Quaternion rotation biradial

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In mathematics, a quaternion biradial[1](Art.93) is the quotient (or product \mathbf{b}/\mathbf{a}, \mathbf{b}\mathbf{a}, \mathbf{a}\mathbf{b}, \mathbf{a}/\mathbf{b}) of two pure quaternion vectors \mathbf{a} and \mathbf{b}, sometimes called rays.

Biradial b/a.
Biradial b/a

Consider the quaternion biradial b=\mathbf{b}/\mathbf{a} (read “b by a”). The biradial b is an operator that turns \mathbf{a} into \mathbf{b} as an operator on \mathbf{a}: \mathbf{b}=b\mathbf{a}=\mathbf{b}\mathbf{a}^{-1} \mathbf{a}=\mathbf{b}. The biradial turning operation is a composition of a rotation and a scaling that rotates \mathbf{a} into the line of \mathbf{b}, followed with scaling by | \mathbf{b} | / | \mathbf{a} |. If | \mathbf{a} | = | \mathbf{b} | then | b | =1 and b acts as only a rotation operator (a.k.a., a versor, rotor, or 2D spinor) in the \mathbf{a}\mathbf{b}-plane through the angle \theta from \mathbf{a} to \mathbf{b}.