# GeodesicEmbedding (GE): A High-Dimensional Embedding Approach for Fast Geodesic Distance Queries

arXiv: https://arxiv.org/abs/2108.13821 · HTML: https://mlsanity.com/paper/2108.13821/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Aug 2021.

## Built on by (most cited first, 3 of 3)

- [Neural Intrinsic Embedding for Non-rigid Point Cloud Matching](https://mlsanity.com/paper/2303.01038/) (2023)
- [Learning the Geodesic Embedding with Graph Neural Networks](https://mlsanity.com/paper/2309.05613/) (2023)
- [Surface-Aware Distilled 3D Semantic Features](https://mlsanity.com/paper/2503.18254/) (2025)

## Abstract (arXiv)

In this paper, we develop a novel method for fast geodesic distance queries. The key idea is to embed the mesh into a high-dimensional space, such that the Euclidean distance in the high-dimensional space can induce the geodesic distance in the original manifold surface. However, directly solving the high-dimensional embedding problem is not feasible due to the large number of variables and the fact that the embedding problem is highly nonlinear. We overcome the challenges with two novel ideas. First, instead of taking all vertices as variables, we embed only the saddle vertices, which greatly reduces the problem complexity. We then compute a local embedding for each non-saddle vertex. Second, to reduce the large approximation error resulting from the purely Euclidean embedding, we propose a cascaded optimization approach that repeatedly introduces additional embedding coordinates with a non-Euclidean function to reduce the approximation residual. Using the precomputation data, our approach can determine the geodesic distance between any two vertices in near-constant time. Computational testing results show that our method is more desirable than previous geodesic distance queries methods.