A discussion on the benefits of the mechanics you suggest, in the thread it was suggested in? Of course it belongs elsewhere. </sarcasm>
It does not really work like that in real life. You can't just know when you first detect something on radar is the factual maximum distance it could have been detected. It will depend on so many things. In reality you use multiple radar installations for a reason to triangular readings. You use multiple sources of aircraft with radar to increase the chances of detecting something etc... things that is not just a result of the curvature of the earth.
Actually, it would. You would detect that object at the maximum distance you could detect that object. If you can't detect it further than that distance, they you didn't detect it while it was further out. Whether it's too small or masking you or your sensors are too crap, you detect it when you detect it and that's the max you could detect it at. Your radar net would detect the same plane at the same time no matter how many times it makes that flight path. Unless it adds/removes shielding or changes it's path, it'll give the same result every time.
Now, detection:
Based on your resolution and sensor strength that you KNOW, using the mathematical equations that your computer KNOWS, you know how far you SHOULD be able to see a ship of X size with each sensor. If that ship is smaller, or is using shielding to make itself seem smaller, then you WON'T pick it up until it's closer. But if something appears on your scanner, you can triangulate it off your own sensors (using multiple receivers on the same ship) or use the parallax of movement, to figure out it's much closer. Based on that, you can figure out how big the ship is (or appears to be) and at least where it is in relation to yourself.
One ship with multiple detection points sufficiently far apart can triangulate with itself, it wouldn't need multiple ships with independent sensors. Remember, these sensors are HUGE. My 80 res, billion km range sensor is 750 tonnes on it's own. The accompanying 129mkm res 1 sensor is another 750tonnes. That's 1500 tonnes of sensor equipment. Assuming that even 40-50% of that is the actual computers, imagine how many antennas you could put using 700 tonnes of light-weight, strong, future-tech materials? I'd put a transmitter for the ping and at least 4-6 antennas (probably on towers too) scattered around the ship. At the distance between ships in a single fleet, it's not enough to give substantial improvement over self-triangulation. The spotter of a sniper team is going to be pretty close to the sniper and they can triangulate fairly well, you don't need to be at crazy angles to triangulate.
What you are describing: using multiple radar stations, would equate to using multiple picket fleets several mkm(or bkm) out to increase coverage, which is a viable tactic. Having multiple ships in the same fleet with scanners would give no benefit unless the scanners are geared differently (AM vs AS).
I also never mentioned anything about curvature of the Earth.
I don't think that Spherical shapes is the most efficient for all intents and purposes (in military application) it all depends on so many other factors where a spherical shaped object is not always the best option. I highly doubt that ships would be truly spherical in space in the future for many different reasons.
A spherical shape doesn't need to worry about breaking along the long axis when making a sharp turn, or having large concentrations of mass swinging wildly during said turn. It also presents the smallest possible TCS at any angle (as opposed to just head-on) and provides maximum volume for mass. A slight elongation along the long axis may be beneficial (Or required to fit specialised equipment), but nothing too extreme otherwise you need to add extra lateral thrust pointing outwards to not snap your ship, which decreases efficiency. Since it has a high surface area to volume ratio, it also means less mass needs to be expended on armoring the same mass ship. It also means that the bridge and critical components have the same amount of armor from any direction, which reduces vulnerabilities. The more spherical, the better. This is ultimately roleplay though as I doubt Steve will ever want to change the code to track TCS from POV of individual objects.
We also know very little what type of drive systems that Aurora actually use and how much thermal energy is actually released, so you can't just assume it is easy to detect. Perhaps ships use a Warp type engine where the heat energy is just a fraction of what a regular thrust engine would use.
It's clearly a non-newtonian drive system (Duh), since the ship stops when it's shut off, but the exact mechanics are irrelevant, since we DO know how much thermal energy is released. Some maths:
My population is 2.6 billion. 4500 various factories, 1.8 million tonne shipyard capacity, various refineries, mines, support structures, etc.
Thermal output is 77500.
My Missile Cruiser has 2x4000EP Internal Confinement Fusion engines, 50% shielded for a total thrust of 8000.
They produce a thermal signature of 4000 units, so 8000 unshielded.
So we can presume that unshielded engines produce 1 thermal per EP.
How visible would my 13,500km/s, 30,000t object be?
5.16% of my colony above, or 10.32% unshielded. At a mass of 0.000000000000005 percent of that same colony.
If you were both sitting still, I suppose you could (maybe) mistake it for a star (If a very odd one). The second one of you moved though, the parallax would show it to be only a few million km away, or the multiple sensors on your hull triangulated between themselves could do the same even without movement.
The fact is, that the sensors as they are work just fine extrapolated from current tech and requiring multiple sensors on multiple ships to get better/more accurate readings is pointless and less realistic than a single ship doing all the work. You're welcome to use multiple fleets to get better coverage of an area, but within the sensor range, the mechanics are just fine as they are.
Unless you can find a new reason for using multiple identical scanners in a fleet, I'm done.