6. The complex half-plane model for the hyperbolic plane

First, review complex numbers! From now on we use the properties of complex numbers!

The model that we start with is called the the upper half-plane model and it is defined to be:

$\displaystyle \mathbb{H}=\left\{z\in \mathbb{C}\vert Im(z)>0\right\}$    

This simply says that we consider all the complex numbers that have the imaginary part strictly positive, i.e. if $ z=a+bi$, then $ b>0$.

The notion of angle in $ \mathbb{H}$ is the one inherited from $ \mathbb{C}$, so the angle between two curves is the angle between their tangent lines.

Before we prove anything, let's do some warm-up exercises.


Exercise 1   Express the equations of the Euclidean line $ ax+by+c=0$ and the Euclidean circle $ (x-h)^2+(y-k)^2=r^2$ in terms of the complex coordinates $ z=x+iy$.

Exercise 2   Let $ \mathbb{S}=\{z\in\mathbb{C}\vert\vert z\vert=1\}$ be the unit circle in the complex plane. Let $ A$ be a Euclidean circle in the complex plane, with Euclidean center $ re^{i\theta}, r>1$ and Euclidean radius $ s>0$. Show that $ A$ is perpendicular to $ \mathbb{S}$ if and only if $ s=\sqrt{r^2-1}$.

You may wonder how this hyperbolic world looks like in this model. A good way to imagine it is to see how lines look like in this new world.

Hyperbolic Lines

There are two seemingly different types of hyperbolic lines, both defined in terms of Euclidean objects in $ \mathbb{C}$. One is the intersection of the half-plane with a Euclidean line in the complex plane perpendicular to the real axis $ \mathbb{R}$. The other is the intersection of $ \mathbb{H}$ with a Euclidean circle centered on the real axis $ \mathbb{R}$. Here is a picture of hyperbolic lines:

Figure 1: Hyperbolic lines in $ \mathbb{H}$
Image Section61

Since the only difference between non-Euclidean and Euclidean geometry is the fifth postulate, the first four should hold. Using just our definition above, we should see that those four postulates hold. The proof is very tedious, so we will only show the first one: there should exist one and only one hyperbolic line passing through any pair of distinct points. Let's see how this follows from our definition:

Proposition 1   For each pair $ p$ and $ q$ of distinct points in $ \mathbb{H}$, there exists a unique hyperbolic line $ l$ passing through $ p$ and $ q$.


This is a typical existence and uniqueness problem. We will first show that such a line exists, and then that it is unique. We divide the proof into two parts: existence and uniqueness.


There are two cases to consider. Suppose first that $ Re(p)=Re(q)$. Then the Euclidean line $ L$ given by the equation $ L=\{z\in\mathbb{C}\vert Re(z)=Re(p)\}$ is perpendicular to the real axis and passes through both $ p$ and $ q$. So the desired line is $ l=L\cap\mathbb{H}$.

Suppose now that $ Re(p)\neq Re(q)$. As the Euclidean line through $ p$ and $ q$ is no longer perpendicular to the real line, we need to construct a Euclidean circle centered on the real axis and passing through the two points.

Denote with $ L_{pq}$ the Euclidean line segment joining the two points, and let $ K$ be the perpendicular bisector of $ L_{pq}$. Then (and this you should know from Euclidean geometry), every Euclidean circle that passes through $ p$ and $ q$ has its center on $ K$. As $ p$ and $ q$ have non-equal real parts, the Euclidean line $ K$ is not parallel to the real axis, so $ K$ intersects the real axis at a unique point $ c$.

Let $ A$ be the Euclidean circle, centered at this point, with radius $ \vert c-p\vert=\vert c-q\vert$ (the two are equal, since the circle passes through both points) (see figure). So we have the line, $ l=A\cap\mathbb{H}$.

Figure 2: Constructing a hyperbolic line
Image Section62


The uniqueness of the line follows from the uniqueness of the Euclidean lines and the Euclidean circle. The proof of the first postulate is complete. $ \square$

Note that since we have chosen the underlying space for this model of the hyperbolic plane to be contained in the complex plane, we can use whatever facts about Euclidean lines and Euclidean circles we already know to analyse the behaviour of hyperbolic lines.

For example, if $ l$ is the hyperbolic line in $ \mathbb{H}$ passing through $ p$ and $ q$, we can express $ l$ explicitly in terms of $ p$ and $ q$. Like we saw above, if the two have the same real parts, then $ l=\{z\in\mathbb{C}\vert Re(z)=Re(p)>0\}$. For the second case, do the following exercise:

Exercise 3   Let $ p$ and $ q$ be distinct points in $ \mathbb{C}$ with non-equal real parts, and let $ A$ be the Euclidean circle centered on $ \mathbb{R}$ and passing through $ p$ and $ q$. Express the Euclidean center $ c$ and the Euclidean radius $ r$ of $ A$ in terms of $ Re(p)$, $ Im(p)$, $ Re(q)$, $ Im(q)$.

Hyperbolic geometry behaves very differently from Euclidean geometry in $ \mathbb{C}$, even though it is expressed in terms of the latter.

Definition 1   Two hyperbolic lines in $ \mathbb{H}$ are parallel if they are disjoint.

Now we can see a first difference: in Euclidean geometry, parallel lines are equidistant, and so there is only one parallel through a given point to a given line, but in hyperbolic geometry things are completely the opposite - there are infinitely many distinct parallel lines through a given point to a given line.

In the figure below you can see how parallel lines look in our model of the hyperbolic plane: lines 1, 3, 4, 5, 6 are parallel, but 2, 3, 6, 7, 8 are not. Try to find all the pairs of parallel lines.

Figure 3: Several parallel hyperbolic lines
Image Section63

You may wonder how polygons, circles and other figures look in hyperbolic geometry. The half-space model is not very good to visualise these objects, it is mostly used for computation or proving different properties (even if there are different models, the properties are the same for all, since they all describe the same concept, hyperbolic geometry). This is the reason why the next model, the Poincaré disk, is used for visualisation.