Hypergeometric Bridges for Multidimensional Grid Visibility

Abstract

Established interpolation-based methods efficiently compute visibility from one source to every grid cell using values computed closer to the source. Interpolation yields fractional values even on binary grids of free or fully blocking cells. Two-dimensional path-counting theory interprets these values as fractions of unobstructed grid paths, but three-dimensional methods use different predecessor structures. Interpreting these values consistently across dimensions and on nonbinary grids remains unresolved. We provide a unified mathematical account that gives every visibility value a precise meaning in any dimension. We establish when grid refinement yields eventual agreement with continuous visibility, implement parallel solvers, and compare four three-dimensional methods in accuracy and throughput. We define cell transmission: zero blocks visibility, one preserves it, and intermediate values attenuate it proportionally. We prove each computed value equals the uniform finite-path average of transmission products over finitely many source-to-cell grid paths. It is the fraction of clear paths on binary grids and the mean path transmission on nonbinary grids. For a source--target query, the fractions of paths passing through cells at each step form a hypergeometric distribution in 2D, which we call the hypergeometric bridge. On the last step into the target, these probabilities equal the interpolation weights. Higher-dimensional interpolation weights arise from products of hypergeometric bridges. For any fixed dimension, the full-field algorithm remains linear in grid size. Exact path enumeration reproduces every checked visibility value. NumPy, OpenMP, and CUDA agree numerically, and our OpenMP solver achieves the highest throughput in the four-method CPU comparison at every tested resolution.