__Go to Index__In video games, or other areas that involve some movement and interactive actions, there is often a need to smooth out movement of some object. As a video game programmer we often stumble across an article or real-life code that tackles the problem in a way that is not satisfactory enough for us. Many people misuse linear interpolation (lerp) call for this purpose. Lerp is defined as follows: $$ lerp(A, B, t) = A + (B-A)t. $$ The problem goes like this: we want to move an object from current position to $X$ smoothly. What some people do, e.g. Unity developers in their Mecanim Animation Tutorial is they call the following code every frame: \begin{equation} \label{wrong} newPosition = lerp(oldPosition, X, a \cdot \Delta t),~~~~~~~~~~~~~~~~(1) \end{equation} where $\Delta t$ is the time last frame lasted measured in seconds and $a$ is a constant that state how fast the transition to $X$ should occur.

Note that given scenario allows $X$ to be changed while the time progresses, so that the object follows the $X$ position. Also, even though the code uses lerp function call, the movement is not linear in any sense of that word.

The thing that caught our attention and made us wonder is that interpolation parameter (the last argument of lerp function) can sometimes be greater than one, e.g. when $\Delta t$ is big because last frame lasted a long time for some reasons. We don't want that.

Given solution is wrong because the movement coded in such way is not independent from the frame rate. To see it clearly, let us rewrite equation $1$ by parametrizing it with $\Delta t$ and make it recursive sequence of positions at $n$-th frame. \begin{equation} \label{wrong2} P^{\Delta t}_n = lerp(P_{n-1}, X, a \cdot \Delta t)~~~~~~~~~~~~~~~(2) \end{equation} We assume for a moment for simplicity that $\Delta t$ and $X$ do not change during execution of the code. For our purposes, we can also assume that $P^{\Delta t}_0=0$ and compute the explicit form of equation $2$ (using our math skills or Wolfram Alpha). $$ P^{\Delta t}_n = X(1 - (1-a\cdot\Delta t)^n) $$ Now let us consider two special cases. Let $a=1$. One were code runs at $60$ FPS for $60$ frames and second one that runs at $10$ FPS for $10$ frames. $$ P^{1/60}_{60} = X(1 - (1-\frac{1}{60})^{60}) \approx 0.6352 X $$ $$ P^{1/10}_{10} = X(1 - (1-\frac{1}{10})^{10}) \approx 0.6513 X $$ Both cases last one second and it would be desirable if those values were the same. Thus, by example above, we see that this method is not frame rate independent.

Generally speaking, we want the following equality to be met \begin{equation} \label{req} P^{\Delta t}_{n} = P^{\Delta t \cdot 1/\beta}_{n \cdot \beta}~~~~~~~~~~~~~~~(3) \end{equation} for any value of $\beta$.

Once we have described and analysed the problem, we can try to come up with a better solution. We propose to use the following code: $$ newPosition = lerp(oldPosition, X, 1 - e^{-a \cdot \Delta t}) $$ where $e$ is the base of the natural logarithm and $a$ is a constant that denote how fast we want to interpolate the movement. Rewriting to a more concise form as we did before, we get $$ P^{\Delta t}_n = lerp(P_{n-1}, X, 1 - e^{-a \cdot \Delta t}). $$ Assuming initial conditions $P^{\Delta t}_0=0$, the explicit version can be computed (we used some math software again). \begin{equation} \label{good} P^{\Delta t}_n = X (1 - e^{-a\cdot\Delta t\cdot n})~~~~~~~~~~~~~~~(4) \end{equation}

Equation $4$ meets the requirement given by equation $3$: $$ P^{\Delta t \cdot 1/\beta}_{n \cdot \beta} = X(1 - e^{-a\cdot\frac{1}{\beta}\cdot\Delta t \cdot n \cdot \beta}) = X (1 - e^{-a\cdot\Delta t\cdot n}) = P^{\Delta t}_n $$ and therefore is independent from the frame rate.

To sum up, we propose to use the $1-e^{-a\cdot\Delta t}$ instead of $a\cdot\Delta t$ as the interpolation factor. It makes much more sense and the movement becomes more predictable when the frame rate varies.

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