Rational trigonometry via projective geometric algebra

Abstract

We show that main results of rational trigonometry (as developed by NJ Wildberger, "Divine Proportions", 2005) can be succinctly expressed using projective geometric algebra (PGA). In fact, the PGA representation exhibits distinct advantages over the original vector-based approach. These include the advantages intrinsic to geometric algebra: it is coordinate-free, treats lines and points in a unified framework, and handles many special cases in a uniform and seamless fashion. It also reveals structural patterns not visible in the original formulation, for example, the exact duality of spread and quadrance. The current article handles only a representative (euclidean) subset of the full content of Wildberger's work, but enough to establish the value of this approach for further development. The metric-neutral framework of PGA makes it especially promising also to handle universal geometry, which extends rational trigonometry to the hyperbolic plane.