Möbius invention of homogeneous coordinates was one of the most far-reaching ideas in the history of mathematics: comparable to Leibnitz's invention of differentials.
H.S.M. Coxeter, Introduction to Geometry, p. 221
... the inhabitants of a hyperbolic world would also study horospherical geometry, which is the same as Euclidean geometry!
H.S.M. Coxeter, Introduction to Geometry, p. 304
Abstract
As part of his program to unify linear algebra and geometry using the language of Clifford algebra, David Hestenes has constructed a (well-known) isomorphism between the conformal group and the orthogonal group of a space two dimensions higher, thus obtaining homogeneous coordinates for conformal geometry.(1) In this paper we show that this construction is the Clifford algebra analogue of a hyperbolic model of Euclidean geometry that has actually been known since Bolyai, Lobachevsky, and Gauss, and we explore its wider invariant theoretic implications. In particular, we show that the Euclidean distance function has a very simple representation in this model, as demonstrated by J. J. Seidel.(18)
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Dress, A.W.M., Havel, T.F. Distance geometry and geometric algebra. Found Phys 23, 1357–1374 (1993). https://doi.org/10.1007/BF01883783
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DOI: https://doi.org/10.1007/BF01883783