## 2 Sampling Representation over the Upper Hemisphere
Due to the geometry of the equivalent problem, the here adopted one bounds a central cylindrical body of radius \(a\) and height \(h_{c}\), which ends into two circular bowls with same aperture radii \(a\) and lateral bending radii \(c\) (see Figure 2b). These parameters can be properly set to minimise the volumetric redundancy, and accordingly, the number of required samples. Note that, by setting \(h_{c}=0\) and \(c=a\), the adaptable surface \(\Sigma_{M}\) reduces to the spherical one; by choosing \(h_{c}=0\), \(\Sigma_{M}\) becomes the surface bounding a double bowl; and by setting \(c=a\), the surface \(\Sigma_{M}\) reduces to that bounding a rounded cylinder. Now, having properly modelled the combined AUT, each of the curves (meridian and parallel) of the observation surface has to be described according to the non-redundant representations framework [33] by choosing the bandwidth \(W_{\tau}\) and determining the optimal parameter \(\tau\) and phase function \(\gamma\). In the case of a meridian, the bandwidth is chosen to be equal to  
\[W_{\tau}=\frac{\ell^{\prime}}{\lambda} \tag{2}\]  
where \(\ell^{\prime}\) is the length of the curve \(C^{\prime}\), obtained by intersecting the meridian plane passing through \(Q\) with \(\Sigma_{M}\) and equal to \(2[h_{c}+(a-c)+\pi c]\), and \(\lambda\) the wavelength. Then, the optimal parameter \(\tau\) and phase function \(\gamma\) are [33; 34]:  
\[\tau=\big{(}\pi/\ell^{\prime}\big{)}\big{[}\,\overline{QQ_{1}}+\sigma_{1}^{ \prime}-\overline{QQ_{2}}+\sigma_{2}^{\prime}\big{]} \tag{3}\]  
\[\gamma=(\pi/\lambda)\big{[}\,\overline{QQ_{1}}+\sigma_{1}^{\prime}+\overline{QQ _{2}}-\sigma_{2}^{\prime}\big{]} \tag{4}\]  
where \(\overline{QQ}_{1}\) and \(\overline{QQ}_{2}\) are the distances from \(Q\) to tangency points \(Q_{1,2}\) on \(C^{\prime}\) and \(\sigma_{1,2}^{\prime}\) the corresponding curvilinear abscissae. Note that relations (3) and (4) depend on the position of \(Q\) when moving from \(0\) to \(\pi\), since the distances \(\overline{QQ}_{1,2}\) and the corresponding curvilinear abscissae \(\sigma_{1,2}^{\prime}\) change accordingly. Five cases occur in such a case, whose details can be found in [34]. In the case of a parallel, it can be easily shown [33; 34] that \(\gamma\) is constant and that the optimal parameter \(\tau\) to be used to represent it is the azimuthal parameter \(\varphi\). The corresponding bandwidth is:  
\[W_{\varphi}=\frac{\pi}{\lambda}\underset{z^{\prime}}{\max}D(_{M}-D_{m})=\frac {\pi}{\lambda}\underset{z^{\prime}}{\max}\bigg{(}\sqrt{\big{(}z-z^{\prime} \big{)}^{2}+\big{(}\rho+\rho^{\prime}(z^{\prime})\big{)}^{2}}-\sqrt{\big{(}z-z ^{\prime}\big{)}^{2}+\big{(}\rho-\rho^{\prime}(z^{\prime})\big{)}^{2}}\bigg{)} \tag{5}\]  
where \(D_{M},D_{m}\) are the maximum and minimum distances from the considered parallel to the circumference of \(\Sigma_{M}\) at \(z^{\prime}\), \(\rho=R_{hs}\sin\theta\), and \(\rho^{\prime}(z^{\prime})\) represents the equation of \(\Sigma_{M}\) in cylindrical coordinates. It can be easily proved [34] that the wanted maximum is in the zone of the surface \(\Sigma_{M}\) lying on the same side of the considered parallel with regard to its maximum transverse circumference. Such a maximum is reached at \(z\prime=z\) if \(|z|<h_{c}/2\). If \(|z|>h_{c}/2\), it is sought as the value of the angular coordinate \(\alpha\) such that  
\[\left\{ \begin{array}{ll}z^{\prime}=h_{c}/2+c\cos\alpha;&\rho^{\prime}=a-c+c \sin\alpha&\text{if $z>h_{c}/2$}\\ z^{\prime}=-(h_{c}/2+c\cos\alpha);&\rho^{\prime}=a-c+c\sin\alpha&\text{if $z<-h_{c}/2$} \end{array} \right. \tag{6}\]  
for which \(\partial(D_{M}-D_{m})/\partial\alpha=0\). The use of the above non-redundant parameters allows one to determine the arrangement of the sampling points over the whole observation sphere, since they have been derived by considering the equivalent problem. Anyhow, only the NF sampling points belonging to the upper hemisphere can be reached in practice due to the presence of the infinite PEC ground plane.