Hi, I only join today and I can't make posts yet. I have a solution to your Curve Movement problem.
So you have you want to go from A to C via B, so you want to start at A at time 0, then at B at time 1/2 and finally C at time 1.
So if you make a chart you want it to look like this
T = 0 P = AX1+BX0+CX0
T = 1/2 P = AX0+BX1+CX0
T = 1 P = AX0+BX0+CX1
To get the correct distributions of 0 and 1 you now require three factors, (t), (t-1/2) and (t-1), these factors equal 0 at t=0, t=1/2 and t=1 respectivly. So If you use them like this you get the 0's at the right points.
A*(t-1/2)*(t-1) + B*(t)*(t-1) + C*(t)*(t-1/2)
But this doesn't yet give you the values of 1 where you need them so you need three magic numbers, M1, M2 and M3. You can prove that for this case M1 = M3 = 2 and M2 = 4.