  Multiplication by the pseudoscalar $\mathbf{I}$ has been traditionally used
in geometric algebra to perform non-metric operations such as calculating
coordinates and the regressive product. In algebras with degenerate metrics,
such as euclidean PGA $P(\mathbb{R}^*_{3,0,1})$, this approach breaks down,
leading to a search for non-metric forms of duality. The article compares the
dual coordinate map $J: G \rightarrow G^*$, a double algebra duality, and Hodge
duality $H: G \rightarrow G $, a single algebra duality for this purpose. While
the two maps are computationally identical, only $J$ is coordinate-free and
provides direct support for geometric duality, whereby every geometric
primitive appears twice, once as a point-based and once as a plane-based form,
an essential feature not only of projective geometry but also of euclidean
kinematics and dynamics. Our analysis concludes with a proposed duality-neutral
software implementation, requiring a single bit field per multi-vector.

General Mathematics