The mathematics of special effects

When you think of Hollywood you might not think of maths, but maths can help us to create some pretty neat special effects. If you’ve ever wanted to fake your own moon landing, or film giants, then this is the post for you.

Before we get stuck into the fun stuff, the proper name for the particular bit of maths we’re using is the Buckingham \pi theorem.

The end result of using some physical intuition and the Buckingham \pi theorem, is the following equation:

{R}/{\sqrt{\frac{g}{L}}}=constant,

where R is the frame rate, g is the acceleration due to gravity and L is a length scale (i.e. it represents the size of the object in the film). As long as we are filming people on earth, we can’t change gravity or the length scale, but we can change the frame rate. Now we’ve covered the background, let’s create human shaped giants, and simulate being on the moon.

Filming giants

To create a giant we need to change the length scale L . This means that the frame rate R needs to be adjusted, since gravity, g, is constant. Equating the special effects and model variables give this equation:

R_{fx}\sqrt{L_{fx}}=R_{m}\sqrt{L_{m}}

where subscript m means it is the value for the model and fx refers to the special effects value seen in the film.

To make our giant ten times bigger than a normal person, we need L_{fx}=10\times L_{m}. This means we end up with:

R_{fx}\times\sqrt{10}=R_{m}

To create a giant we need to film a standard sized person at a frame rate that is approximately 3.2 times faster than normal and then play the film back at normal speed. You can see this technique being used in Jason and Argonauts.

Fly me to the moon

NASA didn’t fake their moon landings, but can we fake one? This time we want to change gravity, but keep the length scale the same. Gravity on the surface of the moon is about one sixth of the gravity here on earth. Using the same approach as last time, but cancelling out L instead of g , we get:

\frac{R_{fx}}{\sqrt{g_{fx}}}=\frac{R_{m}}{\sqrt{g_{m}}}

To make our moon landing footage look realistic, we need to set g_{fx}=g_{m}/6, which gives:

R_{fx}\times\sqrt{6}=R_{m}

So, rather than spending all that money on rockets and scientific advancement all NASA needed to do was film some guys in space suits at a frame rate 2.5 times faster than normal and then play it back at normal speed.

Here’s a video of one particularly cool astronaut John Young (you should really check him out) with me clumsily spliced into the film, and it looks like I’m on the moon! Sort of. The sandstone wall and parking spaces might give the game away…

P.S. just in case you think the moon landing was faked, it was not. Also, giants aren’t real.

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