In Section 7.1 of the course notes, the “bivector” model defines the norm as

\Vert \mathbf x\Vert = \sqrt{|\mathbf x^2|},

where the squaring inside the radical is done with the geometric product. It defines the “ideal norm” (or “infinity norm”) in a case-by-case manner for points, lines, and pseudoscalars without providing a general formula or addressing planes, motors, and reflection operators, and it utterly fails to recognize the symmetry at work.

First, the norm as defined in the course notes isn’t quite correct. It should be

\Vert \mathbf x\Vert = \sqrt{\mathbf x\mathbf{\widetilde x}},

which gets rid of the embarrassing need for the absolute value.

In the “Lengyel” model, the “bulk norm” and “weight norm” are defined as

Bulk norm: \Vert\mathbf x\Vert_\bullet = \sqrt{\mathbf x \mathbin{\char"27D1} \mathbf{\widetilde x}}

Weight norm: \Vert\mathbf x\Vert_\circ = \sqrt{\mathbf x \mathbin{\char"27C7} \mathbf{\utilde x}\vphantom{\mathbf{\widetilde x}}} ,

where \char"27D1 is the geometric product, and \char"27C7 is the geometric antiproduct. For the objects in PGA that we care about, the bulk norm is a scalar, and the weight norm is an antiscalar (pseudoscalar).

The “ideal norm” is mathematically the same as the weight norm, but (incorrectly) dualized to be a scalar instead of an antiscalar. Unfortunately, it cannot be expressed quite as symmetrically because the “bivector” model lacks the antiproduct, but you could write

Ideal norm: \Vert\mathbf x\Vert_\infty = \sqrt{\mathbf x^*\widetilde{\mathbf x^*}} ,

where * is any dualization operator.

Note that because the “bivector” model describes geometries and isometry operators with empty dimensions instead of full dimensions, the meaning of the two norms is reversed. So the “bivector” norm gives you the magnitude of the weight of an object, and the ideal norm gives you the magnitude of its bulk. (Or to use the terminology suggested by the “bivector” crowd, the norm gives you the magnitude of the tangent space aspect of the object, and the ideal norm gives you the magnitude of its positional aspect.)

The projected magnitude or geometric norm of an object \mathbf x, which is the distance from the origin for something regarded as a geometry and half the distance the origin is moved for something regarded as an isometry operator, is given by the ratio of the norms. Using the notation m(\mathbf x) for the projected magnitude here to avoid conflicts with other notation, in the “bivector” model,

m(\mathbf x) = \dfrac{\Vert\mathbf x\Vert_\infty}{\Vert\mathbf x\Vert} = \dfrac{\sqrt{\mathbf x^*\widetilde{\mathbf x^*}}}{\sqrt{\mathbf x\mathbf{\widetilde x}}} ,

and in the “Lengyel” model,

m(\mathbf x) = \dfrac{\Vert\mathbf x\Vert_\bullet}{\Vert\mathbf x\Vert_\circ} = \dfrac{\sqrt{\mathbf x \mathbin{\char"27D1} \mathbf{\widetilde x}}}{\sqrt{\mathbf x \mathbin{\char"27C7} \mathbf{\utilde x}\vphantom{\mathbf{\widetilde x}}}} .

(Note: with the appropriate definitions of wedge and dot products involving scalars, all of the geometric products above can be replaced with dot products and antidot products.)