I have posted my initial question question "Arc-Length parameterization of a cubic bezier curve" here.
I was a bit confused with the mathematics but now i have understood most of it, only one question remains which i think is better to posted in the Mathematics forums.
The paper i use as a basic, describes the calculation of the Arc-Length of a curve segment as follows: $l_i = \int_{t_i}^{t_{i+1}}\sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2}dt$, where i is the i varies from $0$ to $n-1$ and $n$ is is the number of segments in the curve. Therefore the total Arc-Length of the curve would be $\sum_{i=0}^{n-1}l_1$
Now to calculate that integral and implement it, i found a website which tells me to use the Gauss Quadrature and shows:$\int_{0}^{z}\sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2}dt = \frac{z}{2} [C_1 f(\frac{z}{2} t_1+\frac{z}{2})+\cdots+C_n f(\frac{z}{2} t_n+\frac{z}{2})]$, where z is from $0$ to a value smaller or equal to $1$.
In order to implement it, the second solution seems what want to go for. The problem is it always calculates the Arc-length for $0$ to the time specified in $z$ of the original curve.
My question can i use the second formula or modify it so that i can use $\int_{t_i}^{t_{i+1}}$ where $t_i$ would be the time of a segment point and $t_{i+1}$ the time of the next segment point on the bezier curve.