Euler angles


Animation example of an intrinsic Euler rotation

Intrinsic Euler rotations means that the successive rotations occur about the axes of a moving coordinate system


Extrinsic Euler rotations, where the rotations occur about the axes of a fixed coordinate system

Explanation of the Gimbal lock problem

… from a mathematically viewpoint for Z-X-Z Euler angles as the loss of 1 DOF in 3D rotation since alpha and gamma have the same meaning, if beta=0: see here


Axis+angle & Euler angles explained

… very nicely!

How to get unique Euler angles?

restrict the angle ranges!

see here

As a consequence, there is a one-to-one correspondence between
Euler angles and rotation matrices only if the Euler angle domains
are restricted, e.g. to ...
alpha in [0,2PI]
beta in [0,PI]
gamma in [0,2PI]

Find equivalent Euler angles

see here:

There are three equivalences, one obvious, another less obvious
and a third only applicable in certain circumstances.

The obvious one is that you can always add multiples of 2π to any of the
 angles; if you let them range over R, which you must if you want to 
 get continuous curves, this corresponds to using R3 as the parameter
 space instead of the quotient (R/2πZ)3. This equivalence is easy to
 handle since you can change the three angles independently, that is,
 if you change one of them by a multiple of 2π, you directly get the
 same rotation without changing the other two parameters.

What's less obvious is that (referring to this image) the transformation
 (α,β,γ)→(α+π,−β,γ+π) leads to the same rotation. (This is why, in order
 to get unique angles, β has to be limited to an interval of length π,
 not 2π.)

A third equivalence comes into play only if β≡0(modπ), since in this 
case α and γ apply to the same axis and changing α+γ doesn't change
 the rotation. If your rotations are arbitrary and have no reason
 to have β≡0, you won't need to consider this case, though it may
 cause numerical problems if you get close to β≡0 (which is one
 good reason to use quaternions instead of Euler angles).

These three transformations generate all values of the Euler angles
 that are equivalent to each other. Remember that you also have
 to consider combinations of them, e.g. you can add multiples
 of 2π in (α+π,−β,γ+π) to get further equivalent angles.
public/euler_angles.txt · Last modified: 2014/01/19 11:40 (external edit) · []
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