# Kenton's Programming Blog

Posts tagged with Math:

# Building a Collision Engine Part 3: 3D GJK Collision Detection

Now that I've conquered 2D collision detection and intersection calculation, it's time to move onto 3D. Thankfully, GJK is relatively easy to extend into 3D once we have the base framework that we developed before.

# Building a Collision Engine Part 2: 2D Penetration Vectors

In my last post, I discussed how to determine if any two convex shapes in two dimensions overlap. The result of this calculation is a boolean state—either the two are colliding or they aren't. This is great for things such as triggers, but if you want to use the collision engine for any type of physics calculations, you'll need at least one more crucial piece of information: the penetration vector. This post discusses how to calculate the penetration vector using the EPA algorithm.

# Building a Collision Engine Part 1: 2D GJK Collision Detection

I've previously written about using the Minkowski Difference to detect collisions of 2D AABBs, but I now want to expand this into creating a fully fleshed out and flexible collision engine for my own purposes. The engine will detect collisions using the GJK method, and calculate intersections using the EPA method. This post details how 2D GJK works, which will serve as a basis for getting the rest of the engine up and running.

# Quaternions as Four-Dimensional Complex Numbers

Although I have a pretty solid background in math (especially vectors, matrices, and even tensors), I've always somewhat struggled with quaternions. Most sources focus on quaternions as some tool for performing rotations in three-dimensions while avoiding gimbal lock. Which is true, they are/ that, but they're also more. After reading several articles about quaternions over the past several days, quaternions finally clicked and made sense! I'll try to share that insight with you here, though be warned that my description may be just as confusing (if not more so) than anywhere else.

# Swept AABB Collision Detection Using the Minkowski Difference

Continuing on from yesterday’s post where I explored detecting discrete collisions using Minkowski differences, today I’m going to talk about detecting continuous collisions using Minkowski differences (again, focusing solely on axis-aligned bounding boxes). Continuous collision detection is essential in any game where you have fast-moving objects and/or low frame rates. It adds slightly more complexity to the discrete collision detection algorithm, but the advantages far outweigh the costs in this case!

# Simple AABB Collision Detection Using the Minkowski Difference

Since I’ve started on an adventure to start creating my games with Haxe and OpenFL, I found myself in need of some collision detection. I don’t really need anything as fancy or extensive as Nape, and although the HxCollision library is a pretty solid Separating Axis Theorem implementation, it doesn’t deal with swept-collisions, which is a bit of an issue for games (without swept collisions, any lag spikes can easily cause objects to pass right through objects!).

# Simple Bézier Curves in Matlab

I've always been curious about how [Bézier] cubic splines are generated and I how I can use them in various projects (game development probably being the most immediately obvious). If you don't know what I'm talking about, Wikipedia has a decent if somewhat tedious description. After digging through the math, I came up with these results which I'll share for the simplicity of it all (they really are simpler than I ever though). I found most sources started at the beginning and gave huge mathematical backgrounds, which although are nice, are typically just not what I was looking for. What I was looking for (and what is presented here) is just the end result - given a set of input weights, how do I calculate the actual spline?

# Super Mario World Physics

Arguably one of the greatest video game series of all time, I spent countless hours playing these games as a kid. One thing I always particularly loved about the games is that the kinematics / physics of Mario's motion were just so.. fun! Not necessarily that they felt real, but when you ran, you ran fast. When you jumped, you jumped high, etc. I think this is a problem that I have with numerous other platforming games.. yea, sure, the gameplay can be fun, but moving around just seems so... boring in comparison to Mario.

# Solving Equations Using the Newton-Raphson Method

Computers are great, but as it turns out - they're not always the smartest of folks. However, they are great at doing simple math! Today, I'll show you how to exploit these silicon monsters to do something that sometimes humans even fail at: solving a simple non-linear equation.

# Solving Systems of Partial Differential Equations

Systems of partial differential equations crop up all the time in engineering, especially when examining real-world complicated problems that vary in time (such as a ballistic trajectory with drag forces non-negligable), or in various process control systems (ex: relating flow conditions in systems of tanks with the height in those tanks).