There exist many excellent books on non-Euclidean geometry. To add another one is motivated by the fact that these books contain very little about the geometry in hyperbolic space which has found various applications. Most of what is known is to be found in very old, not easily accessible papers. Many of these have shortcomings. In general the proofs do not cover exceptional cases, often of
There exist many excellent books on non-Euclidean geometry. To add another one is motivated by the fact that these books contain very little about the geometry in hyperbolic space which has found various applications. Most of what is known is to be found in very old, not easily accessible papers. Many of these have shortcomings. In general the proofs do not cover exceptional cases, often of special interest. Sometimes one refers to a passage to the limit, an exact proof of which may be tedious. There are also cases in which it is not even obvious what can be expected to be valid in the limit. Further, it had already been noted in old papers that satisfactory results in the geometry of lines, which actually covers the elementary geometry of the hyperbolic plane, can only be obtained if oriented lines are considered. The proofs applied yield however often only squares of the relations aimed at, and the determination of the correct signs is not convincing.