Rotation formalisms in three dimensions

Okay kiddo, so let's say you have a toy car and you want to turn it around. You can turn it in different ways, right? For example, you can turn it by moving your hand in a circular motion, or by tilting it to the side and then turning it.

Now imagine that you have a shape that you want to turn around in three dimensions (which means it can move in three different directions). To describe this turn, we use something called a "rotation formalism."

There are different types of rotation formalisms, but some of the most common ones are called Euler angles, quaternions, and rotation matrices. Each of these formalisms helps us describe a turn in a different way, kind of like different ways to turn your toy car.

Euler angles are like turning your toy car by tilting it and then turning it. We use three numbers to describe this type of rotation: one for each axis (x, y, and z) that we can turn around. Each number tells us how much we've turned the shape around that axis. So if we turned it 30 degrees around the x-axis, 60 degrees around the y-axis, and 90 degrees around the z-axis, we would write it as (30,60,90).

Quaternions are like turning your toy car in a circular motion. They use four numbers to describe the rotation, but they're a bit more complicated than Euler angles. They're like magic numbers that help us turn the shape around smoothly without any weird twists or turns.

Finally, rotation matrices describe a turn by using a matrix (which is like a grid of numbers) to show how the shape's coordinates change as it turns. This one also has three numbers (one for each axis), but it can be harder to understand because it uses a lot of math.

So, rotation formalisms are different ways to describe how a shape turns around in three dimensions. Just like you can turn your toy car in different ways, these formalisms help us describe turns in different ways depending on what we need them for.