Complementary Orientations in Geometric Algebras

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. Dualization in geometric algebras can be used to encode them; or vice versa, orientation types inform the interpretation of dualization. We employ the Hodge dual, to include important algebras with null elements like PGA. Oriented elements can be combined using the meet operation, and the dual join (which is here introduced for that purpose). Software written to process one orientation type can be employed to process the complementary type consistently.