## 2 NR Representation of the Probe Voltage on the Spiral
Thus, by properly taking advantage of the unified theory of spiral scannings for non-volumetric AUTs [36], it is possible to devise an NR voltage representation on the measurement spherical surface \(M\) from a minimum number of samples gathered on an appropriate spiral wrapping of this sphere. In order to determine the spiral trajectory, the spiral wrapping \(\Sigma\) is projected onto the scan sphere \(M\). The spiral step is chosen to be equal to the sample spacing required for interpolating the NF data on any meridian of the sphere \(M\), which still represents a meridian curve in the offset coordinate system \(S^{\prime}\). According to [31; 36], the spatial bandwidth when considering a meridian curve is:  
\[W_{\eta}=\frac{\ell(C^{\prime})}{\lambda}, \tag{5}\]  
where \(C^{\prime}\) is the curve obtained as the intersection between \(\Sigma\) and the meridian plane containing the observation point \(P\), \(\ell(C^{\prime})\) is its length, and \(\lambda\) is the wavelength. The related optimal parameter and optimal phase function are obtained as:  
\[\eta=\left(\pi/\ell(C^{\prime})\right)\,\left[\,\,R_{1}+s^{\prime}(P_{1})-R_{2 }+s^{\prime}(P_{2})\right] \tag{6}\]  
\[\psi=\left(\pi/\lambda\right)\,\left[\,\,R_{1}+s^{\prime}(P_{1})+R_{2}-s^{ \prime}(P_{2})\right], \tag{7}\]  
where \(R_{1}\) and \(R_{2}\) are the distances from \(P\) to the tangency points \(P_{1}\) and \(P_{2}\) on \(C^{\prime}\), respectively, and \(s^{\prime}(P_{1})\) and \(s^{\prime}(P_{2})\) are the corresponding arclength abscissas. It is important to highlight that the parameters in (5), (6), and (7) depend on the specific choice of the modeling surface \(\Sigma\). When a prolate spheroid (see Figure 2a) is chosen as the suitable modeling surface, \(\ell(C^{\prime})=4a^{\prime}E\left(\pi/2\big{|}\sigma^{2}\right)\) and the corresponding expressions of (5), (6), and (7) are [10]:  
\[W_{\eta}=\frac{4a^{\prime}E\left(\pi/2\big{|}\sigma^{2}\right)}{\lambda} \tag{8}\]  
Figure 2: Spherical spiral NF setup for an elongated offset-mounted antenna: (a) prolate spheroidal modeling; (b) rounded cylinder modeling. \[r^{\prime}=\sqrt{d^{2}-\left(d_{s}\sin(\vartheta^{\prime})\right)^{2}}-d_{s} \cos(\vartheta^{\prime}) \tag{2}\]0  
\[r^{\prime}=\sqrt{d^{2}-\left(d_{s}\sin(\vartheta^{\prime})\right)^{2}}-d_{s} \cos(\vartheta^{\prime}) \tag{2}\]1  
where \(E(\cdot|\cdot)\) is the second kind elliptic integral; \(\sigma=f/a^{\prime}\) is the eccentricity of the ellipse, attained as the intersection between a meridian plane and \(\Sigma\); \(2f\) is the focal distance; \(u^{\prime}=(d_{1}+d_{2})/2a^{\prime}\) and \(v^{\prime}=(d_{1}-d_{2})/2f\) are the corresponding elliptic coordinates; and \(d_{1,2}\) are the distances between \(P\) and the foci of the ellipse. On the other hand, when the chosen modeling surface is represented by a rounded cylinder (see Figure 2b), \(\ell(C^{\prime})=2(h_{c}+\pi a_{c})\). Accordingly:  
\[r^{\prime}=\sqrt{d^{2}-\left(d_{s}\sin(\vartheta^{\prime})\right)^{2}}-d_{s} \cos(\vartheta^{\prime}) \tag{2}\]2  
Regarding the parameter \(\eta\) and the phase function \(\psi\), they can be determined by substituting, in (6) and (7), the involved geometrical parameters \(R_{1,2}\) and \(s^{\prime}(P_{1,2})\), whose expressions change depending on the tangency points position. Their explicit expressions are reported in Appendix A. The parametric equations of the spiral, attained by projecting the spiral that wraps \(\Sigma\) on the measurement sphere through the curves at \(\eta=const\), are [36]:  
\[\left\{ \begin{array}{l}x^{\prime}=r^{\prime}\sin[\theta^{\prime}(\eta)]\cos (\phi^{\prime})\\ y^{\prime}=r^{\prime}\sin[\theta^{\prime}(\eta)]\sin(\phi^{\prime})\\ z^{\prime}=r^{\prime}\cos[\theta^{\prime}(\eta)]\end{array} ;\right. \tag{12}\]  
where \(\phi^{\prime}\) is the angular parameter that describes the spiral and \(\eta=k\phi^{\prime}\), \(k=1/(2N^{\prime\prime}+1)\). \(N^{\prime\prime}=\lfloor\chi N^{\prime}\rfloor+1\), and \(N^{\prime}=\left\lfloor\chi^{\prime}W_{\eta}\right\rfloor+1\).