Hypercycle (geometry)

Last updated
A Poincare disk showing the hypercycle HC that is determined by the straight line L (termed straight because it cuts the horizon at right angles) and point P Hypercycle (vector format).svg
A Poincaré disk showing the hypercycle HC that is determined by the straight line L (termed straight because it cuts the horizon at right angles) and point P

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).

Contents

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. [1]

The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity.

Properties 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:

Other properties

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. [2]

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 LT(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 |P| < 1. 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.

References

The alternated octagonal tiling, in a Poincare disk model, can be seen with edge sequences that follow hypercycles. Uniform tiling 433-t0 edgecenter.png
The alternated octagonal tiling, in a Poincaré disk model, can be seen with edge sequences that follow hypercycles.
  1. Martin, George E. (1986). The foundations of geometry and the non-euclidean plane (1., corr. Springer ed.). New York: Springer-Verlag. p. 371. ISBN   3-540-90694-0.
  2. Smogorzhevsky, A.S. (1982). Lobachevskian geometry . Moscow: Mir. p.  68.