Spatial Algebra

Uniqueness of Orientation

We know that we need at least 4 numbers to describe an orientation to not suffer from Gimbal Lock.

Quaternion

1. Dual Representation: In quaternions, the orientation represented by a quaternion qq is also represented by −q−q. This is because both $q$ and $-q$ produce the same rotation when applied to a point or vector. Hence, quaternions have a dual representation for each unique orientation.

Rotation Matrix

1. Unique Matrix: A rotation matrix is generally unique for a given orientation. However, if the matrix includes additional transformations like scaling, then it might not uniquely represent a rotation. In a pure rotation matrix, each orientation corresponds to exactly one matrix.

Euler Angle

1. Ambiguity: Euler angles suffer from a lack of uniqueness because multiple sets of angles can represent the same orientation. This is called “aliasing,” and it can be problematic when you’re interpolating between two sets of Euler angles, as the path taken may not be what’s expected.
2. Normalization: Some systems normalize Euler angles to a specific range to maintain uniqueness, but this doesn’t entirely solve the problem and can introduce other issues like gimbal lock.

Axis-Angle

• Dual representation:The axis-angle representation is not entirely unique. The same rotation can be represented by rotating θθ degrees around an axis $v$ or by rotating $-\theta$ degrees around the opposite axis $-v$.

So both Quaternion and Axis-Angles have this dual-representation / dual-cover property.