2,717 research outputs found
NeuroGF: A Neural Representation for Fast Geodesic Distance and Path Queries
Geodesics are essential in many geometry processing applications. However,
traditional algorithms for computing geodesic distances and paths on 3D mesh
models are often inefficient and slow. This makes them impractical for
scenarios that require extensive querying of arbitrary point-to-point
geodesics. Although neural implicit representations have emerged as a popular
way of representing 3D shape geometries, there is still no research on
representing geodesics with deep implicit functions. To bridge this gap, this
paper presents the first attempt to represent geodesics on 3D mesh models using
neural implicit functions. Specifically, we introduce neural geodesic fields
(NeuroGFs), which are learned to represent the all-pairs geodesics of a given
mesh. By using NeuroGFs, we can efficiently and accurately answer queries of
arbitrary point-to-point geodesic distances and paths, overcoming the
limitations of traditional algorithms. Evaluations on common 3D models show
that NeuroGFs exhibit exceptional performance in solving the single-source
all-destination (SSAD) and point-to-point geodesics, and achieve high accuracy
consistently. Moreover, NeuroGFs offer the unique advantage of encoding both 3D
geometry and geodesics in a unified representation. Code is made available at
https://github.com/keeganhk/NeuroGF/tree/master
Goedel-type Universes and the Landau Problem
We point out a close relation between a family of Goedel-type solutions of
3+1 General Relativity and the Landau problem in S^2, R^2 and H_2; in
particular, the classical geodesics correspond to Larmor orbits in the Landau
problem. We discuss the extent of this relation, by analyzing the solutions of
the Klein-Gordon equation in these backgrounds. For the R^2 case, this relation
was independently noticed in hep-th/0306148. Guided by the analogy with the
Landau problem, we speculate on the possible holographic description of a
single chronologically safe region.Comment: Latex, 21 pages, 1 figure. v2 missing references to previous work on
the subject adde
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