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Direction cosines are two parameters (angles). Unit vectors include a module. What we have here is not that, but Euler Angles |
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Spatial rotations in three dimensions can be [[Coordinate system|parametrized]] using both [[Euler angles]] and [[Quaternions and spatial rotation|unit quaternions]]. This article explains how to convert between the two representations. Actually this simple use of "quaternions" was first presented by [[Leonhard Euler|Euler]] some seventy years earlier than [[William Rowan Hamilton|Hamilton]] to solve the problem of [[magic square]]s. For this reason the dynamics community commonly refers to quaternions in this application as "Euler parameters".
==Definition==
A unit quaternion can be described as:
:<math>\mathbf{q} = \begin{bmatrix} q_0 & q_1 & q_2 & q_3 \end{bmatrix}^T</math>
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:<math>\mathbf{q}_2 = \sin(\alpha/2)\cos(\beta_y)</math>
:<math>\mathbf{q}_3 = \sin(\alpha/2)\cos(\beta_z)</math>
where α is a simple rotation angle and cos(β<sub>''x''</sub>), cos(β<sub>''y''</sub>) and cos(β<sub>''z''</sub>) are the "[[Euler Angles]]
===Relationship with Tait-Bryan angles===
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