Rotation representation: Quaternions

• Quaternions: generalization of complex numbers.

• \(q = x\,{\bf i}+ y \, {\bf j} + z \, {\bf k} + w\) \(w\) real part, \((x,y,z)\) imaginary (or pure quaternion) part.

• We write in short \(q=(x,y,z,w)\)

• Properties of imaginary basis vectors

• \( \left\{ \begin{array}{l} {\bf i}^2={\bf j}^2={\bf k}^2=-1 \\ {\bf i} {\bf j} = -{\bf j} {\bf i} = {\bf k} \\ {\bf j} {\bf k} = -{\bf k} {\bf j} = {\bf i} \\ {\bf k} {\bf i} = -{\bf i} {\bf k} = {\bf j} \\ {\bf i} {\bf j} {\bf k} = -1 \\ \end{array} \right. \)

• • - Provides the algebraic properties