Projective duality encodes complementary orientations in geometric algebras

Abstract

Oriented elements are part of geometry, and they come in two complementary types: intrinsic and extrinsic. Those different orientation types manifest themselves by behaving differently under reflection. Projective dualization in geometric algebras can encode them, or conversely, orientation types inform the interpretation of dualization. Oriented elements may be combined using the meet operation, and the dual‐join (which is here introduced for that purpose). We discuss algebra‐based implementation techniques on how to double the representational power of a geometric algebra, without explicitly doubling the algebra.