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Mathematical model of the Rolling Shutter Effect on propellers (2 propellers)

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:

daum_equation_1429012013318

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

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:

Rolling Shutter Effect

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):

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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.

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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 \({v}_{s} = 1\)

deleteSince \({v}_{s} = 1\), 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 \(w \times t + { \theta  }_{ i }\).

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

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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 \(t\):

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

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

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

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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:

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

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Done. If you have any questions you can email me at nailuj.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.

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