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A ''quaternion'' is an operator that performs a rotation about the origin in 3D space. Conventionally, a quaternion $$\mathbf q$$ is ...ector representing the axis of rotation, and $$\phi$$ is half the angle of rotation. ...

A ''rotation'' is a proper isometry of Euclidean space. ...the case that $$R_{mw} = 0$$. The line $$\boldsymbol l$$ and its [[weight dual]] $$\boldsymbol l^\unicode["segoe ui symbol"]{x2606}$$ are invariant under ...

...the FGED series. However, I knew at the time that I didn’t want to present dual quaternions in the usual ways that they have always been introduced in the ...thbf a)\mathbf b = \mathbf{ba} x \mathbf {ab}$$. Two reflections produce a rotation that we represent by the quaternion $$\mathbf q = \mathbf{ba}$$. ...

...ich is a line when $$n = 3$$. Geometrically, these two interpretations are dual to each other, and their distances to the origin are reciprocals of each ot ...responds to the one-dimensional space of the point that it represents. The dual of a point materializes when we consider all of the directions of space tha ...

...represents a proper Euclidean isometry, which can always be regarded as a rotation about a line $$\boldsymbol l$$ and a displacement along the same line.]] ...ion operator'' or ''moment vector''. Motors are equivalent to the set of ''dual quaternions'' used in conventional theories, and the functionality is prope ...

...Plücker coordinates, [[quaternions]], and screw theory (which makes use of dual quaternions). This makes rigid geometric algebra a natural fit for areas of ...de of its inputs and outputs, we can make the same statement about how the dual operation relates to the antigrade of its inputs and outputs. ...

...tion of both the product and antiproduct allows us to calculate a complete dual of any element using only the multiplication and reversion operations of th ...int at the origin is moved by the motor $$\mathbf Q$$, accounting for both rotation in the plane perpendicular to the axis and translation along the axis. I wi ...