The following analysis assumes a general impedance surface. To treat an object with an arbitrary geometry using PO, the object is first decomposed into many small elementary patches or cells, which have a simple geometry such as a rectangle or triangle. Then, using the tangent plane approximation, the electric and magnetic surface currents, '''J(r)''' and '''M(r)''', on the lit region of the scatterer are approximated by:
:<math> \mathbf{J(r)} = (1+\alpha) \mathbf{\hat{n} \times H(r)} </math>Â :<math> \mathbf{M(r)} = -(1-\alpha) \mathbf{\hat{n} \times E(r)} </math><!--[[File:PO1(1).png]]-->
where '''E(r)''' and '''H(r)''' are the incident electric and magnetic fields on the object and '''n''' is the local outward normal unit vector as shown in the figure below. a is a parameter related to the impedance Z of the surface (expressed in Ohms), which is defined in the following way:
:<math> \alpha = \frac{1-Z/\eta_0}{1+Z/\eta_0} </math><!--[[File:PO2.png]]-->
where <math>\eta_0 = 120\pi \; \Omega</math> is the intrinsic impedance of the free space. Then, the electric and magnetic currents reduce to:
:<math> \mathbf{J(r)} = \frac{2\eta_0}{\eta_0 + Z} \mathbf{\hat{n} \times H(r)} </math>Â :<math> \mathbf{M(r)} = - \frac{2Z}{\eta_0 + Z} \mathbf{\hat{n} \times E(r)} </math><!--[[File:PO3.png]]-->
Two limiting cases of an impedance surface are perfect electric conductor (PEC) and perfect magnetic conductor (PMC) surface. For a PEC surface, Z = 0, α = 1, and one can write:
:<math> \mathbf{J(r)} = 2 \mathbf{\hat{n} \times H(r)} </math>Â :<math> \mathbf{M(r)} = 0 </math><!--[[File:PO4.png]]-->
while for a PMC surface, Z = 8, α = -1, and one can write:
:<math> \mathbf{J(r)} = 0 </math>Â :<math> \mathbf{M(r)} = -2 \mathbf{\hat{n} \times E(r)} </math><!--[[File:PO5.png]]-->
Another special case is a Huygens surface with equivalent electric and magnetic surface currents. In that case, Z = η<sub>0</sub>, α = 0, and one can write:
:<math> \mathbf{J(r) = \hat{n} \times H(r)} </math>Â :<math> \mathbf{M(r) = -\hat{n} \times E(r)} </math><!--[[File:PO10.png]]-->
[[File:po_manual_1.png]]