A Generalized Algorithm for the Gravitational Potential of a Homogeneous Cuboid

Abstract

This paper presents a mathematical formulation for computing the gravitational potential of a cuboid (or right rectangular prism) with uniform density. The resulting analytical expressions are valid inside, on and outside the cuboid. The methodology systematically organizes logarithmic and arctangent terms using a hexagonal graph approach, promoting numerical stability and correct sign allocation. We provide a structured pseudocode algorithm to facilitate computational implementation, and actual codes written in Python, Matlab and Julia. We also introduce a test, that we call a Laplacian test, used to validate our algorithm and codes. Keywords: Gravitational Potential, Cuboid, Hexagonal Graph, Finite Differences, Laplacian Test.