Hypercycle (geometry)

A Poincaré disk showing the hypercycle Template:Mvar that is determined by the straight line Template:Mvar (termed straight because it cuts the horizon at right angles) and point Template:Mvar

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 Template:Mvar and a point Template:Mvar not on Template:Mvar, one can construct a hypercycle by taking all points Template:Mvar on the same side of Template:Mvar as Template:Mvar, with perpendicular distance to Template:Mvar equal to that of Template:Mvar. The line Template:Mvar is called the axis, center, or base line of the hypercycle. The lines perpendicular to Template:Mvar, which are also perpendicular to the hypercycle, are called the normals of the hypercycle. The segments of the normals between Template:Mvar 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.

Properties similar to those of Euclidean lines

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

• In a plane, given an axis (line) and a point not on that axis, there is only one hypercycle through that point with the given axis (compare with Playfair's axiom for Euclidean geometry).
• No three points of a hypercycle are on a circle.
• A hypercycle is symmetrical to each line perpendicular to it. (Reflecting a hypercycle in a line perpendicular to the hypercycle results in the same hypercycle.)

Properties similar to those of Euclidean circles

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

• A line perpendicular to a chord of a hypercycle at its midpoint is a radius and it bisects the arc subtended by the chord.

  Let Template:Mvar be the chord and Template:Mvar its middle point.

  By symmetry the line Template:Mvar through Template:Mvar perpendicular to Template:Mvar must be orthogonal to the axis Template:Mvar.

  Therefore Template:Mvar is a radius.

  Also by symmetry, Template:Mvar will bisect the arc Template:Mvar.

• The axis and distance of a hypercycle are uniquely determined.

  Let us assume that a hypercycle Template:Mvar has two different axes Template:Math.

  Using the previous property twice with different chords we can determine two distinct radii Template:Math. Template:Math will then have to be perpendicular to both Template:Math, giving us a rectangle. This is a contradiction because the rectangle is an impossible figure in hyperbolic geometry.

• Two hypercycles have equal distances if and only if they are congruent.

  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.

• A straight line cuts a hypercycle in at most two points.

  Let the line Template:Mvar cut the hypercycle Template:Mvar in two points Template:Mvar. As before, we can construct the radius Template:Mvar of Template:Mvar through the middle point Template:Mvar of Template:Mvar. Note that Template:Mvar is ultraparallel to the axis Template:Mvar because they have the common perpendicular Template:Mvar. 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 Template:Mvar inside Template:Mvar will have distance from Template:Mvar smaller than the common distance of Template:Mvar and Template:Mvar from Template:Mvar, while the points of Template:Mvar outside Template:Mvar will have greater distance. In conclusion, no other point of Template:Mvar can be on Template:Mvar.

• Two hypercycles intersect in at most two points.

  Let Template:Math be hypercycles intersecting in three points Template:Mvar.

  If Template:Math is the line orthogonal to Template:Mvar through its middle point, we know that it is a radius of both Template:Math.

  Similarly we construct Template:Math, the radius through the middle point of Template:Mvar.

  Template:Math are simultaneously orthogonal to the axes Template:Math of Template:Math, respectively.

  We already proved that then Template:Math must coincide (otherwise we have a rectangle).

  Then Template:Math have the same axis and at least one common point, therefore they have the same distance and they coincide.

• No three points of a hypercycle are collinear.

  If the points Template:Mvar of a hypercycle are collinear then the chords Template:Mvar are on the same line Template:Mvar. Let Template:Math be the radii through the middle points of Template:Mvar. We know that the axis Template:Mvar of the hypercycle is the common perpendicular of Template:Math.

  But Template:Mvar is that common perpendicular. Then the distance must be 0 and the hypercycle degenerates into a line.

Other properties

• The length of an arc of a hypercycle between two points is
  • longer than the length of the line segment between those two points,
  • shorter than the length of the arc of one of the two horocycles between those two points, and
  • shorter than any circle arc between those two points.
• A hypercycle and a horocycle intersect in at most two points.
• A hypercycle of radius Template:Mvar with Template:Math 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 Template:Mvar in an open half-plane of the axis inverts to Template:Mvar whose angle of parallelism is the complement of that of Template:Mvar. 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 Template:Mvar and the distance between the points where the normals intersect with the axis Template:Mvar using the formula Template:Math.

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 Template:Mvar of a given axis. In an incidence geometry, the Steiner conic at a point Template:Mvar produced by a collineation Template:Mvar is the locus of intersections Template:Math for all lines Template:Mvar through Template:Mvar. 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 Template:Mvar between Template:Mvar and Template:Math and the angle of rotation Template:Mvar induced by Template:Mvar about Template:Math. Each Steiner parabola is the locus of points whose distance from a focus Template:Mvar is equal to the distance to a hypercycle directrix that is not a line. Assuming a common axis for the hypercycles, the location of Template:Mvar is determined by Template:Mvar as follows. Fixing Template:Math, the classes of parabolas are in one-to-one correspondence with Template:Math. In the conformal disk model, each point Template:Mvar is a complex number with Template:Math. Let the common axis be the real line and assume the hypercycles are in the half-plane Template:Mvar with Template:Math. Then the vertex of each parabola will be in Template:Mvar, and the parabola is symmetric about the line through the vertex perpendicular to the axis. If the hypercycle is at distance Template:Mvar from the axis, with $\tanh d=\tan \frac{\varphi }{2},$ then $$F=\left(\frac{1-\tan \varphi }{1+\tan \varphi }\right)i.$$ In particular, Template:Math when Template:Math. In this case, the focus is on the axis; equivalently, inversion in the corresponding hypercycle leaves Template:Mvar 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 Poincaré disk model, can be seen with edge sequences that follow hypercycles.

• Martin Gardner, Non-Euclidean Geometry, Chapter 4 of The Colossal Book of Mathematics, W. W. Norton & Company, 2001, Template:ISBN
• M. J. Greenberg, Euclidean and Non-Euclidean Geometries: Development and History, 3rd edition, W. H. Freeman, 1994.
• George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer-Verlag, 1975.
• J. G. Ratcliffe, Foundation of Hyperbolic Manifolds, Springer, New York, 1994.
• David C. Royster, Neutral and Non-Euclidean Geometries.
• J. Sarli, Conics in the hyperbolic plane intrinsic to the collineation group, J. Geom. 103: 131-138 (2012)