論 / 曲面論 / 曲線論 / 射影 / 曲面復元問題 / 離散微分幾何学 / 負定曲率曲面 / 特徴線

Outline of Final Research Achievements

During the research period, I conducted [A] Differential geometric studies of curves and surfaces, and [B] Research on projections of curves and surfaces. In particular, the following are notable research achievements:
(1) Regarding the study of 2-solitons of surfaces with negative Gaussian curvature, through collaborative research we obtained classification results on the distribution of singularity-theoretically important feature lines (ridge lines, flecnodal curves), and obtained good graphics that clearly visualize the results.　(2) Through collaborative research, we found a new approach to the surface reconstruction problem for projections of surfaces with special patterns, and wrote a paper on it.　(3) Through collaborative research, we studied discrete surfaces with constant principal curvatures and wrote a paper on the results (already accepted for publication).

Free Research Field

特異点論

Academic Significance and Societal Importance of the Research Achievements

曲面論は古くから研究されているが，現代の特異点論を用いた新しい観点での研究が近年は盛んになっている．本研究もその流れを汲んでおり，特異点論でよく研究されているridgeやflecnodal curveに注目することで負定曲率曲面のわかりやすい分類を得ることができた．また，主曲率一定離散曲面に関する研究は材料科学の問題に端を発しており，新規の物質の性質の説明に寄与できる可能性がある．さらに，模様を持った曲面の射影に関する研究は，コンピュータービジョンで重要な問題である曲面復元問題に新しいアプローチを提供しており，今後さまざまな応用の可能性があると考えている．