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**closed**Feb 08, 2014 at 11:16 AM by Benproductions1 for the following reason:

Question is off-topic or not relevant

# About the derivation of a composite quaternion

Hi guys,

This problem has been bothering me for several days, hence I decided to ask you for help.

I am reading the book "Quaternions and Rotation Sequence" written by Jack B. Kuipers. In section 6.4, the author derives a formula of a composite rotation quaternion. One of the steps of this derivation is difficult for me to understand.

I would like to briefly describe the derivation process as follow:

Consider a tracking problem as in this picture:

In the picture, XYZ is a global, reference frame. 2 successive rotations are performed: The first one is a rotation about the Z axis through an angle alpha, transforming frame XYZ into a new frame x1y1z1. The second one is a rotation about the y1 axis through an angle beta, transforming frame x1y1z1 into a new frame x2y2z2.

The goal is to find a single composite rotation quaternion which is equivalent to the two rotations above.

The author does this as follow. The first rotation can be represented by the following quaternion p:

p = cos(alpha/2) + k*sin(alpha/2) (1)

In this formula, k is a standard basis vector (we have vectors i, j, k in R3 corresponding to the axes x, y, z respectively).

The second rotation can be represented by the following quaternion q:

q = cos(beta/2) + j*sin(beta/2) (2)

The composite quaternion we are looking for is the product of these 2 quaternions: qp. The formula of this product is:

In order to derive this final formula, the author uses 2 assumptions about the standard basis vectors i, j, k, which are: k.j = 0 and k x j = -i. And this is where I dont understand.

We all know that, for a set of 3 mutually orthogonal vectors i, j, k, these 2 assumptions above are correct. However, vector k in (1) and vector j in (2) don't belong to the same coordinate frame. In other words, k in (1) corresponds to Z in frame XYZ, and j in (2) corresponds to y1 in x1y1z1. And these are 2 different, distinguish frames, so I think the second assumption used by the author is incorrect.

What do you think about this? Any answer would be appreciated. Thank you.

This has nothing to do specifically with Unity. Ask it on math.stackoverflow, where people actually know maths past year 12.

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