On the unification of hyperbolic and Euclidean geometry. (English) Zbl 1068.30038

The author has noticed in 1988 a curious coincidence spanning the two geometries, leading toward their unification in a super-theory. Paving the way of unifying the two geometries, he presents the story of the introduction of the vector notion into hyperbolic geometry, that followed the 1988 year discovery and that unlocked the keys of Möbius addition. Identifying complex numbers with two-dimensional vectors in the usual way, Möbius addition can be written vectorially, obtaining the vector-Möbius addition \[ u \oplus v = \frac{(1 + 2uv + \| v \| ^{2}) u + (1 - \| u \| ^{2})v}{1 + 2uv + \| u \| ^{2} \| v \| ^{2}}, \] which remains valid in the open unit ball \[ {\mathbf B} = \{v \in {\mathbf V} : \| v \| < 1 \} \] of any real inner product space \({\mathbf V}.\) Moebius (Einstein) addition governs the Poincaré (Beltrami) ball model of hyperbolic geometry just as vector addition governs the standard model of Euclidean geometry. Accordingly, the author shows in this article that resulting analogies enable Euclidean and hyperbolic geometry to be unified.


30G35 Functions of hypercomplex variables and generalized variables
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
51M10 Hyperbolic and elliptic geometries (general) and generalizations
83A05 Special relativity
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