Euler Systems

One method of describing 3D rotation is Euler Angles. An Euler Angle is a tuple of 3 elements (denoted here as α, β, γ) each of which are meant to rotate a vector around a particular axis. Though the system is easy to understand mentally there are two main problems with using Euler Angles.

1. Order

Each angle in the tuple represents an angle around an axis. Implicitly most people will assume this means (x, y, z) as they are familiar with in Math classes, but in reality this system is not common. The main systems and their uses are enumerated here.

2. Relative vs. Absolute Rotations

If we were to take X as North, Z as East, and Y as Up and were on the ground directing a plane flying west to change pitch by 15deg (that is rotate 15deg around X) then both the plane would easily be able to adjust to what we were describing. From there we could say to adjust roll (in this case the Y axis) by 15deg. However we have a problem, a geometer would mean rotate about Y (AKA East) by 15deg while a pilot would take that to mean rotate about the forward direction by 15deg. Because the plane is pitched by 15deg these would not describe the same rotation. We call this relative vs absolute/static frames of rotation. In relative systems the next rotation in an Euler transform is around the axis of the object being rotated while absolute frames are from a global perspective.