Analytical solutions for the internal and external gravitation of a homogeneous cuboid

Abstract We extend our previous formulation of the gravitational potential of a homogeneous cuboid (right rectangular prism) by deriving uniform analytical expressions for the gravitational acceleration and the full gravitational tensor. The formulation is valid in the interior and exterior of the cuboid and also on its boundary. Classical rectangular–prism formulas express the potential and its derivatives through logarithmic and arctangent kernels whose arguments vanish at faces, edges, and vertices. Although these singularities are removable analytically, their direct numerical evaluation introduces instabilities near the boundaries of the body. We introduce a unified closed-form expressions based on oriented face–distance coordinates combined with analytically regularized sp_log and sp_arctan functions and an alternating vertex-sign summation. This representation produces continuous expressions for the gravitational potential, acceleration, and tensor that remain well defined at interior, boundary, and exterior field points without case distinctions or limiting procedures. Validation using acceleration-field analysis and an analytic Laplacian test confirms exact agreement with Poisson’s equation inside the cuboid and Laplace’s equation outside. The resulting expressions provide stable analytic first and second derivatives suitable for high-accuracy modeling of gravity and gravity-gradient measurements.