# Appendix 19.A quaternions, Axis-Angle Vectors, and Rotations

## 19.A.1 Conversions Between Rotation Representations

Let be used as an axis-angle representation of rotation where the direction of a specifies the axis of rotation and specifies the magnitude of rotation (in radians).

Let , where H is the space of quaternions, be used as a unit quaternion representation of rotation, where qs is real and qx, qy,qz are coefficients of distinct imaginary numbers, and .

Let be a 3 × 3 rotation matrix, such that M is orthonormal, and det M = + 1.

To convert from an axis-angle vector a to a unit quaternion q, we define the function as

(19.161) To convert from a unit quaternion q to an axis-angle vector a we define the function as

(19.162)

To convert from a unit quaternion q to a rotation matrix

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