One frequent mannequin of the hyperbolic aircraft is the Poincaré higher half aircraft ℍ. That is the set of factors within the advanced aircraft with optimistic imaginary half. Straight strains are both vertical, a set of factors with fixed imaginary half, or arcs of circles centered on the actual axis. The actual axis just isn’t a part of ℍ. From the attitude of hyperbolic geometry these are ideally suited elements, infinitely distant, and never a part of the aircraft itself.
We will outline a metric on ℍ as follows. To search out the space between two factors u and v, draw a line between the 2 factors, and let a and b be the best factors on the finish of the road. By a line we imply a line as outlined within the geometry of ℍ, what we might see from our Euclidean perspective as a half circle or a vertical line. Then the space between u and v is outlined as absolutely the worth of the log of the cross ratio (u, v; a, b).
Cross ratios are unchanged by Möbius transformations, and so Möbius transformations are isometries.
One other frequent mannequin of hyperbolic geometry is the Poincaré disk. We will use the identical metric on the Poincaré disk as a result of the Möbius transformation
maps the higher half aircraft to the unit disk. That is similar to how the Smith chart is created by mapping a grid in the appropriate half aircraft to the unit disk.
