## Appendix B Obtaining the Generic Form \(\sigma\) for Collectors with Cuts  
In this appendix, the procedure for obtaining the flat curve \(\Sigma\) associated with a collecting surface \(\Pi\) with a generic shape is explained in detail so that it is valid for all the collector shapes considered in this work.  
Letting a plane \(\psi\) and a reference system OXY be in this plane, we considered a polygon \(\Pi\) contained in \(\psi\) with the shape of a rectangle with sides \(L_{x}\) and \(L_{y}\) with the vertices trimmed, the coordinates of these vertices being: \((X_{ul},Y_{ul})\), \((X_{ur},Y_{ur})\), \((X_{dl},Y_{dl})\), and \((X_{dr},Y_{dr})\) (Figure A3).  
The analysis of all the translations allows us to obtain the coordinates of the perimeter of \(\Sigma\). In general, \(\Sigma\) is a polygon with 28 vertices that depend on the dimensions reflected in Figure A3 according to expression (A1).  
\[\Sigma=\Sigma\big{(}L_{x},L_{y},\ X_{ul},Y_{ul},\ X_{ur},\ Y_{ur},\ X_{dl},\ Y_{dl},\ X_{dr},\ Y_{dr}\big{)}\] (A1)  
In this way, these coordinates can be structured in two arrays of 28 elements Xenv (28) and Yenv (28). Next, the VBA programming code that allows obtaining each of these coordinates is shown.  
Xenv(1) = Lx \(-\) Xur \(-\) Xdl: Yenv(1) = Ly  
If Yur<Ydl Then  
Xenv(2) = Xenv(1): Yenv(2) = Yenv(1) \(-\) Yur  
Xenv(3) = Xenv(1) + Xur: Yenv(3) = Yenv(1) \(-\) YurElse  
Xenv(2) = Xenv(1): Yenv(2) = Yenv(1) - Ydl  
Xenv(3) = Xenv(1) + Xdl: Yenv(3) = Yenv(1) - Ydl  
End If  
Xenv(7) = Lx: Yenv(7) = Ly - Yur - Ydl  
If Xur<Xdl Then  
Xenv(6) = Xenv(7) - Xur: Yenv(6) = Yenv(7)  
Xenv(5) = Xenv(7) - Xur: Yenv(5) = Yenv(7) + Yur  
Else  
Xenv(6) = Xenv(7) - Xdl: Yenv(6) = Yenv(7)  
Xenv(5) = Xenv(7) - Xdl: Yenv(5) = Yenv(7) + Ydl  
End If  
Xenv(4) = Xenv(3): Yenv(4) = Yenv(5)  
Xenv(11) = Lx: Yenv(11) = -Ly + Ydr + Yul  
If Xdr<Xul Then  
Xenv(10) = Xenv(11) - Xdr: Yenv(10) = Yenv(11) - Ydr  
Else  
Xenv(10) = Xenv(11) - Xul: Yenv(10) = Yenv(11) - Yul  
End If  
Xenv(11) = Xenv(12): Yenv(11) = Yenv(10)  
Xenv(14) = Lx - Xdr - Xul: Yenv(14) = -Ly  
If Ydr<Yul Then  
Xenv(13) = Xenv(14): Yenv(13) = Yenv(14) + Ydr  
Xenv(12) = Xenv(14) + Xdr: Yenv(12) = Yenv(14) + Ydr  
Else  
Xenv(13) = Xenv(14): Yenv(13) = Yenv(14) + Yul  
Xenv(12) = Xenv(14) + Xul: Yenv(12) = Yenv(14) + Yul  
End If  
For i = 15 To 211  
Xenv(i) = -Xenv(i - 14)  
Yenv(i) = -Yenv(i - 14)  
Next i  
However, when there is some kind of symmetry or when any of the dimensions of the cutouts is null, some coordinates contained in the Xenv () and Yenv () arrays will be repeated. In these cases, the number of vertices of \(\Sigma\) will be less than 28. After eliminating the possible repeated points, it is mathematically verified that there are four combinations or symmetries that generate the same polygon \(\Sigma\) Equation (A2).  
\[\begin{array}{l}\Sigma\big{(}L_{x},L_{y},\,X_{ul},\,X_{ur},\,Y_{ur},\,X_{dl},\,Y_{dl},\,X_{dr},\,Y_{dr}\big{)}=\\ \Sigma\big{(}L_{x},L_{y},\,X_{dr},\,Y_{dr},\,X_{ur},\,Y_{ur},\,X_{dl},\,Y_{dl}, \,X_{ul},\,Y_{ul}\big{)}=\\ \Sigma\big{(}L_{x},L_{y},\,X_{ul},\,Y_{ul},\,X_{dl},\,Y_{dl},\,X_{ur},\,Y_{ur}, \,X_{dr},\,Y_{dr}\big{)}=\\ \Sigma\big{(}L_{x},L_{y},\,X_{dr},\,Y_{dr},\,X_{ur},\,Y_{ur},\,X_{dl},\,Y_{dl}, \,X_{ul},\,Y_{ul}\big{)}\end{array}\] (A2)  
In this way, it is possible to obtain the envelope \(\Sigma\) to the family of polygons \(\Pi\) on which the dichotomous criterion of inter-shading between collectors will be based.