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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 theEuclidean distance in the high-dimensional space can induce the geodesicdistance in the original manifold surface. However, directly solving thehigh-dimensional embedding problem is not feasible due to the large number ofvariables and the fact that the embedding problem is highly nonlinear. Weovercome the challenges with two novel ideas. First, instead of taking allvertices as variables, we embed only the saddle vertices, which greatly reducesthe problem complexity. We then compute a local embedding for each non-saddlevertex. Second, to reduce the large approximation error resulting from thepurely Euclidean embedding, we propose a cascaded optimization approach thatrepeatedly introduces additional embedding coordinates with a non-Euclideanfunction to reduce the approximation residual. Using the precomputation data,our approach can determine the geodesic distance between any two vertices innear-constant time. Computational testing results show that our method is moredesirable than previous geodesic distance queries methods.

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