The gravitational field of a homogeneous polyhedron

Abstract We present a uniform closed-form boundary-continuous formulation for the gravitational potential, acceleration, and gravitational gradient tensor of a homogeneous polyhedron. Classical polyhedral gravitational models, though exact in form, become numerically unstable at faces, edges, and vertices and are conventionally restricted to exterior points. Our method eliminates these limitations through analytic regularization of the logarithmic and arctangent terms, ensuring continuity and stability across interior, boundary, and exterior domains. A dyadic tensor formulation provides compact expressions for accelerations and gravitational gradients, guaranteeing compliance with Poisson’s and Laplace’s equations. The implementation is fully vectorized and supports multi-threaded parallelism, delivering high computational efficiency and precision. Validation on convex, concave, and multiply connected geometries demonstrates strict physical consistency and numerical robustness. The proposed framework establishes a universal, high-precision computational standard for polyhedral gravitational modelling, applicable to planetary dynamics, spacecraft navigation, solid Earth geophysics, and physical geodesy.