用网格嵌入展示双曲曲面
Illustrating Hyperbolic Surfaces with Mesh Embeddings
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中文总结 AI 辅助
本文通过将双曲曲面离散化为网格并最小化畸变能量嵌入欧几里得空间，使面积增长等双曲现象可见，并展示了多种插图及其在科研、科普和教育中的应用。
中文摘要 AI 辅助
双曲几何展现出诸如面积快速增长等几何现象，这些现象在欧几里得空间中难以被忠实地可视化，而庞加莱圆盘等标准模型可能会掩盖这些现象。为了使双曲几何生动起来，我们将双曲曲面离散化为网格，并通过最小化畸变能量将其嵌入欧几里得空间，使得嵌入中的边长与双曲平面中的边长相匹配。由此产生的曲面会弯曲和起皱以适应额外的面积，从而使平面模型所隐藏的内容变得可见。我们展示了示例性插图，如嵌入的圆盘、等距条带、发散测地线，以及艺术化的有机风格渲染图。我们讨论了这些模型（作为渲染图和3D打印品）在研究讲座、公众参与、科普推广和教育中的使用。
英文摘要
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.
发表机构
- Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数理科学研究所）
- University of San Francisco（旧金山大学）
机构由 AI 辅助整理，请以论文原文为准。
