Hi again,
Some question of quadrants is what gate polarity can fire the triac with a given main terminal polarity. There are four possibilites (quadrants):
1. +,-
2. +,+
3. -,-
4. -,+
the first sign above is the gate, the second is the main terminal (i think it's MT1 but i'd look at the data sheet) just for convention..
Some of the triacs can operate in all 4 modes, while others may only operate in 3.
The min current is the current required to reliably turn the triac on, and there may be a different spec for each quadrant.
The voltage of the gate you dont worry about. You just make sure you have enough overhead drive voltage available and use a series resistor to drop the excess. If the data sheet says 1.3v then you make sure you have 1.3v, but you do this really by making sure you have MORE than 1.3v and let the resistor drop the rest. For example, if your triac gate is 1.3v and the min current is 10ma and you have a 5v pulse to drive it with, you subtract 1.3 from 5 (which is 3.7) and divided by the a current somewhat greater than the min (0.015) and that gives you 247 ohms. Now when you drive the gate through this resistor you'll get the required 10ma. You dont really think about the voltage though, except in the calculation of the resistor (and later the power). Once you do this calculation you can forget about the voltage. You can forget about it because it wont be 1.3v anyway
It will go to whatever the internals of the triac force it to be, but it will not go over a certain limit like around 1.5v or something like that. Because of this, you do have to check to make sure with the max gate current you still get the min current flowing into the gate.
There is one other consideration, and that is that we dont overdrive the gate. Although there is a min current and max current, if we go close to the max current we could inadvertently exceed the gate power dissipation. This means we might have to drive it with a pulse rather than a constant drive current.
In other words, we drive the gate like an LED with the added restriction that we dont exceed the gate power dissipation.
If you post the triac part number or a link to the data sheet we can take a look and probably help more with this.