I think it's pretty obvious now we had two different thoughts about how to go about things, so I'll start with yours since in theory it would work, save for some practical limitations.
If I had such a material – assuming that you’re talking about a type of material that can measure distances as a function of electrical resistance, then I might be able to determine more than one piece’s location by first identifying three imaginary points on the material. I believe that it is then possible to identify the location of any point on the game board - occupied by one of these pieces, as a function of diameters of three imaginary circles defined as having a center coincident with these three points - by considering a point where the circumferences of the three imaginary circles intercept.
Are you saying you want to turn one piece (and only one piece at any one time) into an electrode and send a current through it and measure the current arriving each of the 3 contacts?
I can see how that might work in theory, but in practicality you are going to need wires coming off of every playing piece. YOu still have to line the edge of the board with contacts so this method is no simpler than a grid method. It is probably more complicated since you now need to monitor 3 contacts with 3 very sensitive current monitors, as well as driving the current with an accurate driver (as opposed to monitoring 2 contacts with simple go/no go comparators.)
Also remember you can only inject current through one piece at any one time, or else you won't be able to differentiate between the positions of the various pieces.
You also need (fancy) math to map out the circles as well as to find the intersection. You also need math to account for how materials and measurements are never perfect so you will never get a single intersection between the 3 circles. It is much more likely that you will get 3 intersections between each pair of the three circles. These 3 points would form a triangle, somewhere within would be your playing piece, and you would need math to approximate where the best location in that area is (probably the center of the triangle, but that still takes some fairly fancy math to do).
BTW there are two ways to do that, you could use a current source and inject a known current into the board through the playing piece and measure how much current is exiting the board through each of the 3 contacts and compare that to the original amount injected. Or you could use a voltage source and measure the current exiting the board through each of the 3 contacts and then calculate the effective resistance between the playing piece and each contact. Both have the problem of non-linearity because...(see next answer)
If I’m correct, a Gaussian distribution is continuous. However, if I understand what you mean by array correctly, then such a design would not produce an electric output – indicative of the location of a game piece, that could be measured as having a continuous change. What type of circuit would allow me to work with a measurement of change in electricity that is continuous - and could therefore allow me to measure the location of game pieces continuously?
Hypothetically, if you implenented your design you shouldn't bother with look-up tables. If your board was infinite in size you could use a single "simple" current distribution model. But it isn't. Your board is finite sized and the current will distribute differently if the current is injected into the board near the edge than if it is injected near the center. The result is that the effective resistance (if a voltage source is used to inject current into the board through the playing piece) or current at the edge contact (if a current source is used to inject current into the board through the playing piece) that is measured is non-linear with respect to the distance between contact and playing piece.
So you would have to use many different simple models for each area of the board and evaluate them all every time you get a reading or you would have to use one big complicated model. And then you would still have to follow up on the circle intersection math. Either way, this stuff is best left to MATLAB and to do it in real time requires lots of resources and time. It's much easier to map out a grid of points on the board and then place inject current into the board at every point and record the readings at the edge contacts, then put it into a look-up table. If the matrix is fine enough, then you can probably ignore the difference in current distribution effects between two adjacent points and just use interpolation to figure out where a piece is with more resolution than the matrix of test points (ie. if it's sitting between points rather than on a point). Still a lot of math and memory resources.
What is capacitive touch?
It's what is used on touch screens or laptop touchpads (or other things like that).
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Now if you wanted still want to talk about how my method worked (and I clearly explained why it does not). It's concept in one dimension is just if you have a long strip of resistive material (ie. a 1D board so to speak) and place a smaller resistive piece of material on it (ie. the playing piece) this would almost be like connecting two resistors in parallel or in series and would thus modify the resistance seen between the two contacts of the larger resistor. In 1D you can only tell if a playing piece is there or not, in 2D you would be able to tell where a piece was on the resistor.
But this is where I start with the problem of current distributing evenly throughout the smaller resistor due to surface contact and volume because of it's distance away from the sight-line between the two contacts of the larger resistor. THen there is the parallel resistance dominance effect made between the small resistor and the path of large resistor that it is in contact with (which acts like a parallel resistance connection), as well as the series resistance dominance effect (which is formed by the aformentioned parallel connection effectively being in series with the remainder of the larger resistor that is not in contact with the small resistor).
THe reason I though this was the only method was because I figured you wanted simple playing pieces with no wires or power sources in them. Either way, this method is not workable with more than one piece, and is barely workable in theory with just one piece.