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As mentioned above a significant challenge is modeling 3 empires. The above design will work to model three empires, but the modeling becomes a little more complicated (which is why I suspect the devs may have avoided this implementation).
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I don't understand your need to talk about vectors here.
This "modelling" is actually incredibly easy if you forget the vector nonsense and just think ternary diagram and percentages. Then give the capture point rate per control point as a function of percentage adjacent controlled.
You're also forgetting that territory control is discrete and based on a hex system. So you can only hold edges in 1/6th (~17%) increments. The whole thing can be done as "If half adjacent controlled, then 5 points per second per capture point controlled" blah blah.
Also, if a facility is controlled, do the points need to be drained from the holding empire first and then added to the new empire, or do all empires points reset to 0 upon a capture? In which case, it's actually beneficial to let a facility be captured, rather than try to struggle to take something during a 3-way.