ℍ3 and Euclidean Approximations in ℍ2

Abstract

To investigate the fine structure of hyperbolic geometry, we use a method due essentially to Lobachevski. It turns out that the geometry of the horosphere, a certain surface in three-dimensional hyperbolic space ℍ3, is Euclidean if the definitions are made as follows. The “straight lines” of that geometry are the horocycles in that surface, the “distances” are the arclengths between points along horocycles lying in that surface, and the “angle” between two intersecting horocycles is the angle between the tangent vectors. In that sense, the horosphere, together with the geometry that it inherits from the ℍ3 in which it is embedded, is a model of the Euclidean plane. In it, parallel horocycles are equidistant throughout their entire lengths; there are Cartesian coordinates; and so on. All the formulas and relations of Euclidean geometry apply. Therefore, by projecting small figures in that surface onto a tangent plane in ℍ3, we find that in that plane, for small figures near the point of tangency, the Euclidean formulas hold approximately, with relative error terms that tend to zero as the sizes of the figures tend to zero. In order to carry out this approach, we first develop three-dimensional hyperbolic geometry, which, however, has independent interest. We give a set of axioms, which, like those of the hyperbolic plane, are user friendly. The first seven axioms are almost identical with those of the hyperbolic plane and the last three deal with planes in ℍ3. We then develop a set of theorems of the geometry of ℍ3, and we prove that the geometry inherited by a horosphere is Euclidean.