So I decided to buckle down and write this. Hopefully it will be useful, maybe it won't, we'll see.
First some background on controls:
---> Mouse meet Touchpad, Touchpad meet Mouse
As there are both displacement pointing devices that serve the same purpose. Back in the days of mouse pads you could make this statement rather easily:
"Place your (mouse/finger) in the center of the (pad/touchpad) and move your (mouse/finger) in a NW direction. A movement of X units of your (mouse/finger) will result in your pointer/crosshair moving NW by Y units, where the difference between X and Y is the sensitivity." (Optical mice have brought an age where your "pad" can be quite irregular in shape, but the same principle still applies.)
FPS benefited from the use of a mouse in that if you see something onscreen that is X units NW, you brain knows how to translate that into Y units you need to move the pointing device to get there. Once you have displaced those X units, you also stay focused on that area. Also if you ran out of room to track you could always "lift the (mouse/finger), adjust to a new location, re-engage, and continue tracking."
---> Buttons are Buttons
Wether it be a face button on a controller, a thumb controlled D-pad, or an inverted T-formation of keys on a keyboard. They're all just buttons laid out in different fashions to make it easier to handle input.
---> The Analog Stick: Spring Based Pointing Done Right
So unlike a mouse, the analog stick does not follow the same principle of displacement. Don't get me wrong, you still displace the stick, but it doesn't correlate to a displacement on screen. Instead, it correlates to how quickly you would like the displacement to occur and in what direction, and springs are used to center the stick to a neutral position.
---> Touch Screen as a Analog Stick
So there are several immediate ways of attempting to make a touch pad an analog stick equivalent.
1) Touching the pad anywhere is equivalent to "grabbing" a stick in the neutral position. So in essence, you just determined the neutral position of the stick. Any movement away from that point is equivalent to pushing the stick in that direction. Returning to the neutral position either consists of moving your finger back to the point you first touched, or removing your finger (resulting in letting the stick 'snap' back to neutral).
2) Same as one, but letting go doesn't snap the stick to neutral (instead some other input, like a button is needed to return to neutral).
3) The center of the screen is the neutral position of the stick. One then can move left for example by either touching the center and moving their finger left, or just by touching the left side of the screen.
All these approaches rely heavily on having an effective "dead zone" (or area that is considered neutral). It can't be too big, because then it seems unresponsive. Too small and it becomes difficult to control. In addition, the size of that zone is also proportional to what is being used as a stylus.