Interpolation curve for animation

[BigMoosie]

Hi,

 

I am designing an animation library for the web, and I wanted to create an animation interpolation that is more flexible and elegant than using 'sine' or similar.

 

I have spent countless hours trying to solve this, so I'll be extremely thankful for any help offered.

 

My system I devised is this:

 

A partical is at position 0, with velocity 'v' and accelleration 'a'. I want it to move along a function to land at position 'h' with velocity 0 and accelleration 0.

 

Simply put, I want to get a moving object to land somewhere stationry. 'v' and 'a' are provided to the function as attributes so that if the target moves then the curve can adjust without any abruptness.

 

I figured a polynomial of degree 5 would be the simplest curve to do so:

 

[math]f(x) = Px^5 + Qx^4 + Rx^3 + \frac{a}{2} x^2 + vx[/math]

 

If I let the amount of time for the object to land still be t, and using my constraints from above I can get the following equations:

 

[math]Pt^5 + Qt^4 + Rt^3 + \frac{a}{2} t^2 + vt = h[/math]

 

[math]5Pt^4 + 4Qt^3 + 3Rt^2 + at + v = 0[/math]

 

[math]20pt^3 + 12Qt^2 + 6Rt + a = 0[/math]

 

This is a piece of cake to solve if I know what t is. But I want 't' to be automatic, though of course then I have 4 unknowns with 3 equations. The final constraint I want to impose is this:

 

If:

v = 0

a = 0

h = 1

Then t = 1

(see diagram)

 

[math]f(x) = 6x^5 - 15x^4 + 10x^3[/math]

 

And, given that function, if the animation is paused and a new function is calculated using the same current velocity, accelleration and relative displacement from target 'h', then it should still arrive at the same time.

 

That is, if I update the target position on each frame even though the target has not moved, then the total time taken for the animation should not be comprimised.

 

As far as I can see this should be enough to result in a unique solution for 't', that solution is all I need.

 

If there is a more appropriate system, such as some kind of spline, then please mention it.

 

Thankyou,

BigMoosie

Edited January 17, 2009 by BigMoosie
Add math markup