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Quaternion: Difference between revisions
→Double Quaternions
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Dual quaternions are written as <math>q_1 + \varepsilon q_2</math> with <math>\varepsilon^2 = 0</math>. | Dual quaternions are written as <math>q_1 + \varepsilon q_2</math> with <math>\varepsilon^2 = 0</math>. | ||
===Cayley Factorization=== | |||
See [Federico Thomas]. | |||
Any 4D rotation matrix can be decomposed into a right and a left isoclinic rotation matrix: | |||
<math>R = R^L R^R = R^R R^L</math> | |||
<math>R^R</math> and <math>R^L</math> can be seen as a single double quaternion. | |||
The product of left and right isoclinic rotation matrices commute. | |||
Furthermore, the product of two left isoclinic rotation matrices is a left isoclinic rotation matrix. Same with right. | |||
Thus, <math>R_1 R_2 = (R_1^L R_1^R) (R_2^L R_2^R) = (R_1^L R_2^L) (R_1^R R_2^R) = R</math>. | |||
This shows that the composition of two double quaternions will be a double quaternion. | |||
==Dual Quaternions== | ==Dual Quaternions== | ||