I have a spherical harmonic which is a function of $\displaystyle Y_{nm}(\theta,\phi) $.I recently found an equation which is written as follows:$\displaystyle Y_{nm}(\frac{x_2 - x_1}{\left\|x_2 - x_1\right\|})$.I am trying to understand its meaning in order to include it in my simulations. Note that in my case $\displaystyle \theta = \pi/2$ so the vectors become 2-dimensional.I know that in order to evaluate a Spherical Harmonic we need a value for $\displaystyle \theta$ and a value for $\displaystyle \phi$.As I can understand from the equation above, the $\displaystyle \frac{x_2 - x_1}{\left\|x_2 - x_1\right\|}$term will produce a new normalized vector, that is the difference between $\displaystyle x_2 - x_1$.In my case the points/vectors are located on a circle (not inside the circle) of some radius R.
The author of the book mentions that the magnitude is $\displaystyle \left\|x_2 - x_1\right\| = 0.1,0.2,...2$, in steps of 0.1. My problem is that I want to evaluate the Spherical Harmonic for all the values of $\displaystyle \left\|x_2 - x_1\right\| = 0.1,0.2,...2$ but I cannot see how to do it since the input to the spherical harmonic is the angle $\displaystyle (\theta=\pi/2, \phi)$. Knowing that $\displaystyle \left\|x_2 - x_1\right\| = 0.1,0.2,...2$does not tell me anything about $\displaystyle x_2 - x_1$ or DOES IT???

I would appreciate someone's help