Direct Transformation of Laplace Equation’s Solution from Spherical to Cartesian Representation

The description of the Earth’s gravitational field, governed by the fundamental potential equation (the Laplace equation), is conventionally expressed using spherical harmonics, yet the Cartesian formulation, using a Taylor series representation, offers significant algebraic advantages. This paper proposes a novel Direct Cartesian Method for generating spherical basis functions and coefficients directly within the Cartesian coordinate system, utilising the partial derivatives of the inverse distance ( 1 / R ) function. The present study investigates the structural correspondence between the Cartesian form of spherical basis functions and the high-order partial derivatives of 1 / R . The study reveals that spherical basis functions can be categorised into four distinct groups based on the parity of the degree n and order m . It is demonstrated that each spherical basis function is equivalent to a weighted summation of the partial derivatives of the inverse distance ( 1 / R ) with respect to Cartesian coordinates. Specifically, the basis functions are combined with those derivatives that share the same order of Z-differentiation and possess matching parities in their orders of differentiation with respect to X and Y. In order to facilitate the practical calculation of these high-degree derivatives, a recursive numerical algorithm has been developed. The method generates the polynomial coefficients for the numerator of the 1 / R derivatives. A pivotal innovation is the implementation of a step-wise normalization scheme within the recursive relations. The integration of the recursive ratios of global normalization factors (including full Schmidt normalization) into each step of the algorithm effectively neutralises factorial growth, rendering the process immune to numerical overflow. The validity and numerical stability of the proposed method are demonstrated through a detailed step-by-step derivation of a sectorial basis function ( n = 8 , m = 2 ).