One of the most interesting properties of Maxwell’s equations is that they are form-invariant under any general coordinate transformation. It is this property that gave rise to the field of TO. For example, consider the transformation from an unprimed coordinate system to a primed coordinate system (x, y, z) ^ (X, y , z'), we have:
It is important to note that the curl in the right-hand side is primed, i.e., it must be performed in the transformed space. Given a Jacobian A defined as:
we can find the primed electromagnetic fields:
In order to emulate the coordinate transformation, the required constitutive material parameters are given via the following relations:
By using these equations, the primed electric fields of the transformed space are recreated in the original space by constructing a material whose constitutive parameters are prescribed by Eq. 6.9. If the transformation is a conformal map, the medium is isotropic.
As an example, consider a 2D transformation from (x,y, z) to (X , / , z):
If the starting medium is isotropic (e.g., free space) and the transformation satisfies the Cauchy-Riemann equations, then the expressions for the constitutive parameters are very simple. For simplicity in notation, we define:
The Jacobian is given by: whose determinant is
and the expression for the permittivity is given by:
Similarly, the primed permeability is given by:
For the TMZ polarization, where E = Ez and H • z = 0, a nonmagnetic, inhomogeneous isotropic dielectric could be used since fizz would have no impact and ezz would be the only parameter that has any polarizing effect.
An important thing to note is that for these equations to be valid, they need to be applied for all space. However, this is not possible in practice, as the area where the coordinate transformation takes place has to be physically restricted. If the fields are to be manipulated external to the designed media, discontinuities in the coordinate system across the outer boundary need to be introduced. Such a transformation features a material embedded in a finite region of space and is called a finite embedded coordinate transformation [45, 46]. But due to impedance mismatches, these designs may have reflections at the boundaries. An example that can be continuous across material boundaries is the well-known invisibility cloak [30, 31, 33]. The invisibility cloak can hide the presence of a PEC object such that an incident field illuminating the PEC and cloak will result in the same incident field outside the cloak. For such a design, it is required that the coordinate system be continuous across material boundaries, since reflections need to be eliminated. In contrast, there are designs that explicitly require discontinuities across material boundaries. One such example is a flat focusing lens [29, 37]. Traditional homogeneous lenses require curvature for focusing, but lenses with large curvatures can be difficult to manufacture. A flat focusing lens can be accomplished by mapping a curved surface to a flat planar surface which embeds the refractive properties of the surface into the bulk of the lens through a TO medium. To focus an incident plane wave to a point, the flat lens must modify the phase distribution of the electric field as it passes through the medium, so the field exiting the lens must be different from the incident field. Therefore, the transformation must be discontinuous and reflections will occur. Such reflections are not unique to TO lenses; homogeneous lenses are susceptible to such reflections as well and both designs can have their reflections mitigated through the introduction of anti-reflective coatings, which will be discussed in Section 6.3.4.
While none of the designs discussed so far involve sources in the transformation regions, charges and currents may be transformed in a similar way to the constitutive material properties, using what is called an optical source transformation, as described by Kundtz et al. .
Using these equations, a source distribution can be geometrically transformed without changing its radiation characteristics. Zhang et al.  used an elliptical transformation to map a line current to a spherical surface current. When combined with the appropriate medium, the radiation characteristics are similar to the original line current. This can easily be extended to mapping a linear phased array to a conformal surface, as described by Popa et al. .
These equations provide an analytical method of determining the permittivity and permeability tensors to achieve a certain geometrical transformation. To physically achieve these material parameters, anisotropic metamaterials must be utilized, which can be lossy and resonant. To achieve isotropic material parameters, the transformation must satisfy the Cauchy-Riemann equations, and it can be very difficult to derive an analytical expression for such a transformation. A more general numerical algorithm called qTO can be employed to find the required transformation between two general geometries.