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.