By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are introduced here, and the main formulas extend those of rational trigonometry in the plane. This gives a unified, computational model of both spherical and hyp...

the aim of this paper is to show the importance of analytic hyperbolic geometry introduced in [9]. in [1], ungar and chen showed that the algebra of the group sl(2,c) naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the lorentz group and its underlying hyperbolic geometry. they defined the chen addition and then chen model of hyperbolic geometry. in this paper,...

This will be a description of a few highlights in the early history of non-euclidean geometry, and a few miscellaneous recent developments. An Appendix describes some explicit formulas concerning volume in hyperbolic 3-space. The mathematical literature on non-euclidean geometry begins in 1829 with publications by N. Lobachevsky in an obscure Russian journal. The infant subject grew very rapidl...

460 NOTICES OF THE AMS VOLUME 47, NUMBER 4 I n the fall semester of 1988, I taught an undergraduate course on Euclidean and nonEuclidean geometry. I had previously taught courses in projective geometry and algebraic geometry, but this was my first time teaching Euclidean geometry and my first exposure to non-Euclidean geometry. I used the delightful book by Greenberg [8], which I believe my stu...

Our goal is to study image processing techniques for 360-degree images, which are obtained from omni-directional sensors [11, 17]. Having a curved mirror, i.e. hyperbolic, spherical or parabolic, as a lens for the corresponding catadioptric system, we obtain non-Euclidean images. One way of processing such an image is to perform a panoramic projection of this image onto a cylinder. In this way ...

We propose a method to construct quantum theory of matter fields in a topology changing universe. Analytic continuation of the semiclassical gravity of a Lorentzian geometry leads to a non-unitary Schrödinger equation in a Euclidean region of spacetime, which does not have a direct interpretation of quantum theory of the Minkowski spacetime. In this Euclidean region we quantize the Euclidean ge...