particle - interpolate

Question

I want to make a particle that interpolates in Alpha and velocity.

public class Particle extends GameObject implements Updateable
{

    int time;

    int lifetime;

    int Alpha;

    double dx;

    double dy;

    int[][] alphaEnvelope;

    double[][] speedEnvelope;


    @Override
    public void update()
    {
        time++;

    }


}

this is my particle class.

alphaEnvelope[x][y] changes the Alpha over time where X is the time and Y is the Alpha Value.

For example:

[1][100]

[60][70]

[70][0]

the Alpha is 100 at lifetime 1, 70 at lifetime 60 and 0 at lifetime 70.

How do I interpolate between these values fluently?

Answer 1

Linear interpolation you can do with following code:

int l = 0;
for(int i = 0; i < keys.Length; i++)
{ 
if(time > keys[i]) l = i;
}
double interval_len = keys[l + 1] - keys[l]; //r = l + 1
double k2_a = (time - keys[l]) / interval_len; //k1_a = 1 - k2_a
return k2_a * values[l+1] + (1 - k2_a) * values[l];

It is very similar to your code - you were on the right track. Code explanation: first we find between which keys the time lies, then we find the interval length, after that we subtract left value from time to get "local time on interval". And lastly we find "how much close" it is to right boundary by dividing local time by interval length. We multiply right value by the result and finally we multiply the left value by complement and sum it for desired result.
If you wanted more smooth values you could for example interpolate by cubic spline solving system of linear equations (not hard to find, you will be likely looking for chord parameterization and natural spline bounding conditions).

Answer 2

You can try some polinomial interpolation : i.e lagrangian interpolation

In your example it becomes (if I calculated it right ):

L(x) = 100 * ((x-60)/(1-60)) * ((x-70)/(1-70)) +
       70 * ((x-1)/(60-1)) * ((x-70)/(60-70)) +
       0 * ((x-1)/(70-1)) * ((x-60)/(70-60)) +