Magnitude
A magnitude is a quantity that represents a concrete distance of some kind. In rigid geometric algebra, a magnitude $$\mathbf z$$ is composed of two components, a scalar and an antiscalar, as follows:
- $$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$
Magnitudes are homogeneous just like everything else in a projective geometric algebra. This means it has both a bulk and a weight, and it is unitized by making the magnitude of its weight one.
Examples
- The geometric norm produces a magnitude that gives the perpendicular distance between an object and the origin. This is also half the distance that the origin is moved by an object used as an operator.
- Euclidean distances between objects are expressed as magnitudes given by the sum of the bulk norms and weight norms of commutators.
- Exponentiating the magnitude $$d\mathbf 1 + \phi {\large\unicode{x1d7d9}}$$ produces a motor for which $$d/\phi$$ is the pitch of the screw transformation.