## 4 Results
### Investigation of the Solution Behavior for Various Geometry Modifications  
To check the robustness of the method described above, several modifications of the geometry were carried out and the respective results were extracted. In all cases below, lossy dielectric material with relative permittivity \(\varepsilon_{r}=2.56-0.102\)\(j\) is considered, whereas the incidence angle is always \(\varphi_{inc}=0\) with an arc radius of \(\rho=3\lambda\). First, a scatterer that deviates only slightly from a circle was studied, i.e., a layout with small arc displacement \(d=0.1\rho=0.3\lambda\), followed by a scatterer with a moderate deviation from a circle, i.e.,  
Figure 9: (**a**) Symmetric geometry with non-symmetric auxiliary points allocation, (**b**) RCS, and (**c**) \(\Delta\)E and \(\Delta\)H error. a layout with mediocre arc displacement \(d=0.5\rho=1.5\lambda\), and, finally, a scatterer with a significant distortion with respect to a circle, which, therefore, encompasses sharp wedges, i.e., a layout with large arc displacement \(d=0.7\rho=2.1\lambda\). All three geometries and their computed results are depicted in Figures 10-12, respectively, while their MAS parameters and the quantified error in the generic boundary condition of the electromagnetic field are presented in Table 1.  
Figure 10: Small arc displacement \(d=0.1\rho=0.3\lambda:\) (**a**) geometry and conformal auxiliary surfaces; (**b**) geometry and improved auxiliary surfaces; (**c**) conformal \(\Delta E\) and \(\Delta H\) error; (**d**) improved \(\Delta E\) and \(\Delta H\) error; (**e**) conformal RCS; and (**f**) improved RCS. As a verification of the broad applicability of the technique, an asymmetric scatterer was also analyzed. Namely, two unequal circular arcs were connected, forming a \"sorrowful\" eye. The geometry parameters were chosen as follows: radius \(\rho=4\lambda\) and arc displacement \(d=0.5\rho=2\lambda\) for the upper arc, radius \(\rho=5\lambda\) and arc displacement \(d=0.5\rho=2.5\lambda\) for the lower arc, originally 160 CPs (hence 160 inner and 160 outer ASs), \(1/2.5\) of the inner ASs and \(1/4\) of the outer ones were displaced, with proximity factor set equal to \(s=0.6\) and \(s=0.6\), respectively. No densification was implemented and no extra ASs were added. The maximum \(E\) field error at MPs was reduced from \(4.19*10^{-2}\) to \(1.57*10^{-2}\) and the maximum \(H\) field error from \(4.36*10^{-1}\) to \(1.38*10^{-1}\). The results are depicted in Figure 13.  
Figure 11: Mediocre arc displacement \(d=0.5\rho=1.5\lambda\): (**a**) geometry and conformal auxiliary surfaces; (**b**) geometry and improved auxiliary surfaces; (**c**) conformal \(\Delta E\) and \(\Delta H\) error; (**d**) improved \(\Delta E\) and \(\Delta H\) error; (**e**) conformal RCS; and (**f**) improved RCS. Judging from the outcome of the aforementioned tests, the method proposed herein is very robust, producing reliable results for blunt, acute, or non-symmetric wedges. Figure 12: Large arc displacement \(d=0.7\rho=2.1\lambda:\) (**a**) geometry and conformal auxiliary surfaces; (**b**) geometry and improved auxiliary surfaces; (**c**) conformal \(\Delta E\) and \(\Delta H\) error; (**d**) improved \(\Delta E\) and \(\Delta H\) error; (**e**) conformal RCS; and (**f**) improved RCS. \begin{table}
\begin{tabular}{c c c c} \hline Arc displacement \(d\) & \(0.1\rho=0.3\lambda\) & \(0.5\rho=1.5\lambda\) & \(0.7\rho=2.1\lambda\) \\ \hline number of points (ASs, CPs) & 160 & 160 & 156 \\ displaced inner ASs fraction & 1/2 & 1/5 & 1/6 \\ displaced outer ASs fraction & 1/2 & 1/8 & 1/2 \\ inner proximity factor \(s\) & 0.55 & 0.75 & 0.74 \\ \hline \end{tabular}
\end{table} 
Table 1: Geometry definition, method of auxiliary sources (MAS) parameters, and computed BC error of the electromagnetic field.