Illustrating Hyperbolic Surfaces with Mesh Embeddings
Abstract
Hyperbolic geometry exhibits geometric phenomena, such as fast area growth, that are difficult to visualize faithfully in Euclidean space, and which standard models like the Poincaré disk can obscure. To bring hyperbolic geometry to life, we embed hyperbolic surfaces in Euclidean space by discretizing the surfaces into meshes, and minimizing a distortion energy so that the edge lengths in the embeddings match those in the hyperbolic plane. The resulting surfaces buckle and ruffle to accommodate the extra area, making visible what flat models hide. We present exemplary illustrations, such as embedded disks, equidistant strips, diverging geodesics, and also artistic organic-like renders. We discuss our use of these models, as renders and 3D prints, in research talks, public engagement, outreach, and education.
Related
- Isometric Embeddings of Asymptotically Hyperbolic Manifolds into Hyperbolic Space
- Non-uniqueness of optimal $1$-embeddings on surfaces up to weak equivalence
- Complex hyperbolic surfaces whose closed geodesics are all simple
- Holonomies of hyperbolic surfaces with one branch point
- Resonance counting and spectral gaps for hyperbolic surfaces via Fourier decay
BibTeX
@misc{lander2026illustrating,
title = {{Illustrating Hyperbolic Surfaces with Mesh Embeddings}},
author = {Fabian Lander and Erik Löffelholtz and Diaaeldin Taha and Steve Trettel and Anna Wienhard},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06766},
eprint = {2609.06766},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06766},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Illustrating Hyperbolic Surfaces with Mesh Embeddings
AU - Fabian Lander
AU - Erik Löffelholtz
AU - Diaaeldin Taha
AU - Steve Trettel
AU - Anna Wienhard
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06766
UR - https://arxiv.org/abs/2609.06766
AB - Hyperbolic geometry exhibits geometric phenomena, such as fast area growth, that are difficult to visualize faithfully in Euclidean space, and which standard models like the Poincaré disk can obscure. To bring hyperbolic geometry to life, we embed hyperbolic surfaces in Euclidean space by discretizing the surfaces into meshes, and minimizing a distortion energy so that the edge lengths in the embeddings match those in the hyperbolic plane. The resulting surfaces buckle and ruffle to accommodate the extra area, making visible what flat models hide. We present exemplary illustrations, such as embedded disks, equidistant strips, diverging geodesics, and also artistic organic-like renders. We discuss our use of these models, as renders and 3D prints, in research talks, public engagement, outreach, and education.
ER -