Measuring length of a circuit by signal attenuation (or of the like)
No, not like you imagine it would work:

What you could measure is the effective resistance from your corner to your contact point. That's probably what you've meant with attenuation.

That resistance is going to be very, very small. You need a very sensitive (read: expensive) measurement device to measure it. An Arduino cannot do this on its own.

The resistance of the contacts will be hard to determine. So, they add uncertainty to your problem. Sadly, that uncertainty will be much larger than the measurable resistance, so, this will also not work out.

Now, you will quickly notice that if you put your contact pen on a place on the edge of your metal plane, then the current between the pen and the other contact has much less area to flow through then when you put it somewhere in the middle of the plane.

This idea alone shows that the "points of identical resistance" will not be on a circle; the shape of your metal plane plays an important role here!

So, what is the shape of the "points of identical resistance"? That's hard to answer. 

In fact, you will need to realize that the question "where does current flow in a large conductor" needs a consideration where current already flows – because where current flows throug a non-perfect conductor, there is a voltage (potential difference), as well as a magnetic field (induction!). That magnetic field will "push away" other currents that produce a similar magnetic field. But then you get a different potential. And then you get different currents.

So, this leads to a system of equations where one equation depends on the integral of another equation. We call such equations differential equations. 

The shape of the metal plane gives you boundary conditions. For example, there can't be current flowing "out of" the plate.

To solve such systems of differential equations, one can sometimes still find a closed form solution (i.e. a formula). But usually, that is impossible, and the best we can do is do a simulation. Such a simulation typically breaks down that plane into thousands of smaller planes, and then does what we call finite elements methods to simulate how things behave in total.

None of this is possible without a very fast computer and without very deep knowledge of how electrical and magnetic fields work.

I'm afraid the answer is:


  Yes, calculating possible places on your plane based on measuring the resistance is possible. It is not possible in any way that you think of, and it will take years of study to learn how its possible and very expensive equipment to do so.