[toxik]

My thoughts are the same direction. This seems like a lot of analysis for something you ultimately don’t want, you want to integrate the camera motion together with the other physics integrations. Especially if you want the camera to be repulsed by objects in the game, which is typically what you want. This then is an n body problem with no analytical solution.

[regularfry]

I was sort of hoping that the closed form analysis was going to be aimed at getting the critical damping factor to then plug it into a dumb Euler simulation so the behaviour is correct. Kinda disappointing that the final thing is so complicated.

[EsportToys]

if you are only every gonna use critical damping, you can do something like:

    function crit_response(dt,pos,vel,rate)
       local dissipation = math.exp(-dt*rate)
       local disspatePos = pos*dissiptation
       local disspateVel = vel*dissiptation
       local posCarried = 1 + dt*rate
       local velCarried = 1 - dt*rate
       local posFromVel =     dt 
       local velFromPos =    -dt*rate*rate
       local newpos = posCarried*dissipatePos + posFromVel*dissipateVel
       local newvel = velFromPos*dissipatePos + velCarried*dissipateVel
      return newpos , newvel
    end

(intentionally verbose for self-didactic purposes to show how you can actually split the dissipation step into its own loop out from the elastic calculation, meaning that you can use this as a more accurate initial guess in numeric simulations compared to just constant-acceleration approximations)

[regularfry]

I can see an argument for wanting to go slightly to one side or the other of the critical factor for aesthetic reasons, but yes. If you want "critically-damped spring behaviour" you don't have to go the long way round to get it.