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 L and a point P not on L, one can construct a hypercycle by taking all points Q on the same side of L as P, with perpendicular distance to L equal to that of P. The line L is called the axis, center, or base line of the hypercycle. The lines perpendicular to L, which are also perpendicular to the hypercycle, are called the normals of the hypercycle. The segments of the normals between L and the hypercycle are called the radii. Their common length is called the distance or radius of the hypercycle. The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity.

WHATTHESIGMAAREUDOINGHERE similar to those of Euclidean lines

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

Properties similar to those of Euclidean circles

Hypercycles in hyperbolic geometry have some properties similar to those of circles in Euclidean geometry: L1, L2 . R1, R2 . R1, R2 will then have to be perpendicular to both L1, L2 , giving us a rectangle. This is a contradiction because the rectangle is an impossible figure in hyperbolic geometry. C1, C2 be hypercycles intersecting in three points A, B, C. R1 is the line orthogonal to AB through its middle point, we know that it is a radius of both C1, C2 . R2 , the radius through the middle point of BC. R1, R2 are simultaneously orthogonal to the axes L1, L2 of C1, C2 , respectively. L1, L2 must coincide (otherwise we have a rectangle). C1, C2 have the same axis and at least one common point, therefore they have the same distance and they coincide. R1, R2 be the radii through the middle points of AB, BC. We know that the axis L of the hypercycle is the common perpendicular of R1, R2 .

Other properties

sinh 2r = 1 induces a quasi-symmetry of the hyperbolic plane by inversion. (Such a hypercycle meets its axis at an angle of π/4.) Specifically, a point P in an open half-plane of the axis inverts to P' whose angle of parallelism is the complement of that of P. This quasi-symmetry generalizes to hyperbolic spaces of higher dimension where it facilitates the study of hyperbolic manifolds. It is used extensively in the classification of conics in the hyperbolic plane where it has been called split inversion. Though conformal, split inversion is not a true symmetry since it interchanges the axis with the boundary of the plane and, of course, is not an isometry.

Length of an arc

In the hyperbolic plane of constant curvature −1, the length of an arc of a hypercycle can be calculated from the radius r and the distance between the points where the normals intersect with the axis d using the formula l = d cosh r .

Construction

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.

Congruence classes of Steiner parabolas

The congruence classes of Steiner parabolas in the hyperbolic plane are in one-to-one correspondence with the hypercycles in a given half-plane H of a given axis. In an incidence geometry, the Steiner conic at a point P produced by a collineation T is the locus of intersections L ∩ T(L) for all lines L through P. 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 s between P and T(P) and the angle of rotation φ induced by T about T(P) . Each Steiner parabola is the locus of points whose distance from a focus F is equal to the distance to a hypercycle directrix that is not a line. Assuming a common axis for the hypercycles, the location of F is determined by φ as follows. Fixing sinh s = 1 , the classes of parabolas are in one-to-one correspondence with φ ∈ (0, π/2) . In the conformal disk model, each point P is a complex number with . Let the common axis be the real line and assume the hypercycles are in the half-plane H with Im P > 0 . Then the vertex of each parabola will be in H, and the parabola is symmetric about the line through the vertex perpendicular to the axis. If the hypercycle is at distance d from the axis, with then In particular, F = 0 when φ = π/4 . In this case, the focus is on the axis; equivalently, inversion in the corresponding hypercycle leaves H invariant. This is the harmonic case, that is, the representation of the parabola in any inversive model of the hyperbolic plane is a harmonic, genus 1 curve.