I have set of points for ex. $A_0 (0,0); A_1 (1,2); A_3 (3,3);$ I need an object to travel between these points during some period of time.
I was able to construct this trajectory with Bezier curve and move the object by changing $x$-value from range $[0,1]. But I need that object to reach each point, so I'm guessing that the solution is some interpolation method, but not sure which one and how to calculate it :(
With three points you get a unique quadratic curve interpolating them. So, write $\gamma(t)=(x(t),y(t))$ with $x(t)$ and $y(t)$ quadratic functions of $t$ and solve the system $\gamma(0)=A_0$, $\gamma(1/2)=A_1$, $\gamma(1)=A_3$. You have a choice for the $t$ corresponding to $A_1$; choosing $t=1/2$ seems natural.
It is simpler to express $x(t)$ and $y(t)$ in the Bernstein basis instead of the monomial basis. Indeed, writing $\gamma(t) = P_0 (1-t)^2 + 2P_1 t(1-t) + P_2t^2$, we find $A_0=\gamma(0)=P_0$, $A_3=\gamma(1)$, and $A_1=\gamma(1/2)=(1/4)(P_0+2P_1+P_2)=(1/4)(A_0+2P1+A_3)$, from which we get $P_1=(1/2)(4A_1-A_0-A_3)$.