Determining Coordinate Transformations for Cloaking Device

[Lizard]

I was reading this article: https://arxiv.org/ftp/arxiv/papers/1005/1005.5206.pdf , regarding the mathematical description of a diamond-shaped cloaking device, and am struggling to understand how the authors found the coordinate transformations in equations (1) and (5).

What is the process for determining this type of transformation?

 

[Stephen Tashi]

What is the process for determining this type of transformation?

The authors didn't mention that they set out to determine a transformation of a specific mathematical type, but I assume they were looking for a transformation given by linear equations ( which cannot technically be called a "linear transformation" due to the fact these equations may have a non-zero constant term).

So if we assume the transformations are:
$x' = A_1 x + A_2 y + A_3 z + A_4$
$y' = B_1 x + B_2 y + B_3 z + B_4$
$z' = C_1 x + C_2 y + C_3 z + C_4$

Then if we establish enough particular pairs points that we wish to map to each other (e.g. (a,0,0) to (b,0,0) ) each mapping of a point to another point gives 3 simultaneous equations that must be satisfied. Once we get enough equations, we can solve for the unknowns $A_i, B_i, C_i$.

A feature of the article not found in most scenarios for transforming coordinates is that the authors want some of their transformations to be many-to-one in order to collapse figures that have an area onto a line.

For example in eq. 1 of the article, the z-coordinate is not to be mapped to a different value and the transformation of the other coordinates is supposed to be independent of z. This implies $C_3 = 1$, $C_1=C_2=C_4=A_3=B_3=0$

So the equations we must solve can be simplified to
$x' = A_1 x + A_2 y + A_4$
$y' = B_1 x + B_2 y + B_4$

As I interpret the figure, goals of the transformation are to transform
$(c,0,0)$ to $(b,0,0)$
$(0,d,0)$ to $(0,d,0)$

This implies the equations:
$A_1c + A_4 = b$
$B_1c + B_4 = 0$

$A_2d + A_4 = 0$
$B_2d + B_4 = d$

I haven't solved these simultaneous equations. I note that the coefficients in eq 1 appear to be the solutions.

For example, using the coefficents from the article,
$A_1c + A_4 = (\frac{(b-c)}{(a-c}) c + (\frac{(a-b)}{(a-c)})c = \frac{ bc - c^2 + ac - bc}{(a-c)} = \frac{(ac - c^2)}{(a-c)} = c$

Someone with a good geometric intuition might be able to deduce the coefficients "by inspection". (I wonder why the authors write the equations with the term involving "y" preceeding the term involving "x".)