Rotation

A rotation is a proper isometry of Euclidean space performed by the operator


 * $$\mathbf R = \boldsymbol l\sin\dfrac{\phi}{2} + {\large\unicode{x1d7d9}}\cos\dfrac{\phi}{2}$$ ,

where $$\boldsymbol l$$ is a unitized line corresponding to the axis of rotation. This operator is identical to the rotation operator in rigid geometric algebra but with the extra factor of $$\mathbf e_5$$. It rotates an object $$\mathbf x$$ through the angle $$\phi$$ about the line $$\boldsymbol l$$ when used with the sandwich antiproduct $$\mathbf R \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}}$$.