Conversion between quaternions and Euler angles: Difference between revisions

Revision as of 17:22, 9 March 2023

Revision as of 17:25, 9 March 2023

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== Vector rotation ==

Let us define scalar <math>q_w</math> and vector <math>\vec{q}</math> such that quaternion <math>\mathbf{q} = (q_w,\vec{q}) = q_w+iq_x+jq_y+kq_z</math>.

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

:<math>\mathbf{v}^{\,\prime} = \mathbf{qvq}^\ast</math>

where <math>\mathbf{v} = (0,\vec{v}) = 0+iv_x+jv_y+kv_z</math> is a quaternion containing the embedded vector <math>\vec{v}</math>, <math>\mathbf{q}^\ast=(q_w,-\vec{q})</math> is a [[Quaternion#Conjugation, the norm, and reciprocal|conjugate quaternion]], and <math>\mathbf{v}^{\,\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_w \vec{t} + \vec{q} \times \vec{t}</math>