B.1 Derivatives of the surface function r($\theta$, $\phi$)

The values of the various derivatives of the surface function r($r(\theta,\phi)$, ):w are given below, in terms of the spherical harmonic expansion and using the recursion relations given in Ref. [36], where the auxiliary parameter f_nm is:

$$f_{nm}=\sqrt{\left (\frac{(2 n +1) (n-m)!} {4 \pi (n+m)!} \right )}$$ (31)

Note that for negative values of m, one uses the definitions of associated Legendre functions with negative values of m given above in Appendix A.

$$r_{\phi} \sum_{n=0}^{\infty} \sum_{m=-n}^n (i m)\: a_{nm} Y_n^m(\theta,\phi)$$ =

$$r_{\phi\phi} \sum_{n=0}^{\infty} \sum_{m=-n}^n -m^2 \: a_{nm} Y_n^m(\theta,\phi)$$ =

$$r_{\theta} \sum_{n=0}^{\infty} \sum_{m=-n}^n \frac{-a_{nm} f_{nm}}{\sin{\theta}} \, \left [(n+1)\cos{\theta} P_n^m-(n-m+1) P_{n+1}^m \right ] \, e^{im\phi}$$ =

$$r_{\theta \theta} \sum_{n=0}^{\infty} \sum_{m=-n}^n \frac{a_{nm} f_{nm}}{\sin^2{\theta}} \,$$ =

$$[(n+1+(n+1)^2\cos^2{\theta}) P_n^m- 2\cos{\theta}(n-m+1)(n+2)P_{n+1}^m$$

$$\sum_{n=0}^{\infty} \sum_{m=-n}^n \frac{-im \, a_{nm}\, f_{nm}}{\... ...ta}} \, \left [(n+1)\cos{\theta} P_n^m-(n-m+1) P_{n+1}^m \right ] \, e^{im\phi}$$ = = (32)