Performing spherical interpolation to obtain interpolated intensities at each point on the spiral is not the most appropriate method in this case. Spherical interpolation techniques, such as linear interpolation or spherical harmonics, are commonly used for interpolating values on the surface of a sphere. However, they are typically applied to interpolate values between known points on the sphere, rather than mapping values from a 2-dimensional function (pole density function) to a 1-dimensional function (spiral function) as you described.

You need a different approach to map the pole density function onto the spiral function. Here's a suggested method:

Note that the success of this mapping process depends on the suitability of the relationship between the internal spiral parameter and the coordinates (Chi, Phi) on the sphere. Ensure that the relationship accurately represents the desired mapping between the two functions.

Additionally, remember that this mapping approach assumes that the spiral and pole density functions are compatible and can be directly related. Additional considerations and adjustments may be necessary if the functions have significantly different characteristics or underlying principles.