The thread about ultrasonic position sensing is interesting, and it "set me off". Rather than make that thread "fray", here's a note of an alternative system.
In either case (indeed any measuring problem!) of course tow questions have to be asked and answered before much design work is done: What range are we dealing with, and what resolutions (spacial, temporal... etc?) are sought.
I THINK (little real work done yet!) the following would stand a chance of working well over the area of a large room, or even, say, a basketball court. (The room suffers from the problem of obstructions, the basketball court.. sans game.. is not available to many of us!
The idea is inspired by how I THINK the FAA's VOR radio navigation beacons work, the system pilots used before GPS, and still use today to pass tests and when their GPS fails.
You need....
Two beacons. (More in a moment)
A sensor on the object that you want to know the location of. It must be able to distinguish between the signals from the two beacons. (For the Arduino, I think this can be done by using different colors of light, and having TWO sensors, one for each beacon. Each sensor would have an optical filter, making it "see" only one of the beacons.)
(In aircraft VOR, the signal from the beacon is a radio signal. We can use a visible (or IR) light signal)
The beacons....
Can be made lots of ways. I'll describe ONE way to make one, and you can do anything that is functionally equivalent.
"My" design would have two light bulbs. One would flash briefly every, say, second. We'll call that the "pulse" blink. The pulse blink goes out from the beacon in every direction. The second light bulb is in a mechanism that makes it shine in one direction only, and that direction sweeps around in a horizontal plane, like the beam from a lighthouse. We'll call this the "radial" beam. From anywhere in the room, it will appear to wink briefly.
The timing of the pulse blink is sychronized with the sweep of the radial beam, so that the pulse blink occurs when the radial beam is shining to the north of the beacon. Thus, if the sensor sees the two blinks at very nearly the same moment, the sensor is north of the beacon. (Making it able to see the two blinks at the SAME moment is an interesting design challenge... as is dealing with it being "blind" when EXACTLY north of the beacon, if you choose to "go around" the other problem.) If the sensor sees the radial blink exactly half way between two pulse blinks, then the sensor is due south of the beacon.
The rest is details! Have fun!
) as a radial beam...