What are the equations/algorithms used by the VVVV Damper node?

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the mathematical details are in here: mathematica notebook] (by gregsn).
it can be opened with the freely avaliable mathematica player app.
the document was the actual working notebook to develop the formulas and is not embedded into a explaining paper.

so a few words to main idea here:

Develop an damped oscillating system where you can specify the position to go to with some intuitive parameters for the behaviour of the system.

The main requirement was like in Newton (Animation) or DeNiro (Animation): respect the current position AND VELOCITY to get smooth transition when the new destination is picked in the middle of a running animation. This can frequently occur in interactive systems...

So we have an intial position p0, an initial velocity v0, a position to go to: p1.

Since all the people outside tell that a springs behaviour can be expressed by a differential equation, where acceleration at time t is dependend from velocity and position at that time t. mathematica seems to be a good option to the hard work. And it works: you can work with equations like that without having studied them.

this is the setup of the differential equation. it should be a curve going up and down around p1 (the goal). so p1 - p[t] is a kind of normalization.

we want that that curve starts at the current position and current velocity. we want to be able to specify them. lets name them p0 and v0. so at time t=0:

p[0] == p0

p'[0] == v0

we get a function t] and v[t describing the position and velocity over time. so we need to store those two functions in a way that we can sample the functions in the future at time t.

at the end of the document we have developed some simple imperative commands / terms which deconstruct the complicated functions into some simple functions depending on variables which only need to be calculated once (when a new curve is generated).

at the end we need to get a new funtion curves each time a parameter or the destination position is changed.