On the other hand, if the completion is elliptic, as the name suggests, then how does it square with the hypothesis of the acute angle? But if we remove all lines asymptotically parallel to some line in the hyperbolic plane, nothing will be left other than that line itself. If the semi-elliptic plane is isometric to a subset of a hyperbolic one then we should be able to obtain it by removing some points from the latter. In other words, no two lines are asymptotically parallel. It satisfies the Lambert's hypothesis of the acute angle (in any quadrilateral with three right angles the fourth angle is acute), but any two non-intersecting lines in it have a unique common perpendicular. One of them, called the semi-elliptic plane, is quite peculiar. Also, there is an algebraic classification of Archimedean H-planes due to Pejas described in the Greenberg's paper (p.760). With extra points added perhaps some lines intersect more than once. It is easy to check by limit arguments that most axioms still hold in the completion, but not that two lines intersect at no more than one point, for example. Will their completions still always be Euclidean or hyperbolic? In other words, if we remove the axiom ensuring completeness, and then take the completion, do we end up with what we started with? Without the continuity however Archimedean H-planes may not be metrically complete. Incidence, order and congruence imply that perpendicular is shorter than oblique, so the triangle inequality holds. ![]() ![]() The distance between two points is defined to be the length of the line segment connecting them, unique by incidence. Repeatedly bisecting a picked "unit segment" and laying its pieces off of any other gives a binary fraction (possibly infinite) that is assigned as the segment's length. We can still define a metric on them in the usual way. For an elementary version we also drop the (Cantor's) axiom of continuity, Greenberg calls such geometries Archimedean H-planes in his survey paper. It is well-known that it is either the Euclidean or a hyperbolic plane. Absolute geometry is any one that satisfies Hilbert's axioms of plane geometry without the axiom of parallels.
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