structural:reformatting, structural:scholar_edition
Hypercycles and horocycles

In hyperbolic geometry, there is no line all of whose points are equidistant from another line. Instead, the points that all have the same orthogonal distance from a given line lie on a curve called a hypercycle.

Another special curve is the horocycle, a curve whose 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 points 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 the 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 of a hypercycle is between 0 and 1.