EULER ANGLE TO CARTESIAN COORDINATES

If you work with technical documentation from skilled laborers or engineers, you will often find jargon that makes the math of the trade the easiest to understand in implement. Unfortunately this usually leaves those outside the trade (e.g. software developers, journalists) with a learning curve. If you have watched Top Gun, or played a video game that included a flight sim, you have been exposed to terms like: yaw, pitch, and roll; these are an example of Euler Angle notation. Below is a table that describes a few scenarios with their respective notation:

Scenario Airplane Directions Angular velocity Telescope Position Associated Symbol
Rotation about z heading yaw azimuth θ
Rotation about y attitude pitch elevation φ
Rotation about x bank roll tilt ψ

Visual Example of yaw, pitch and roll from wikipedia user Auawise

It may be more convenient mathematically to use Cartesian coordinates. In order to do this, simply apply the rotation matrices. If you have forgotten your linear algebra here they are below:

The final rotation being

 

 









https//www.staff.city.ac.uk/~sbbh653/publications/euler.pdf

https//www.euclideanspace.com/maths/geometry/rotations/euler/