Chapter 1
The euclidean plane
We are all very familiar with the geometry of the euclidean plane R2.
We will encounter a new type of 2-dimensional geometry in the next
chapter, that of the hyperbolic plane H2. In this chapter, we first
list a series of well-known properties of the euclidean plane which,
in the next chapter, will enable us to develop the properties of the
hyperbolic plane in very close analogy.
Before proceeding, you are advised to briefly consult the Tool
Kit in the appendix for a succinct review of the basic definitions and
notation concerning set theory, infima and suprema of sets of real
numbers, and complex numbers.
1.1. Euclidean length and distance
The euclidean plane is the set
R2
= {(x, y); x, y R}
consisting of all ordered pairs (x, y) of real numbers x and y.
If γ is a curve in
R2,
parametrized by the differentiable vector-
valued function
t
(
x(t), y(t)
)
, a t b,
1
http://dx.doi.org/10.1090/stml/049/01
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