The Use of Calculus in Video Games

Math is one of the foundations of video games. To put it in simple words. No Math = No Game development.

However, math is very diverse and has a lot of  branches. Some of these branches are extremely important to video games while others are rarely needed.

Where does calculus stand on this spectrum though? Well, Calculus is on the “Extremely important” side when it comes to game development.

Calculus is used as the foundation of some of the most important building blocks in video games. Video games’ lighting, collisions, physics engines, simulations and so much more rely on calculus to produce a good gaming experience.

Before we dive into more details about where calculus is used in the different areas of game development, I would like to point out that many of the heavy calculus calculations are usually handled by the “Game Engines” creators.

If you don’t know what a game engine is, it is basically a set of tools that game developers can use to make their development journey much easier than it would be compared to building their own tools from scratch.

That said, if you are planning to become an indie game developer yourself, and you’re worried about having to deal with calculus, I can assure you that you will be fine. Most of the heavy calculus calculations will be handled by the game engine that you use.

You will still need to know calculus to solve certain problems, but not knowing calculus won’t prevent you from building video games in general.

That said, the rest of this article will just list some of the areas that need calculus in order to function properly.

Where is calculus used in video games?

1- Calculus is used in collision detection.

In real life, if you try to push a wall, your hand won’t just go inside the wall. Who would have thought right? In video games, that can absolutely happen if there is no collision detection in place.

Collision detection in video games is the main reason behind objects not going through each other.

With the right collision detection algorithms in place, a bullet knows when it has hit an object and it reacts accordingly, the main character knows when it has hit the ground and it stops, and so on.

Calculus is used in some collision detection algorithms. Certain 3D collision detection algorithms use shape calculus to be able to detect which objects are colliding with one another, and lets the game react accordingly.

2- Calculus is used in lighting calculations.

Lighting is one of the fundamental concepts in video games. Video games will have to calculate how much light is emitted and/or reflected from the polygons that make up the objects in the game in order to properly display the game world on your screen.

Some of these calculations require calculus. Have you ever heard of the “Rendering Equation”?. It is basically an integral equation that deals with the light reflected and/or radiated from surfaces/objects.

Usually, rendering algorithms try to estimate the values of this equation to make lighting simulation/calculation as realistic as possible while keeping performance in mind. This involves a lot of calculus.

3- The physics engines in video games use calculus.

What are physics engines in the first place? Well, a physics engine is a set of tools that simulate physical interactions in a game.

For example, physics engines can handle forces, velocities, acceleration and so on.

Since calculus is basically all about the “rate of change of properties”, you can easily tell that it is heavily involved in the physics engines used by video games.

4- Calculus is used in fluid simulations

If you’ve ever played a game that contains some cool fluid simulations, these simulations probably involve a lot of calculus.

Fluid mechanics generally involves many of the fundamental calculus concepts.

5- Splines and curve paths use calculus.

In video games, splines are usually used to guide the AI in the game. Enemies usually follow paths and trajectories in the game. They might change their behavior when they encounter the player, but they usually go back to their planned routes / paths.

Dealing with these splines generally requires some calculus. I have never had to deal with the calculus behind these splines myself though since I usually use 3rd party assets that deal with the math behind the scenes, and I am very grateful that I can do that !

There are many more applications of calculus in video games. This is just a very small list compared to how much calculus is used in video games.

However, I’ll end this article here since I think we’ve covered enough applications to at least let you know how deeply integrated calculus is with the technologies behind your favorite video games.

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