Hypercycle (geometry)

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: 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.[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 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 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).

Let the common axis be the real line and assume the hypercycles are in the half-plane H with Im P > 0.

In this case, the focus is on the axis; equivalently, inversion in the corresponding hypercycle leaves H invariant.