Hypercycles and Horocycles

There is a curve peculiar to hyperbolic geometry, called the horocycle  . Consider two limiting parallel lines, and m, with a common direction, say . Let P be a point on one of these lines . If there exists a point such that the trilateral , , has the property that

then we say that Q corresponds to P. If the trilateral has the above property we shall say that it is equiangular . Note that it is obvious from the definition that if Q corresponds to P, then P corresponds to Q. The points P and Q are called a pair of corresponding points .

Theorem 18.1:   If points P and Q lie on two limiting parallel lines in the direction of the ideal point, , they are corresponding points on these lines if and only if the perpendicular bisector of PQ is limiting parallel to the lines in the direction of .

Theorem 18.2:   Given any two limiting parallel lines, there exists a line each of whose points is equidistant from them. The line is limiting parallel to them in their common direction.

Proof: Let and m be limiting parallel lines with common direction . Let and . The bisector of in the trilateral meets side in a point X and the bisector of meets side AX of the triangle in a point C. Thus the bisectors of the angles of the trilateral meet in a point C. Drop perpendiculars from C to each of and m, say P and Q, respectively. By Hypothesis-Angle (M is the midpoint of AB) and . Thus, . Thus, by SAS for trilaterals, we have that

and thus the angles at C are congruent. Now, consider the line and let F be any point on it other than C. By SAS we have . If S and T are the feet of F in and m, then we get that and . Thus, every point on the line is equidistant from and m.

This line is called the equidistant line. 

Theorem 18.3:   Given any point on one of two limiting parallel lines, there is a unique point on the other which corresponds to it.

Theorem 18.4:   If three points P, Q, and R lie on three parallels in the same direction so that P and Q are corresponding points on their parallels and Q and R are corresponding points on theirs, then P, Q, and R are noncollinear.

Theorem 18.5:   If three points P, Q, and R lie on three parallels in the same direction so that P and Q are corresponding points on their parallels and Q and R are corresponding points on theirs, then P and R are corresponding points on their parallels.

Consider any line , any point , and an ideal point in one direction of , say . On each line parallel to in the direction there is a unique point Q that corresponds to P. The set consisting of P and all such points Q is called a horocycle , or, more precisely, the horocycle determined by , P, and . The lines parallel to in the direction , together with , are called the radii  of the horocycle. Since may be denoted by , we may regard the horocycle as determined simply by P and , and hence call it the horocycle through P with direction , or in symbols, the horocycle  .

All the points of this horocycle are mutually corresponding points by Theorem 19.5 , so the horocycle is equally well determined by any one of them and . In other words if Q is any point of horocycle other than P, then horocycle is the same as horocycle . If, however, P' is any point of other than P, then horocycle is different from horocycle , even though they have the same direction and the same radii. Such horocycles, having the same direction and the same radii, are called codirectional horocycles. 

There are analogies between horocycles and circles. We will mention a few.

Lemma 18.1: There is a unique horocycle with a given direction which passes through a given point. (There is a unique circle with a given center which passes through a given point.)

Lemma 18.2: Two codirectional horocycles have no common point. (Two concentric circles have no common point.)

Lemma 18.3: A unique radius is associated with each point of a horocycle. (A unique radius is associated with each point of a circle.)

A tangent to a horocycle  at a point on the horocycle is defined to be the line through the point which is perpendicular to the radius associated with the point.

No line can meet a horocycle in more than two points. This is a consequence of the fact that no three points of a horocycle are collinear inasmuch as it is a set of mutually corresponding points, cf. Theorem 19.4.

Theorem 18.6:   The tangent at any point A of a horocycle meets the horocycle only in A. Every other line through A except the radius meets the horocycle in one further point B. If is the acute angle between this line and the radius, then d(A,B) is twice the segment which corresponds to as angle of parallelism. 

Proof Let t be the tangent to the horocycle at A and let be the direction of the horocycle. If t met the horocycle in another point B, we would have a trilateral with two right angles, since A and B are corresponding points. In fact the entire horocycle, except for A, lies on the same side of t, namely, the side containing the ray .

Let k be any line through A other than the tangent or radius. We need to show that k meets the horocycle in some other point. Let be the acute angle between k and the ray . Let C be the point of k, on the side of t containing the horocycle, such that AC is a segment corresponding to as angle of parallelism. (RECALL: ). The line perpendicular to k at C is then parallel to in the direction . Let B be the point of k such that C is the midpoint of AB. The trilaterals and are congruent. Hence , B corresponds to A, and .

A chord  of a horocycle is a segment joining two points of the horocycle.

Theorem 18.7:   The line which bisects a chord of a horocycle at right angles is a radius of the horocycle.

 
Figure 19.1: A horocycle in the Poincaré model

We can visualize a horocycle in the Poincaré model as follows. Let be the diameter of the euclidean circle whose interior represents the hyperbolic plane, and let O be the center of . It is a fact that the hyperbolic circle with hyperbolic center P is represented by a euclidean circle whose euclidean center R lies between P and A.

As P recedes from A towards the ideal point , R is pulled up to the euclidean midpoint of , so that the horocycle is a euclidean circle tangent to at and tangent to at A. It can be shown that all horocycles are represented in the Poincaré model by euclidean circles inside and tangent to . Moreover, all the Poincaré lines passing through the ideal point are orthogonal to the horocycle .

In the Poincaré model two horocycles tangent to at are said to be concentric. It is by studying the ratio of corresponding arcs on concentric horocycles that the Bolyai-Lobachevsky formula can be derived.  

 
Figure 19.2: Concentric horocycles in the Poincaré model

Another curve found specifically in the hyperbolic plane and nowhere else is the equidistant curve, or hypercycle.   Given a line and a point P not on , consider the set of all points Q on one side of so that the perpendicular distance from Q to is the same as the perpendicular distance from P to .

The line is called the axis,  or base line,  and the common length of the perpendicular segments is called the distance. The perpendicular segments defining the hypercycle are called its radii. The following statements about hypercycles are analogous to statements about regular euclidean circles.

1. Hypercycles with equal distances are congruent, those with unequal distances are not. (Circles with equal radii are congruent, those with unequal radii are not.)
2. A line cannot cut a hypercycle in more than two points.
3. If a line cuts a hypercycle in one point, it will cut it in a second unless it is tangent to the curve or parallel to it base line.
4. A tangent line to a hypercycle is defined to be the line perpendicular to the radius at that point. Since the tangent line and the base line have a common perpendicular, they must be hyperparallel. This perpendicular segment is the shortest distance between the two lines. Thus, each point on the tangent line must be at a greater perpendicular distance from the base line than the corresponding point on the hypercycle. Thus, the hypercycle can intersect the hypercycle in only one point.
5. A line perpendicular to a chord of a hypercycle at its midpoint is a radius and it bisects the arc subtended by the chord.
6. Two hypercycles intersect in at most two points.
7. No three points of a hypercycle are collinear.

In the Poincaré model let P and Q be the ideal end points of . It can be shown that the hypercycle to through P is represented by the arc of the euclidean circle passing through A, B, and P. This curve is orthogonal to all Poincaré lines perpendicular to the line .

 
Figure 19.3: A hypercycle in the Poincaré model

In the Poincaré model a euclidean circle represents:

1. a hyperbolic circle if it is entirely inside ;
2. a horocycle if it is inside except for one point where it is tangent to ;
3. an hypercycle if it cuts nonorthogonally in two points;
4. a hyperbolic line if it cuts orthogonally.

It follows that in the hyperbolic plane three non-collinear points lie either on a circle, a horocycle, or a hypercycle accordingly, as the perpendicular bisectors of the triangle are concurrent in an ordinary point, an ideal point, or an ultraideal point.