Question

for unit quaternions used to represent rotations for 3D prove that :

a) quaternion conjugate corresponds to rotation inverse ( Approach 1: negate the angle in the formula for converting from the axis-angle form and apply trigonometric identities. Approach 2: convincingly discusses what negating and axis description does in terms of rotation effects and in terms of the formula)

b) negated quaternions represent the same rotation ( hint: add 2PI to the rotation angle in the axis angle formula, explain what that means in terms of orientation, and apply trigonometric identities

Answer 1

We first solve this using a group-theoretic approach:

The group is isomorphic to the quaternions of unit norm via the following map:

Now, identify with the span of . Now, if and , then . And we know from the properties of quaternions that is a rotation of . This is true because is the double cover of , which represents rotation of .

(a) Now, note that if , then the composition of these rotations is:

Let . Then the composition is:

And hence the result follows.

(b) Now, mapping , we have which is just the same rotation as before. And the result follows.

The following is matrix based approach:

Given a unit quaternion , the rotation matrix looks like:

This is a general element of , and hence the matrix is orthogonal by definition.

(a) Now, , and hence the matrix becomes:

But note that:

And since :

And the quaternion conjugate represents the opposite rotation.

(b) Here, , and thus the matrix becomes:

But then:

And hence the negated quaternion represents the same rotation.