Self-Reverse Elements and Lines in an Algebra for 3D Space

Robert J. Cripps, Ben Cross, Glen Mullineux

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A geometric algebra provides a single environment in which geometric entities can be represented and manipulated and in which transforms can be applied to these entities. A number of versions of geometric algebra have been proposed and the aim of the paper is to investigate one of these as it has a number of advantageous features. Points, lines and planes are presented naturally by element of grades 1, 2, and 3 respectively. The self-reverse elements in the algebra form a field. This allows an equivalence relation between elements of grade 2 to be defined so that, although not every grade 2 element corresponds to a line, each equivalence class does, and vice versa. Examples are given to illustrate the ease in which geometric objects are represented and manipulated.
Original languageEnglish
Article number50
JournalAdvances in Applied Clifford Algebras
Issue number4
Early online date15 Jul 2020
Publication statusPublished - 1 Sept 2020


  • Geometric algebra
  • Line
  • Plane
  • Point
  • Self-reverse element


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