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Appendix C: Quaternions

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This Appendix presents an alternate representation of rotation matrices using quaternions. This representation is extremely useful in reducing rounding-error problems found when combining rotation matrices. Also, the interpolation between two quaternions representing the orientation of an object is easier than using rotation matrices. This is especially useful when backtracking in time the object’s motion to determine the instant just before a collision.


  • Rotation Matrix Representation
  • Unit Quaternions
  • Pure Quaternion
  • Quaternion Result
  • Computer Graphics Industry

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Fig. 8.1
Fig. 8.2


  1. 1.

    Unless otherwise stated, whenever we mention ℝn, we are referring to the n-dimensional Euclidean space and its associated properties.


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Coutinho, M.G. (2013). Appendix C: Quaternions. In: Guide to Dynamic Simulations of Rigid Bodies and Particle Systems. Simulation Foundations, Methods and Applications. Springer, London.

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