Position

Let the camera position be the three-dimensional vector $\mathbf{p}$, the position at the start of the transition be ${\mathbf{p}}_0$, and the $k$-th frame's target position be ${\mathbf{p}}_T^{(k)}$, with corresponding progress $p_k$.

Fixed-start interpolation

The fixed-start algorithm saves ${\mathbf{p}}_0$ and linearly interpolates it with the current target position each time:

$${\mathbf{p}}_k=(1-p_k){\mathbf{p}}_0+p_k{\mathbf{p}}_T^{(k)}$$

When ${\mathbf{p}}_T^{(k)}$ is constant, ${\mathbf{p}}_k$ parameterizes the line segment between the start and the target; when the target changes, only the current frame's target is substituted into the same formula.

Feed-forward with error correction

The feed-forward-with-error-correction algorithm first applies the target's inter-frame displacement directly to the previous frame's camera position, then corrects the remaining error. Let the previous frame's camera position be ${\mathbf{p}}_{k-1}$. First compute:

$$\mathrm{\Delta }{\mathbf{p}}_T={\mathbf{p}}_T^{(k)}-{\mathbf{p}}_T^{(k-1)},{\mathbf{p}}_k^{\prime }={\mathbf{p}}_{k-1}+\mathrm{\Delta }{\mathbf{p}}_T$$

Then compute the error between the temporary position and the current target:

$${\mathbf{e}}_k={\mathbf{p}}_T^{(k)}-{\mathbf{p}}_k^{\prime }$$

The correction ratio is the share of this frame's progress increment in the previous frame's remaining progress:

$$c_k=\frac{p_k-p_{k-1}}{1-p_{k-1}}$$

After clamping $c_k$ to $[0,1]$, the final position is:

$${\mathbf{p}}_k={\mathbf{p}}_k^{\prime }+c_k{\mathbf{e}}_k$$

When $p_k=1$, $c_k=1$, so ${\mathbf{p}}_k={\mathbf{p}}_T^{(k)}$. When the target is stationary, the recurrence is the same as fixed-start interpolation; when the target changes, $\mathrm{\Delta }{\mathbf{p}}_T$ carries the target's inter-frame displacement, while the error term gradually changes the remaining positional difference.