Chapter 31. Clifford Algebras

The quaternion-based formalism for handling and visualizing rotations works well in dimensions 2, 3, and 4 because in these dimensions the Spin group (the double covering of the orthogonal group) has simple topology and geometry. It would be natural to expect that this simplicity continues to hold for rotations in any dimension, and that all of our 3D intuitions about labeling frames, interpolating frames, and simple frame-to-frame distance measures continue to be valid. Unfortunately, that is not the case: quaternions are quite unique to 3D, and only a serendipitous accident of topology allows an extension even to 4D.

On the other hand, there is a mathematical formalism that treats N -dimensional rotations in a very ...

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