## Quaternions part 2

posted by harrison on August 31, 2012

This will be a short post explaining why conjugation is rotation, and why the axis-angle formula works. If you didn’t see it, here’s part 1.

### Why is conjugation rotation?

To see why conjugation is rotation, it is helpful to expand the multiplication (assuming q is a unit quaternion):

When multiplying vectors as quaternions, notice that since the real part is 0, .

To finish, we need to talk about parallel and perpendicular components of to . and . Notice that .

And now we can start using trig to get rid of those half angles:

From here, you can see that the parallel component remains unchanged, as expected. Also, notice that is just rotated around , So you can see that the perpendicular part is being rotated around by .

next time: exponentials, logs, and slerp

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