Generalize cross-ratios in n-dimensional Plane-Based Geometric Algebra

TL;DR

This paper develops a comprehensive theory of projective cross-ratios in n-dimensional Plane-Based Geometric Algebra, unifying classical invariants across various geometric objects and configurations.

Contribution

It introduces explicit cross-ratio formulas for all object types in PGA, revealing their dualities and reducing measurement operators to fundamental algebraic commutators.

Findings

Cross-ratios recover classical invariants like distance ratios and sine ratios.

All configurations organize into dual pairs under Hodge duality.

Measurement operators simplify to commutator forms.

Abstract

We develop a complete theory of projective cross-ratios in n-dimensional Plane-Based Geometric Algebra (PGA), R(n,0,1), covering geometric objects of every grade: finite and ideal points, hyperplanes, and intermediate flats. For each object type and configuration, we establish an explicit cross-ratio formula, prove that it recovers the appropriate classical invariant, and identify the canonical pairwise measurement operator. A systematic duality analysis further revealed that all eight configurations organize into four dual pairs under the Hodge dual, and that all measurement operators reduce to either the commutator or the commutator dual, depending solely on the geometric configuration rather than on object grade. In each case the formula recovers the appropriate classical invariant: signed distance ratios for parallel configurations and sine cross-ratios for secant ones. These results establish the cross-ratio as a grade-agnostic projective invariant within PGA, and provide a constructive foundation for defining n-dimensional homographies directly from prescribed invariants.