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:
This simply says that we consider all the complex numbers that have the imaginary part strictly positive, i.e. if , then .
The notion of angle in is the one inherited from , 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 and the Euclidean circle in terms of the complex coordinates .
Exercise 2 Let be the unit circle in the complex plane. Let be a Euclidean circle in the complex plane, with Euclidean center and Euclidean radius . Show that is perpendicular to if and only if .
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.
Proposition 1 For each pair and of distinct points in , there exists a unique hyperbolic line passing through and .
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.
Suppose now that . As the Euclidean line through and 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 the Euclidean line segment joining the two points, and let be the perpendicular bisector of . Then (and this you should know from Euclidean geometry), every Euclidean circle that passes through and has its center on . As and have non-equal real parts, the Euclidean line is not parallel to the real axis, so intersects the real axis at a unique point .
Let be the Euclidean circle, centered at this point, with radius (the two are equal, since the circle passes through both points) (see figure). So we have the line, .
For example, if is the hyperbolic line in passing through and , we can express explicitly in terms of and . Like we saw above, if the two have the same real parts, then . For the second case, do the following exercise:
Exercise 3 Let and be distinct points in with non-equal real parts, and let be the Euclidean circle centered on and passing through and . Express the Euclidean center and the Euclidean radius of in terms of , , , .
Hyperbolic geometry behaves very differently from Euclidean geometry in , even though it is expressed in terms of the latter.
Definition 1 Two hyperbolic lines in 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.
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.