  We consider statistical methods for reduction of multivariate dimensionality
that have invariance and/or commutativity properties under the affine group of
transformations (origin translations plus linear combinations of coordinates
along initial axes). The methods discussed here differ from traditional
principal component and coordinate approaches in that they are origin-centric.
Because all Cartesian coordinates of the origin are zero, it is the unique
fixed point for subsequent linear transformations of point scatters. Whenever
visualizations allow shifting between and/or combining of Cartesian and polar
coordinate representations, as in Biplots, the location of this origin is
critical. Specifically, origin-centric visualizations enhance the psychology of
graphical perception by yielding scatters that can be interpreted as Dyson
swarms. The key factor is typically the analyst's choice of origin via an
initial "centering" translation; this choice determines whether the recovered
scatter will have either no points depicted as being near the origin or else
one (or more) points exactly coincident with this origin.
