Conversion between quaternions and Euler angles: Difference between revisions

Revision as of 22:34, 6 July 2005

Revision as of 00:18, 7 July 2005

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:<math>\mathbf{q} = \begin{bmatrix} q_0 & q_1 & q_2 & q_3 \end{bmatrix}^T</math>

:<math>|\mathbf{q}|^2 = q_0^2 + q_1^2 + q_2^2 + q_3^2 = 1</math>

:<math>q_0\mathbf{q}_0 = \cos(\alpha/2)</math>

:<math>q_1\mathbf{q}_1 = \sin(\alpha/2)\cos(\beta_x)</math>

:<math>q_2\mathbf{q}_2 = \sin(\alpha/2)\cos(\beta_y)</math>

:<math>q_3\mathbf{q}_3 = \sin(\alpha/2)\cos(\beta_z)</math>

where <math>\alpha</math> is a simple rotation angle (radians) and <math>\beta_x</math>, <math>\beta_y</math>, <math>\beta_z, </math> are the "direction cosines" of the axis of simple rotation (Euler's Theorem).