A Structure-Preserving Numerical Method for Harmonic Maps Between High-genus Surfaces
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866916928090537984 |
|---|---|
| author | Zhu, Zhipeng Lam, Wai Yeung Lui, Lok Ming |
| author_facet | Zhu, Zhipeng Lam, Wai Yeung Lui, Lok Ming |
| contents | Motivated by geometry processing for surfaces with non-trivial topology, we study discrete harmonic maps between closed surfaces of genus at least two. Harmonic maps provide a natural framework for comparing surfaces by minimizing distortion. Unlike conformal or isometric maps-which may not exist between surfaces with different geometries-harmonic maps always exist within a fixed homotopy class and yield optimal homeomorphisms when the target surface has negative curvature. We develop a structure-preserving algorithm to compute harmonic maps from a triangulated surface to a reference hyperbolic surface. The method minimizes Dirichlet energy over geodesic realizations of the surface graph into the target hyperbolic surface in the homotopy class of a homeomorphism. A central feature of our framework is the use of canonical edge weights derived from the hyperbolic metric, which generalize the classical cotangent weights from the Euclidean setting. These weights preserve injectivity and ensure that isometries remain harmonic in the discrete theory, reflecting their classical behavior. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01256 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Structure-Preserving Numerical Method for Harmonic Maps Between High-genus Surfaces Zhu, Zhipeng Lam, Wai Yeung Lui, Lok Ming Numerical Analysis Optimization and Control Motivated by geometry processing for surfaces with non-trivial topology, we study discrete harmonic maps between closed surfaces of genus at least two. Harmonic maps provide a natural framework for comparing surfaces by minimizing distortion. Unlike conformal or isometric maps-which may not exist between surfaces with different geometries-harmonic maps always exist within a fixed homotopy class and yield optimal homeomorphisms when the target surface has negative curvature. We develop a structure-preserving algorithm to compute harmonic maps from a triangulated surface to a reference hyperbolic surface. The method minimizes Dirichlet energy over geodesic realizations of the surface graph into the target hyperbolic surface in the homotopy class of a homeomorphism. A central feature of our framework is the use of canonical edge weights derived from the hyperbolic metric, which generalize the classical cotangent weights from the Euclidean setting. These weights preserve injectivity and ensure that isometries remain harmonic in the discrete theory, reflecting their classical behavior. |
| title | A Structure-Preserving Numerical Method for Harmonic Maps Between High-genus Surfaces |
| topic | Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/2509.01256 |