When we determine the natural frequency of free vibration for a system, we start of by defining a reference from where displacements will be measured. I have usually seen in books, taking the reference corresponding to the equilibrium position of the system. Is this compulsory? Or the reference can be arbitrary?
So you need a point, "zero point" as you really want the amplitude of the displacement as the conditions vary.
if you use a moving point to measure another moving point then you have to recalculate to get the simple displacement. As long as you get the equations of movement correct the result will be the same.
As a different example, we were plowing at night. We were told to"line up" with an identifiable point on the horizon to keep the first plow line straight (this was before all the gps in tractors). My mate picked a white dot then proceeded to plow all night. In the morning his first line was found to be a curve... Lots of hilarity as he had a lot of short work to do (this is where you spend more time positioning compared to working - costs you money as you are paid for ground covered). We realised that the white dot he had picked was a sheep gently grazing across another field :) Not a fixed point then...
Any point in a wave can be arbitrarily chosen as the reference and it wouldn't change the frequency and amplitude. but it would change the phase of the wave which is essential in complex analysis and tensor calculus.
y = A sin(B(x + C)) + D
A is th amplitude
Period = 2π/B
phase shift is C (positive is to the left)
vertical shift is D
it is interesting that if you have two identical waves with a half period phase difference and you add them together the sum will be a straight line. This is the basis of noise cancelling.
Other uses of phase angle difference is in electrical motors and the key to Tesla engine.
If you add two identical waves with same phase angle you get the same frequency twice amplitude.