Blade Products and the Angle Bivector of Subspaces

TL;DR

This paper introduces a novel approach to describing the relative inclination of subspaces using an angle bivector derived from principal angles, connecting geometric algebra, Grassmannians, and Plücker coordinates.

Contribution

It presents a new geometric interpretation of the angle bivector and relates it to various blade products, expanding understanding of subspace relations in geometric algebra.

Findings

Defines an angle bivector using principal angles

Links the exponential of the bivector to rotors and geodesics

Provides formulas connecting blade products to subspace angles

Abstract

Principal angles are used to define an angle bivector of subspaces, which fully describes their relative inclination. Its exponential is related to the Clifford geometric product of blades, gives rotors connecting subspaces via minimal geodesics in Grassmannians, and decomposes giving Pl\"ucker coordinates, projection factors and angles with various subspaces. This leads to new geometric interpretations for this product and its properties, and to formulas relating other blade products (scalar, inner, outer, etc., including those of Grassmann algebra) to angles between subspaces. Contractions are linked to an asymmetric angle, while commutators and anticommutators involve hyperbolic functions of the angle bivector, shedding new light on their properties.