An object in 3-space can go from any initial orientation to any final orientation by:
- defining exactly one vector in 3-space
- rotating that object about that vector by a specific angle
The anatomy of a quaternion
A quaternion is defined:
Where , and are all unit vectors along the three spatial axes.
Significance of the 3-Space Component
To understand quaternion rotation intuitively, let’s ignore the “real” part and only pay attention to the 3-space component:
The magnitude defines a fraction from to which represents a to rotation about the vector .
Note: The relationship between is not linear. therefore, a rotation of maps to .
What about a?
The mathematics of quaternions are a little bit more involved than what I described above.
Thier inventer, William Hamilton desctibes them as quotents of two vectors, which requires . Therefore, effectively describes how much a vector should not rotate in order to point in the direction of another vector.
A value of indicates the two vectors making up the quaternion pointed in the same direction.
Calculating a quaternion from a vector and rotation
Given a unit vector:
And an angle which we will rotate around the unit vector by, .