A Geometrical Approach for the Computation and Rotation of Spherical Harmonics and Legendre Functions up to Ultra-High Degree and Order

Abstract

In this section we introduce a new algorithm for the computation and rotation of spherical harmonics, Legendre polynomials and associated Legendre functions up to ultra-high degree and order. The algorithm is based on the geometric rotation of Legendre polynomials in Hilbert space. It is shown that Legendre polynomials can be calculated using geometrical rotations and can be treated as vectors in the Hilbert space leading to unitary Hermitian rotation matrices with geometric properties. We use the term geometrical rotations because although rotation itself is not governed by gravity and it can be used as a proxy to represent a gravity field geometrically. This novel method allows the calculation of spherical harmonics up to an arbitrary degree and order, i.e., up to degree and order 106 and beyond.