1 Basics

1.1 Quaternion

1. A quaternion q is defined as the sum of a scalar q0 and a vector q = (q1, q2, q3); namely,

   1. q = q0 + q = q0 + q1i + q2j + q3k.

2. Addition and Multiplication

   

   

   

3. The complex conjugate of q, denoted q∗, is defined as

   1. q∗ = q0 − q = q0 − q1i − q2j − q3k.

   

4. The norm of a quaternion q, denoted by |q|, is the scalar

   

1.2 Unit quaternion

1. A quaternion is called a unit quaternion if its norm is 1.

2. The norm of the product of two quaternions p and q

   

3. The inverse of a quaternion q is defined as

   

   In the case q is a unit quaternion, the inverse is its conjugate q∗

1.3 Quaternion Rotation Operator

rotate an angle, such as \theta = 90 degree, around unit 3d vector u, such as u=(0i, 0j, 1k), i.e. z axis.

note: q∗ = q0 − q = q0 − q1i − q2j − q3k. Here use -\theta.

1.3.1 Unit quaternion to rotation matrix