Math in Movies: Frames, CGI, and Stunts

Last summer I filmed my kids jumping off the low diving board with my phone set to slow motion. On the screen they hung in the air as if they had forgotten gravity, arms out, water beading off their shoulders. My son watched the replay and asked the question that started this article: did the phone record more, or just play it slower? Both, I told him, and the answer turned out to be a small pile of arithmetic we did together on a scrap of paper by the pool.

A film is not a moving object. It is a stack of still images. Show the eye 24 pictures every second and the brain blends them into motion. That blending is persistence of vision at work, but the real trick is the rate: change the rate and motion feels completely different. The standard for theatrical films has been 24 frames per second for decades, while phone video and TV often run at 30 or 60. Faster rates look smoother but change the feel of a scene, and slower rates turn jumpy. When we got home I checked the clip details with my son: 240 frames per second recorded, played back at 30. One second of jump had become eight seconds of airtime, and he has quoted that number at every pool visit since. The choice of rate is mathematical first and artistic second.

Frame rate is a ratio

Frame rate is a ratio of frames to time. At 24 frames per second, 240 frames make 10 seconds of footage. If you shoot at 240 frames per second and play it back at 24 frames per second, you play each second of recorded motion in 10 seconds. That is why a splash looks impossibly graceful in slow motion. You are just stretching time by a factor of 10.

This is pure multiplication and division. Shooting rate divided by playback rate gives the slow-motion factor. Here is a quick way to see it:

The math tells you exactly how much time you are stretching before you ever look at the footage.

There is also something called shutter angle. On a traditional film camera, a rotating shutter blocks light for part of each frame. A 180 degree shutter means it is open for half of each frame's time, which tends to look natural at 24 frames per second. Change that angle and motion blur changes. More blur looks softer and dreamier, less blur looks crisper and more video-like. It is another ratio, this time between shutter time and frame time.

Early sound films helped lock in 24 frames per second as the theatrical standard. It was a balance: fast enough for smooth motion, slow enough to be affordable on film stock, and just right for recording and playing back optical soundtracks. Audiences grew used to that specific look, and it stuck. That means 24 fps is not just a technical choice, but a historical one that became the default.

Our guide to ratios and proportions makes this kind of scaling obvious, and movies are a perfect place to see ratios create an effect.

Time, distance, and stunts

A car jump looks dangerous for good reason, but it is also a distance-rate-time problem. How far does a car travel in the air? Speed times time in the air. How fast is it going when it leaves the ramp? That depends on the ramp angle and speed. Stunt teams spend enormous effort calculating those numbers because guessing is not an option.

Angles matter too. A small change in ramp angle changes the trajectory a lot. If the ramp is too steep, the car goes up too high and runs out of time in the air. If it is too shallow, it may not clear the gap. Add in wind, weight of the vehicle, and where the wheels will hit, and you are doing a chain of calculations with very little room for error. The stunt might look reckless, but underneath it is careful arithmetic.

Timing is just as important. If an explosion needs to happen a fraction of a second after a person passes a mark, that is down to speed and distance. If someone is running at 5 meters per second and needs to cross a mark 10 meters away, that is 2 seconds. Set the explosion for 1.9 seconds and they miss it. Set it for 2.1 seconds and it is too late. On set, those fractions of a second are measured and rehearsed over and over again until they are right.

Walk a few steps, time them, and you can work out roughly how long it takes to cross a mark. A film set does the same thing, only with far less room for mistakes.

CGI is geometry at scale

Computer generated images are built from geometry. A character model is a mesh of triangles. Move a joint, and the triangles move. Add texture, lighting, and shadows, and it looks more real. None of that happens by drawing every frame by hand. It happens by applying math to those shapes.

Perspective is a big part of it. To make a giant monster look like it is stepping over a real building, the CGI must match the camera angle and lens of the live action shot exactly. The size of objects changes with distance, and a model scaled incorrectly will look tiny or enormous no matter how detailed it is. Our geometry guide shows why angles and positions behave this way. Get the angle wrong by even a little and it looks fake. Get it right and our brains accept it without question.

Motion also has to follow physics. A falling object accelerates, cloth drapes differently than rigid metal, water splashes in predictable ways. Simulating those things uses ideas about motion that show up in our distance-rate-time guide and in geometry. Good CGI is not about adding more detail, it is about getting the math of motion right.

Even blending matters. The edges between real footage and CGI must match in color, focus, and motion blur. If the real plate was shot at 24 frames per second with a certain shutter angle, the CGI has to be rendered to match those same numbers. Consistency of the frame rate and blur is what sells the illusion.

Illusion by numbers

The magic of movies is that we know they are fake, but we still believe them. The reason is consistency. 24 frames per second give just enough change between frames for motion to read. A slow-motion shot keeps proportions consistent because it is a scaled version of the same motion. A stunt works because the timing matches what we expect from real objects moving at those speeds.

Our brains are very good at picking up when something feels off, even if we cannot explain why. If motion looks too smooth or too choppy compared to what we expect, the illusion breaks. If a jump looks like it takes far too long or far too little time for the distance shown, we notice. The math is what keeps all of those expectations aligned.

Break any of those ratios and the illusion falls apart. Play at 6 frames per second and it looks like a slideshow. Get the scale wrong in CGI and the monster looks toy-sized. Mistime an explosion and it feels off even if we cannot say why.

Movies are a constant negotiation between math and art. The math sets the rules for what can look real. The art decides when to follow those rules and when to bend them just enough to be more exciting.

Try it yourself

Take a short video on your phone and play it back at half speed if possible. Count roughly how many frames pass in a second at normal speed, then see what slow motion does to a quick motion like water drops or a ball bounce. You are seeing the ratio in action. Try to work out the slow-motion factor from the numbers.

Or think about a movie chase scene. How long would it actually take to jump between rooftops at those speeds? Pick a rough distance and a rough speed, and do a distance-rate-time estimate. A quick estimate with distance and time will show you where the movie is stretching reality for drama, and where it is keeping the numbers believable.

Once you start asking those questions, you cannot unsee it. The action is still thrilling, but now you can also see the arithmetic holding it together.