1. Rotation Representations
notes

2. Rotations Differ from Translations
Rotations are non-Euclidean
like travelling on a globe vs. a grid
Rotations are not commutative
x-rotate, y-rotate is not equal y-rotate, x-rotate etc.
Rotations are non-linear
true of all parameterizations other than trivial SO(3)
Rotations are not very intuitive, because when it comes to space, we think linearly.
For example, if you get in a rocket ship and start moving to the left, you’ll keep going forever, never revisiting anyplace you’ve been, unless you change directions. This seems correct. However, if I continuously increase a rotation, I will keep coming back to my starting orientation an infinite number of times, because a rotation of zero degrees is equal to a rotation of 360 degrees.
This is similar to moving on a sphere… and is a consequence of rotations being non-Euclidean.
Secondly, if I give someone directions for traveling on a grid, and tell them their destination is 5 blocks to the left and 4 blocks up from here, it doesn’t matter whether they first go 5 blocks over and then 4 blocks up, or first go 4 blocks up and then five blocks over. But rotating first by 40 degrees around the X axis and then 50 degrees around the Y axis is very different from rotating 50 degrees around the Y axis and then 40 degrees around the X axis.
And finally, rotations are non-linear functions of their parameters, whereas translations are linear functions.
OK, so rotations are different from more intuitive translations in a number of ways. Why is that a problem?

3. Rotation Parameterization
Represent rotation space in Euclidean R3
e.g. Euler angles, exponential map
Pros
three parameters for three DOFs
Cons
singularities, potentially poor interpolation
Parameterizations of rotations fall into two major groups - those that use Euclidean params, and those that use non-Euclidean.
The first group represents a non-Euclidean 3 DOF rotation with 3 Euclidean parameters by “flattening” the rotation space. This sounds strange, but it’s very similar to what map-makers do when they attempt to represent the surface of the Earth on a grid by distorting the geometry as you get close to the poles.
We’ll examine two different parameterizations that take this approach: Euler angles and the exponential map.
The advantages to this approach are that we need only three parameters to represent three rotational DOFs, and our tools are happy because our parameterizations are Euclidean.
Unfortunately, there are also problems, namely singularities and potential problems when we interpolate in these parameter spaces. We’ll examine this problems in detail when we talk about the individual parameterizations.

4. Rotation Parameterization
non-Euclidean space
e.g. unit quaternions (S3)
Pros
singularity free
Cons
must take extra measures to stay in legal sub-space
four parameters required for three DOFs
Our other alternative is to use a non-Euclidean parameter space with the same topology as the rotation space.
The most famous of these is unit quaternions - the surface of the unit-radius sphere in four dimensional space.
The strengths of such parameterizations (we will be examining quaternions later) are exactly the weak points of the Euclidean ones - they are singularity-free and interpolate rotations well.
The downside is that we must now take extra, explicit measures to enable our tools to function properly on these non-Euclidean parameterizations. Also, such parameterizations always require more than 3 parameters to represent a three DOF rotation; quaternions require 4 - this means our algorithms will be less efficient.

5. Euler angles (φ,θ,ψ)
An Euler angle is a rotation about a single Cartesian axis
Create multi-DOF rotations by concatenating Eulers
R = Rψ Rθ Rφ
3 DOFs can be obtained by concatenating:
An Euler angle is a rotation about a single Cartesian axis
We can Create multi-DOF rotations by concatenating Eulers
For example, we Can get a three DOF by concatenating X, Y, and Z Euler angles together. In doing so the The order in which the rotations are applied must remain fixed: always rotate first about X, then Y, then Z.
Euler-X
Euler-Y
Euler-Z

6. X-Convention Most commonly used
The rotation given by Euler angles (φ,θ,ψ), where the first rotation is by an angle φ about the z-axis, the second is by an angle θ about the x-axis, and the third is by an angle ψ about the z-axis (again).
R = Rψ Rθ Rφ

7. Yaw-Pitch-Roll Convention

8. Singularities
More than one sets of parameters can create the same rotation matrix.
Gimbal lock - two or more axes align, results in loss of rotational DOFs
For Yaw-Pitch-Roll Convention
As we mentioned previously, the Euler angle parameter space has singularities. The definition of a singularity is a continuous subspace of the entire parameter space, all of whose elements correspond to the same value (in this case, rotation). To continue our analogy to globes and maps, singularities occur at the north and south poles - each is but a single point on the globe, but when the map is unwrapped as we talked about before, the entire line at the top of the map corresponds to the north pole, and similarly for the south pole.
Why is this bad? Because singularities induce gimbal lock - a state familiar to mechanical engineers in which two or more rotational axes lign up, resulting in a loss of a rotational DOF. What does that mean?! Let’s see...

9. Rotation Axis + Angle Euler’s Rotation Theorem:
all rotations can be expressed as axis/angle P Y O O’ Y = ú û ù ê ë é - +
sin ,
cos
and / ) ( ),
where 1 3 2 z h y n R yx x

10. Quaternions
Traditional solution: Use unit quaternions to represent rotations
S3 has same topology as rotation space (a sphere), so no singularities
A member of unit sphere in R4
q=(qx,qy,qz,qw)
a rotation about unit axis v
However, gimbal lock, and thus singularities, are just unacceptable for IK and any form of optimization (including spacetime).
The traditional solution in graphics is to use unit-length quaternions to represent rotations.
The set of all unit-length quaternions forms the surface of a 4 D sphere, and is called S^3. Since the space of rotations has the same topology as a sphere, there are no singularities in the mapping of unit quaternions to rotations.