Maths for Game Development: Quaternions

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That is probably the easiest way to think of quaternions. Rotation around an arbitrary axis. There are two quaternions that represent the same result, where the doppelganger is the original quaternion with all coordinates negated.
For practical use, if you have a quaternion orientation that you want to convert to a look-at direction vector and an up vector, then just start with the basis vector and run it through a quaternion rotation to get the final direction (same for the up vector).

erikjohnson