The representation of rotations boiled down to picking points on and respecting the fact that antipodal points give the same element of . In a sense, this has nothing to do with the algebraic properties of quaternions. It merely means that can be parameterized by picking points in , just like was parameterized by picking points in (ignoring the antipodal identification problem for ).

However, one important reason why the quaternion arithmetic was
introduced is that the group of unit quaternions with and
identified is also isomorphic to
. This means that a sequence of rotations can be multiplied
together using quaternion multiplication instead of matrix
multiplication. This is important because fewer operations are
required for quaternion multiplication in comparison to matrix
multiplication. At any point, (4.20) can be used to
convert the result back into a matrix; however, this is not even
necessary. It turns out that a point in the world,
, can be transformed by directly using quaternion arithmetic.
An analog to the complex conjugate from complex numbers is needed.
For any
, let
be
its *conjugate*. For any point
, let
be the quaternion
. It can be shown (with a lot of
algebra) that the rotated point is given by
. The , , components of the resulting quaternion are
new coordinates for the transformed point. It is equivalent to having
transformed with the matrix .

Steven M LaValle 2012-04-20