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