PGA has two sides: a projective one and a metric one. The projective world involves the incidence relationships; the metric one the measurements of distances, angles, areas, volumes, etc. Each of these worlds has a different relationship to the weight of its elements.

Consider the simple example of the joining line of two euclidean points \bf{A} \vee \bf{B}. If you’re only interested in that line as a geometric entity and not the relationship of \bf{A} and \bf{B} then you can safely ignore the weights, both in the arguments and in the result. Multiplication by non-zero scalars doesn’t change the identity of that line (ignoring issues of orientation, which depend on the underlying metric).

But if you’re doing metric (for example euclidean) geometry and want to measure the distance d between the two points, then you do care about weight. Pick out the representatives of the points with weight 1 by normalizing them (using the metric!) and apply the formula: d = \| \bf{A} \vee \bf{B} \|. In words: the weight of the joining line of normalized points is the euclidean distance between the points. (Without normalized points one has the more general formula d = \dfrac{\| \bf{A} \vee \bf{B} \|}{\|\bf{A}\|\|\bf{B}\|}.) There are other ways to measure distance between points but they all involve calculating a weight.

That’s just one example of many. Look at the cheat sheets for 2D and 3D PGA and you’ll find many more. For example, the. distance from point to line in 2D or point to plane in 3D is given by the weight of the wedge product \bf{P} \wedge \bf{m} – again, under the assumption of normalized arguments!

My favorite example, since it involves both the standard and the ideal norm, arises in 2D when considering the meet of two lines \bf{m} \wedge \bf{n}. When the two lines meet in a euclidean point, then \| \bf{m} \wedge \bf{n}\| = \sin{\alpha} where \alpha is the signed angle between them. And, when they meet in an ideal point (i. e., they are parallel), then \| \bf{m} \wedge \bf{n}\|_\infty = d where d is the signed euclidean distance between them! (The ideal norm \| x \|_\infty is used in EPGA to normalize ideal elements which satisfy \| x\|=0.) This is an example of what I call PGA polymorphicity – the same formula can yield angles or distances, depending on the arguments.

In kinematics and dynamics, the weight of bivectors carry physical significance. Here the situation is doubly interesting since the natural weights for bivectors are not real numbers (as for points and planes) but dual numbers a + b\bf{I}. This is due to the fact that for a bivector \bf{A}, \bf{A}^2 = a+b\bf{I} where b=0 \leftrightarrow \bf{A} is simple. A normalized bivector is one whose square is a unit dual number. These same dual numbers appear as weighting factors when you calculate the logarithm of a motor.

Suppose the weight of a bivector is a+b\bf{I}. This weight is related to measurement as follows: for a velocity/acceleration bivector (a “plane-wise” bivector or “axis”) the real part a gives the angular speed/acceleration (“rotation”) and the dual part b gives the linear speed/acceleration part (“translation”), whereas for an impulse or force bivector (a “point-wise” bivector or “spear”), it’s reversed: the real part gives the strength of the linear impulse/force and dual part gives the strength of the angular impulse/force (aka force couple).

@bellinterlab, I hope this small sample of results makes clear how wonderfully weights are woven into the fabric of PGA. Further details can be found in the SIGGRAPH course notes and the cited references.

Readers who are aware of novel applications of the weight not covered in the literature: please post them!