| Spherical harmonics play a central role in the modelling of spatial and temporal processes in
the system Earth. The gravity field of the Earth and its temporal variations, sea surface
topography, geomagnetic field, ionosphere etc., are just a few examples where spherical
harmonics are used to represent processes in the system Earth. We introduce a novel method
for the computation and rotation of spherical harmonics, Legendre polynomials and
associated Legendre functions without making use of recursive relations. This novel
geometrical approach allows calculation of spherical harmonics without any numerical
instability up to an arbitrary degree and order, e.g. up to degree and order 106 and
beyond.
The algorithm is based on the trigonometric reduction of Legendre polynomials and the
geometric rotation in hyperspace. It is shown that Legendre polynomials can be computed
using trigonometric series by pre-computing amplitudes and translation terms for all
angular arguments. It is shown that they can be treated as vectors in the Hilbert
hyperspace leading to unitary hermitian rotation matrices with geometric properties.
Thus, rotation of spherical harmonics about e.g. a polar or an equatorial axis can be
represented in the similar way. This novel method allows stable calculation of spherical
harmonics up to an arbitrary degree and order, i.e. up to degree and order 106 and
beyond. |