Figure 2: The angle between two hyperbolic lines is defined as
the Euclidean angle between the tangents at their point of intersection.
The Poincaré angle ACB is defined as the Euclidean angle
A’CB’ .
Distance is more complicated in the Poincaré disk model and is
significantly distorted. The ratio of distances is also distorted.
However, the cross ratio defined as a ratio of ratios of distances is
preserved [7].
Definition 3: Suppose that A and B are any two
points in the Poincaré disk and let P and Q be the ideal
endpoints of the line that connects them. Then the cross ratio is
defined as
where the AP, AQ, BP and BQ are Euclidean
lengths.
Theorem 3: Let c be a circle with center O. If
A , B, P, Q are four distinct points
different from O and A’, B’, P’, Q’
are there inverses with respect to c, then their cross ratio is
preserved, .
Definition 4: The distance between points A and B
in the Poincaré disk model is defined in terms of the cross ratio as .
A hyperbolic circle with center A and radius r is defined
as the set of all points X in the Poincaré disk such that
d (A, X) = r. The relationship between
hyperbolic and Euclidean circles in the Poincaré disk model is captured
in the following theorem [1, 7].
Theorem 4: Hyperbolic circles in the Poincaré disk model are
also Euclidean circles but their centers are not the same unless the
hyperbolic circle is centered at O.