The circular inversion
The inversion application
The well know inversion map plays an important role when restricted to the Hyperbolic disk. So the hyperbolic inversion maps circles and lines in circles and/or lines, playing in the hyperbolic world a similar figure that plays the reflection in the euclidean one.
In the Poincaré Disk model, hyperbolic lines (arc perpendicular to de disk, are mapped to the complementary arc and circles tangent to the disk are mapped into also tangent circles by that point, but inside. Straight lines outsides the disc are mapped into circles inside that pases by the center, corresponding by inversion to infinity point