Conversion between quaternions and Euler angles: Difference between revisions

Revision as of 15:25, 16 April 2021

Revision as of 06:54, 15 May 2021

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Note that the canonical way to rotate a three-dimensional vector <math>\vec{v}</math> by a quaternion <math>q</math> defining an [[Conversion between quaternions and Euler angles#Conversion|Euler rotation]] is via the formula

:<math>\mathbf{p}^{\,\prime} = \mathbf{qpq}^\ast</math>

where <math>\mathbf{p} = (0,\vec{v}) = 0+iv_1+jv_2+kv_3</math> is a quaternion containing the embedded vector <math>\vec{v}</math>, <math>\mathbf{q}^\ast=(q_0,-\vec{q})</math> is a [[Quaternion#Conjugation.2C, the norm.2C, and reciprocal|conjugate quaternion]], and <math>\mathbf{p}^{\,\prime} = (0,\vec{v}^{\,\prime})</math> is the rotated vector <math>\vec{v}^{\,\prime}</math>. In computational implementations this requires two quaternion multiplications. An alternative approach is to apply the pair of relations

:<math>\vec{t} = 2\vec{q} \times \vec{v}</math>

:<math>\vec{v}^{\,\prime} = \vec{v} + q_0 \vec{t} + \vec{q} \times \vec{t}</math>