Determining Coordinate Transformations for Cloaking Device

In summary, the article discusses the mathematical description of a diamond-shaped cloaking device and how the authors determined the coordinate transformations in equations (1) and (5). They did not mention setting out to find a specific type of transformation, but rather used linear equations to map specific points to one another. They also wanted some of the transformations to be many-to-one in order to collapse figures onto a line. By solving simultaneous equations, the authors were able to find the coefficients for the transformations. However, someone with a good geometric intuition may be able to deduce these coefficients without solving the equations.
  • #1
Lizard
5
0
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?
 
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  • #2
Lizard said:
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".)
 

Related to Determining Coordinate Transformations for Cloaking Device

1. What is a cloaking device?

A cloaking device is a technology that can make an object or person invisible or undetectable by manipulating electromagnetic radiation. This is achieved by using materials that can bend and redirect light around the object, creating the illusion of transparency.

2. How does a cloaking device work?

A cloaking device works by using a combination of specialized materials, such as metamaterials, and advanced mathematical calculations to manipulate light and other forms of electromagnetic radiation. This creates an invisible shield around the object, making it appear as though it is not there.

3. Why is it important to determine coordinate transformations for a cloaking device?

Determining coordinate transformations is crucial for a cloaking device to work effectively. These transformations are used to calculate the precise placement and orientation of the cloaking materials, ensuring that light is redirected around the object in the desired way. Without accurate coordinate transformations, the cloaking device may not be effective and could potentially be detected.

4. What are some challenges in determining coordinate transformations for a cloaking device?

One of the main challenges in determining coordinate transformations for a cloaking device is the complexity of the mathematical calculations involved. These calculations must take into account the shape and size of the object, as well as the properties of the cloaking materials. Additionally, factors such as the type of light being used and the direction and speed of movement of the object must also be considered.

5. Are there any potential applications for a cloaking device?

Cloaking devices have a wide range of potential applications, including military and defense purposes, in which invisible or undetectable objects could provide a strategic advantage. They could also be used in medical imaging technology, allowing for more precise and non-invasive procedures. Additionally, cloaking devices could have practical applications in consumer products, such as creating invisible screens or windows.

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