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We here consider the transformation of vector components between a cartesian system and spherical-polar coordinate systems. we here consider the typical spherical-polar coordinates, where the position of a point is given in terms of the distance from the origin, the azimuthal angle φ and the polar (zenith) angle θ, as shown in figure 1 below. However, locations on a geoid are frequently given in terms of logitude λ and latitude δ, so we also consider the coordinate system outlined in the right panel of figure 1.

The azimuthal angle φ and the polar (zenith) angle θ are here defined such that:

The azimuthal and polar angles & radial coordinates are related to the Cartesian coordinates by:

or, conversely

Geographical coordinates (logitude λ and latitude δ) are related to the spherical-polar angles by:

The components of a vector in spherical-polar coordinates are:

The unit vectors of the spherical-polar system an be expressed in terms of the unit-vectors of the Cartesian system:

so we can write,

where the transformation matrix is:

Note that the transformation matrix T is self orthogonal, so the reverse transformation is simply: