In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis).
Given a straight line and a point not on, one can construct a hypercycle by taking all points on the same side of as, with perpendicular distance to equal to that of . The line is called the axis, center, or base line of the hypercycle. The lines perpendicular to, which are also perpendicular to the hypercycle, are called the normals of the hypercycle. The segments of the normals between and the hypercycle are called the radii. Their common length is called the distance or radius of the hypercycle.[1]
The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity.
Hypercycles in hyperbolic geometry have some properties similar to those of lines in Euclidean geometry:
Hypercycles in hyperbolic geometry have some properties similar to those of circles in Euclidean geometry:
Let be the chord and its middle point.
By symmetry the line through perpendicular to must be orthogonal to the axis .
Therefore is a radius.
Also by symmetry, will bisect the arc .
Let us assume that a hypercycle has two different axes .
Using the previous property twice with different chords we can determine two distinct radii . will then have to be perpendicular to both, giving us a rectangle. This is a contradiction because the rectangle is an impossible figure in hyperbolic geometry.
If they have equal distance, we just need to bring the axes to coincide by a rigid motion and also all the radii will coincide; since the distance is the same, also the points of the two hypercycles will coincide.
Vice versa, if they are congruent the distance must be the same by the previous property.
Let the line cut the hypercycle in two points . As before, we can construct the radius of through the middle point of . Note that is ultraparallel to the axis because they have the common perpendicular . Also, two ultraparallel lines have minimum distance at the common perpendicular and monotonically increasing distances as we go away from the perpendicular.
This means that the points of inside will have distance from smaller than the common distance of and from, while the points of outside will have greater distance. In conclusion, no other point of can be on .
Let be hypercycles intersecting in three points .
If is the line orthogonal to through its middle point, we know that it is a radius of both .
Similarly we construct, the radius through the middle point of .
are simultaneously orthogonal to the axes of, respectively.
We already proved that then must coincide (otherwise we have a rectangle).
Then have the same axis and at least one common point, therefore they have the same distance and they coincide.
If the points of a hypercycle are collinear then the chords are on the same line . Let be the radii through the middle points of . We know that the axis of the hypercycle is the common perpendicular of .
But is that common perpendicular. Then the distance must be 0 and the hypercycle degenerates into a line.
In the hyperbolic plane of constant curvature −1, the length of an arc of a hypercycle can be calculated from the radius and the distance between the points where the normals intersect with the axis using the formula .[2]
In the Poincaré disk model of the hyperbolic plane, hypercycles are represented by lines and circle arcs that intersect the boundary circle at non-right angles. The representation of the axis intersects the boundary circle in the same points, but at right angles.
In the Poincaré half-plane model of the hyperbolic plane, hypercycles are represented by lines and circle arcs that intersect the boundary line at non-right angles. The representation of the axis intersects the boundary line in the same points, but at right angles.
The congruence classes of Steiner parabolas in the hyperbolic plane are in one-to-one correspondence with the hypercycles in a given half-plane of a given axis. In an incidence geometry, the Steiner conic at a point produced by a collineation is the locus of intersections for all lines through . This is the analogue of Steiner's definition of a conic in the projective plane over a field. The congruence classes of Steiner conics in the hyperbolic plane are determined by the distance between and and the angle of rotation induced by about . Each Steiner parabola is the locus of points whose distance from a focus is equal to the distance to a hypercycle directrix that is not a line. Assuming a common axis for the hypercycles, the location of is determined by as follows. Fixing, the classes of parabolas are in one-to-one correspondence with . In the conformal disk model, each point is a complex number with . Let the common axis be the real line and assume the hypercycles are in the half-plane with . Then the vertex of each parabola will be in, and the parabola is symmetric about the line through the vertex perpendicular to the axis. If the hypercycle is at distance from the axis, with
\tanhd=\tan\tfrac{\phi}{2},