Hypercycle (geometry)

In hyperbolic geometry, an hypercycle or equidistant curve' is a line whose points have the same orthogonal distance from a given straight line.

Given a straight line l and a point P not on l, we can construct an hypercycle by taking all points Q on one side of l with perpendicular distance to l equal to that of P.

The line l is calle the axis or base line of the hypercycle. The orthogonal segments from each point to l are called radii. Their common length is called distance.

Hypercycles in hyperbolic geometry have some properties similar to those of circles in Euclidean geometry:

• Two hypercycles have equal distances iff they are congruent.
• A line cannot cut a hypercycle in more than two points.
• If a line cuts a hypercycle in one point, it will cut it in a second unless it is tangent to the curve or parallel to it base line.
• A tangent line to a hypercycle is defined to be the line perpendicular to the radius at that point. Since the tangent line and the base line have a common perpendicular, they must be hyperparallel. This perpendicular segment is the shortest distance between the two lines. Thus, each point on the tangent line must be at a greater perpendicular distance from the base line than the corresponding point on the hypercycle. Thus, the hypercycle can intersect the hypercycle in only one point.
• A line perpendicular to a chord of a hypercycle at its midpoint is a radius and it bisects the arc subtended by the chord.
• Two hypercycles intersect in at most two points.
• No three points of a hypercycle are colline