Euler angles in latitude longitude space

[meteo student]

In most physics introductions Euler angles(pitch, roll, yaw) are defined with respect to Cartesian coordinate system.

If I chose not to use a Cartersian coordinate system but instead use a latitude, longitude and a proprietary vertical coordinate(and no back transformations to Cartersian coordinate system permitted) basis vector space how would the pitch, yaw, roll Euler angles be defined ?

What I mean by this is the following

Initially I have a point defined in terms of λ,∅ and the vertical coordinate is defined as ζ.

Now I rotate the axes (not the point !) to a new set of axes λ',∅',ξ'.

I want to be able to define the Euler angles with respect to these two sets of orthonormal vectors.

 

[meteo student]

I want to correct what I wrote yesterday. When a proprietary vertical coordinate is used there is no restriction of basis vectors being orthonormal. These are basically curvilinear coordinates in which the rotations are being performed.