Editing Hyperbolic geometry (section)

===Regular apeirogon and pseudogon===
[[File:Hyperbolic apeirogon example.png|thumb|An [[apeirogon]] and circumscribed [[horocycle]] in the [[Poincaré disk model]].]]
{{main article|Apeirogon#Hyperbolic geometry}}

Special polygons in hyperbolic geometry are the regular [[apeirogon]] and '''pseudogon''' [[uniform polygon]]s with an infinite number of sides.

In [[Euclidean geometry]], the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle, or make the interior angles tend to 180° and the apeirogon approaches a straight line.

However, in hyperbolic geometry, a regular apeirogon or pseudogon has sides of any length (i.e., it remains a polygon with noticeable sides).

The side and angle [[bisection|bisectors]] will, depending on the side length and the angle between the sides, be limiting or diverging parallel. If the bisectors are limiting parallel then it is an apeirogon and can be inscribed and circumscribed by concentric [[horocycle]]s.

If the bisectors are diverging parallel then it is a  pseudogon and can be inscribed and circumscribed by [[hypercycle (geometry)|hypercycles]] (since all its vertices are the same distance from a line, the axis, and the midpoints of its sides are also equidistant from that same axis).