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I Determining Coordinate Transformations for Cloaking Device

  1. Feb 9, 2017 #1
    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?
     
  2. jcsd
  3. Feb 12, 2017 #2

    Stephen Tashi

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    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".)
     
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