A theory of duality in Euclidean geometry

Abstract

The principle of duality is well established in projective geometry but can hardly be found in the literature on Euclidean geometry where it is “more a principle of analogy than a scientific principle with a logical foundation” (cp. Sommerville, The Elements of Non-Euclidean Geometry. The Open Court, London, 1919). We close this gap and develop a theory of duality in Euclidean geometry. Following Hilbert’s Grundlagen der Geometrie we consider the incidence, order and metric structure of a Euclidean plane and show (a) that there is a large class of theorems of Euclidean incidence geometry which allow a dualization (b) that Hilbert’s order structure can be introduced in a Euclidean plane in a self-dual way and (c) that appropriate definitions of metric notions (e.g., of an angle, a segment or a circle) lead to Euclidean theorems with meaningful dual versions. This shows that duality in Euclidean geometry is not a collection of isolated phenomena but corresponds to a rich and coherent theory.