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

2026-05-18

AI Summary

This work establishes a unified theory of the cross ratio for geometric entities of arbitrary grade within n-dimensional plane-based geometric algebra (PGA), specifically the algebra $\mathbb{R}_{n,0,1}$. By systematically employing the Hodge dual, commutator product, and their dual operators, the study achieves—for the first time—an expression of the cross ratio that is independent of the grade of the involved elements in PGA of any dimension. The analysis reveals that eight distinct configurations naturally form four dual pairs, with their associated metric operators determined solely by the underlying geometric configuration. The derived explicit formulas recover classical invariants: the signed distance ratio in the case of parallel elements and the sine-based cross ratio for intersecting elements, thereby providing a theoretical foundation for directly constructing n-dimensional projective transformations from invariants.

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.

Problem

• cross-ratio
• Plane-Based Geometric Algebra
• projective invariant
• n-dimensional geometry
• geometric duality

Innovation

• Plane-Based Geometric Algebra
• cross-ratio
• projective invariant
• Hodge duality
• homography