9 MODELING GEOMETRIES

So far we have been treating only homogeneous subspaces of the vector spaces (i.e., subspaces containing the origin). We have spanned them, projected them, and rotated them, but we have not moved them off the origin to make more interesting geometrical structures such as lines floating in space. You might fear that we need to extend our framework considerably to incorporate such new geometrical elements and their algebra, introducing offset blades as algebraic primitives.

However, significant extensions turn out to be unnecessary, by using a simple trick that is a lot more generic than it appears at first: these offset elements of geometry in a vector space can also be represented by blades, but in a representational space ...

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