A theory of duality in Euclidean geometry

  • Rolf Struve
Original Paper


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.


Euclidean geometry Principle of duality Self-dual order structure Circle geometry 

Mathematics Subject Classification

51M05 03B30 51F15 51G05 


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Authors and Affiliations

  1. 1.BochumGermany

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