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== Rotation matrices ==
The [[orthogonal matrix]] corresponding to a rotation by the unit quaternion ''z'q' = ''a'' + ''bi'' + ''cj'' + ''dk'' (with |z| = 1) is given by
:<math>\begin{bmatrix}
a 1- 2(q_2^2 +b^2-c^2-d q_3^2) &2bc-2ad 2(q_1 q_2 - q_0 q_3) &2ac+2bd 2(q_0 q_2 + q_1 q_3) \\
2ad+2bc2(q_1 q_2 + q_0 q_3) & 1 - &a^2-b(q_1^2 +c q_3^2-d^2&2cd-2ab) & 2(q_2 q_3 - q_0 q_1) \\
2bd-2ac2(q_1 q_3 - q_0 q_2) & 2( q_0 q_1 &2ab+2cd q_2 q_3) & 1 - &a^2-b^2-c(q_1^2 +d q_2^2\\)
\end{bmatrix}</math>
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