Английская Википедия:Hypercycle (geometry)
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 Шаблон:Mvar and a point Шаблон:Mvar not on Шаблон:Mvar, one can construct a hypercycle by taking all points Шаблон:Mvar on the same side of Шаблон:Mvar as Шаблон:Mvar, with perpendicular distance to Шаблон:Mvar equal to that of Шаблон:Mvar. The line Шаблон:Mvar is called the axis, center, or base line of the hypercycle. The lines perpendicular to Шаблон:Mvar, which are also perpendicular to the hypercycle, are called the normals of the hypercycle. The segments of the normals between Шаблон:Mvar 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:
- In a plane, given a line and a point not on it, there is only one hypercycle of that of the given line (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 Шаблон:Mvar be the chord and Шаблон:Mvar its middle point.
- By symmetry the line Шаблон:Mvar through Шаблон:Mvar perpendicular to Шаблон:Mvar must be orthogonal to the axis Шаблон:Mvar.
- Therefore Шаблон:Mvar is a radius.
- Also by symmetry, Шаблон:Mvar will bisect the arc Шаблон:Mvar.
- The axis and distance of a hypercycle are uniquely determined.
- Let us assume that a hypercycle Шаблон:Mvar has two different axes Шаблон:Math.
- Using the previous property twice with different chords we can determine two distinct radii Шаблон:Math. Шаблон:Math will then have to be perpendicular to both Шаблон: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 Шаблон:Mvar cut the hypercycle Шаблон:Mvar in two points Шаблон:Mvar. As before, we can construct the radius Шаблон:Mvar of Шаблон:Mvar through the middle point Шаблон:Mvar of Шаблон:Mvar. Note that Шаблон:Mvar is ultraparallel to the axis Шаблон:Mvar because they have the common perpendicular Шаблон: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 Шаблон:Mvar inside Шаблон:Mvar will have distance from Шаблон:Mvar smaller than the common distance of Шаблон:Mvar and Шаблон:Mvar from Шаблон:Mvar, while the points of Шаблон:Mvar outside Шаблон:Mvar will have greater distance. In conclusion, no other point of Шаблон:Mvar can be on Шаблон:Mvar.
- Two hypercycles intersect in at most two points.
- Let Шаблон:Math be hypercycles intersecting in three points Шаблон:Mvar.
- If Шаблон:Math is the line orthogonal to Шаблон:Mvar through its middle point, we know that it is a radius of both Шаблон:Math.
- Similarly we construct Шаблон:Math, the radius through the middle point of Шаблон:Mvar.
- Шаблон:Math are simultaneously orthogonal to the axes Шаблон:Math of Шаблон:Math, respectively.
- We already proved that then Шаблон:Math must coincide (otherwise we have a rectangle).
- Then Шаблон: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 Шаблон:Mvar of a hypercycle are collinear then the chords Шаблон:Mvar are on the same line Шаблон:Mvar. Let Шаблон:Math be the radii through the middle points of Шаблон:Mvar. We know that the axis Шаблон:Mvar of the hypercycle is the common perpendicular of Шаблон:Math.
- But Шаблон: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 Шаблон:Mvar with Шаблон: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 Шаблон:Mvar in an open half-plane of the axis inverts to Шаблон:Mvar whose angle of parallelism is the complement of that of Шаблон: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 Шаблон:Mvar and the distance between the points where the normals intersect with the axis Шаблон:Mvar using the formula Шаблон:Math.[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 Шаблон:Mvar of a given axis. In an incidence geometry, the Steiner conic at a point Шаблон:Mvar produced by a collineation Шаблон:Mvar is the locus of intersections Шаблон:Math for all lines Шаблон:Mvar through Шаблон: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 Шаблон:Mvar between Шаблон:Mvar and Шаблон:Math and the angle of rotation Шаблон:Mvar induced by Шаблон:Mvar about Шаблон:Math. Each Steiner parabola is the locus of points whose distance from a focus Шаблон: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 Шаблон:Mvar is determined by Шаблон:Mvar as follows. Fixing Шаблон:Math, the classes of parabolas are in one-to-one correspondence with Шаблон:Math. In the conformal disk model, each point Шаблон:Mvar is a complex number with Шаблон:Math. Let the common axis be the real line and assume the hypercycles are in the half-plane Шаблон:Mvar with Шаблон:Math. Then the vertex of each parabola will be in Шаблон:Mvar, and the parabola is symmetric about the line through the vertex perpendicular to the axis. If the hypercycle is at distance Шаблон:Mvar from the axis, with <math>\tanh d = \tan\tfrac{\phi}{2},</math> then <math display=block>F = \left(\frac{1-\tan\phi}{1+\tan\phi}\right)i.</math> In particular, Шаблон:Math when Шаблон:Math. In this case, the focus is on the axis; equivalently, inversion in the corresponding hypercycle leaves Шаблон: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
- Martin Gardner, Non-Euclidean Geometry, Chapter 4 of The Colossal Book of Mathematics, W. W. Norton & Company, 2001, Шаблон: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)
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