First tings first, I am going to use LaTeX code to express **important** mathematical expressions along with pictures. One of the reasons is because this post may be better understandable with a graphing calculator, preferably https://www.desmos.com/calculator because it uses LaTeX coding, so feel free to click on the simulators given in this post and typing your own equations for a better understanding of this post. Here is an example of the function in picture and the function in LaTeX:

\(\left(\frac{x}{\left|0.3f\left(g\left(a\right)\right)-2\right|}+0.1f\left(g\left(a+\mod \left(a,2\right)\right)\right)\right)^2+\left(\frac{y}{\left|0.3f\left(g\left(a+2\right)\right)-2\right|}+0.1f\left(g\left(a+\mod \left(a,2.5\right)\right)\right)\right)^2<2\)

This function is taken from another simulator I built a longtime ago https://www.desmos.com/calculator/clr9waooqe.

Occasionally, I will be using only LaTeX code and when I use it, it will be enclosed in \(…\) or \[…\]. If you can’t read basic LaTeX code, you can enter https://www.desmos.com/calculator and paste the code into the text box.

**Back to the topic!**

You might have seen something like this in a photo:

Notice how distorted the propellers are. This is because the propeller is moving very fast, and the shutter takes the picture from left to right/right to left. I would not explain much on the rolling shutter effect because information about it can already be found online.

To get you started, here is the Rolling Shutter in Wikipedia: http://en.wikipedia.org/wiki/Rolling_shutter

And here is a simulator I built using Desmos Graphing Calculator (The second most powerful online graphing calculator I know) to give you a visual sense of this effect: https://www.desmos.com/calculator/kxiqxtbaj3

I will be explaining on the derivation of the functions in the simulator.

**Derivation of the equation of the shape of the distorted propeller** (which is the curved black line in the pic below)**:**

So, lets first assume the picture as a square with corners (1,1), (1,-1), (-1,1), (-1,-1) on the xy coordinate plane and the shutter moves from left to right.

Ok lets define

\({v}_{s}\) as the velocity of shutter

\(w\) as the angular momentum of propeller (Radians per unit time)

\({ \theta }_{ i }\) as the initial displacement of propeller before the picture is taken (in radians)

These definitions are the same as I used in the simulator, so you can check it out above to understand what I mean. Note that I used time as \(a\) in the simulator instead of \(t\) which I would be doing so in this post.

**Essential**: Radians is just another way of expressing angle. E.g:

360 degrees corresponds to 2pi in radians.

180 degrees corresponds to pi in radians.

And so on.

Now, we can eliminate \({v}_{s}\) as what matters is the ratio of \({v}_{s}\) and \(w\). So for now, we can assume \({v}_{s} = 1\)

Since \({v}_{s} = 1\), after time \(t\), the shutter would have moved by \(t\) towards the right from \(x=-1\), so the equation of the shutter would be \(x = t – 1\).

Now for the propeller. So now, after time t, the propeller would be displaced by \(w \times t + { \theta }_{ i }\).

So the equation of the propeller would be \(\tan \left( -wt+\theta _{ i } \right) x\)

Now… here is the interesting part. Every point on the distorted shape of the propeller is where the propeller and the shutter intersect at a certain point in time. Since the shutter displacement is equal to the value of \(t\), we can find the point at which it intersects at time \(t\):

\(y=\tan \left( -wt+\theta _{ i } \right) x\\ x=t-1\\ \boxed { x=t-1\\ y=\tan \left( -wt+\theta _{ i } \right) (t-1) } \)

So now, it is clear that the point at which the propeller and the shutter would intersect at time \(t\) is given as:

\(\left( t-1,\tan \left( -wt+\theta _{ i } \right) (t-1) \right) \)

For those who recognise it, yes it is a parametric equation.

So now for solving the parametric equation:

And so the boxed equation above is the equation of the shape of the distorted propeller with \({v}_{s} = 1\).

However, we are not done yet.

To take \({v}_{s}\) into account, all we have to do is to divide every \(w\) in the equation with \({v}_{s}\), so the full equation would be:

\(y=x\tan { \left( -\frac { w }{ { v }_{ s } } (x+1)+{ \theta }_{ i } \right) } \)

Done. If you have any questions you can email me atnailuj.noop@gmail.com

Thanks! (I think you can guess who I am)

Update: I found a LaTeX to WordPress converter thanks to Mr Damien Boh. I’ll be using that in the future.