Editing Hyperbolic geometry (section)

=== Hypercycles and horocycles ===
[[File:Hyperbolic pseudogon example0.png|thumb|Hypercycle and pseudogon in the [[Poincare disk model]] ]]
{{main article|Hypercycle (hyperbolic geometry)|horocycle}}

In hyperbolic geometry, there is no line whose points are all equidistant from another line. Instead, the points that are all the same distance from a given line lie on a curve called a [[hypercycle (hyperbolic geometry)|hypercycle]].

Another special curve is the [[horocycle]], whose [[normal (geometry)|normal]] radii ([[perpendicular]] lines) are all [[limiting parallel]] to each other (all converge asymptotically in one direction to the same [[ideal point]], the centre of the horocycle).

Through every pair of points there are two horocycles. The centres of the horocycles are the [[ideal point]]s of the [[perpendicular bisector]] of the line-segment between them.

Given any three distinct points, they all lie on either a line, hypercycle, [[horocycle]], or circle.

The '''length''' of a line-segment is the shortest length between two points.

The arc-length of a hypercycle connecting two points is longer than that of the line segment and shorter than that of the arc horocycle, connecting the same two points.

The lengths of the arcs of both horocycles connecting two points are equal, and are longer than the arclength of any hypercycle connecting the points and shorter than the arc of any circle connecting the two points.

If the Gaussian curvature of the plane is −1, then the [[geodesic curvature]] of a horocycle is 1 and that of a hypercycle is between 0 and 1.<ref name="auto"/>