Exploiting Degeneracy in Projective Geometric Algebra

Abstract The last two decades, since the seminal work of Selig [18], has seen projective geometric algebra (PGA) gain popularity as a modern coordinate-free framework for doing classical Euclidean geometry and other Cayley-Klein geometries. This framework is based upon a degenerate Clifford algebra, and it is the purpose of this paper to delve deeper into its internal algebraic structure and extract meaningful information for the purposes of PGA. This includes exploiting the split extension structure to realise the natural decomposition of elements of this Clifford algebra into Euclidean and ideal parts. This leads to a beautiful demonstration of how Playfair’s axiom for affine geometry arises from the ambient degenerate quadratic space. The highlighted split extension property of the Clifford algebra also corresponds to a splitting of the group of units and the Lie algebra of bivectors. Central to these results is that the degenerate Clifford algebra $${{\,\textrm{Cl}\,}}(V)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mspace/> <mml:mtext>Cl</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is isomorphic to the twisted trivial extension $${{\,\textrm{Cl}\,}}(V/\mathbb {F}{e_{0}})\ltimes _\alpha {{\,\textrm{Cl}\,}}(V/\mathbb {F}{e_{0}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mspace/> <mml:mtext>Cl</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>/</mml:mo> <mml:mi>F</mml:mi> <mml:msub> <mml:mi>e</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mo>⋉</mml:mo> <mml:mi>α</mml:mi> </mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>Cl</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>/</mml:mo> <mml:mi>F</mml:mi> <mml:msub> <mml:mi>e</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , where $${e_{0}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>e</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> is a degenerate vector and $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> is the grade-involution.