2025/01/23 by Tim Hoffmann, Hoffmann, Tim, Jannik Steinmeier +3
Computer Science · Engineering · #Computational Geometry and Mesh Generation #Advanced Numerical Analysis Techniques #Digital Image Processing Techniques
paper · pdf · doi:10.48550/arxiv.2501.13477
We propose a notion of discrete elastic and area-constrained elastic curves in 2-dimensional space forms. Our definition extends the well-known discrete Euclidean curvature equation to space forms and reflects various geometric properties known from their smooth counterparts. Special emphasis is paid to discrete flows built from Bäcklund transformations in the respective space forms. The invariants of the flows form a hierarchy of curves and we show that discrete elastic and constrained elastic curves can be characterized as elements of this hierarchy. This work also includes an introductory chapter on discrete curve theory in space forms, where we find discrete Frenet-type formulas and describe an associated family related to a fundamental theorem.