Hyperbolic geometry is the result of replacing the parallel axiom of Euclidean Geometry with the alternative of there being, through a given point, at least two lines parallel to a given line. It's one thing to develop the theorems in such a geometry, but it's another to do constructions -- a euclidean model where the compass and straight edge may be used would be advantageous.
A model (there are others) in the euclidean plane for non-Euclidean
Hyperbolic geometry is due to Henri Poincaré.
The points in the geometry are all the points on one side of a fixed line,
i.e. a half-plane. The line that determines this half-plane can be called
h-bar (h with overscore). The lines in the geometry are of two types. Type-1
h-lines are euclidean rays perpendicular to h-bar.
Type-2 h-lines are semicircles centered on h-bar.
The angle formed by two intersecting h-lines is determined by substituting the tangent to any type-2 h-lines at the point of intersection and measuring the angle as we would between two intersecting e-lines.
To make congruent copies of figures we can use certain transformations. In euclidean geometry we can use translation, rotation, and reflection to make congruent copies of segments and angles. In the Poincaré model, reflection is achieved by euclidean reflection in type-1 h-lines and circle inversion in type-2 h-lines
Here's a Java version of a construction of a circle inversion from the inside of a circle to the outside and vice-versa.
Another use for circle inversions is in the fact that a circle passing through both points of a circle inversion pair will be orthogonal to the circle of inversion. So circle inversion may be used to aid in the construction of a type-2 h-line through a given point that is perpendicular to a given type-2 h-line.