09-02-2026, 10:52 PM
A hypercycle (also called a hypercircle or equidistant curve) is a curve in hyperbolic geometry whose points all lie at the same perpendicular distance from a fixed straight line, called its axis. Given an axis $L$ and a point $P$ not on it, there is exactly one hypercycle through $P$ consisting of all points on the same side of $L$ whose perpendicular distance from $L$ equals that of $P$. The perpendicular segments from the axis to the hypercycle act like “radii,” although unlike an ordinary Euclidean circle, the hypercycle does not have a single center point.
Hypercycles combine properties reminiscent of both Euclidean lines and circles. They are symmetric about every line perpendicular to them, a straight line can intersect a hypercycle in at most two points, and two distinct hypercycles can also meet in at most two points. Their axis and distance from it uniquely determine them, and two hypercycles are congruent exactly when they have the same distance from their axes. In a hyperbolic plane of curvature $-1$, if the hypercycle has radius $r$ and the corresponding points on its axis are separated by distance $d$, the length of the hypercycle arc is $l=dcoshr.l=d\cosh r.$
Thus the farther the curve lies from its axis, the more rapidly its length grows compared with the corresponding segment of the axis.
In common models of hyperbolic geometry, hypercycles have characteristic representations. In the Poincaré disk, they appear as lines or circular arcs meeting the boundary circle at angles other than $90^\circ$, whereas their axes meet the boundary orthogonally. The same idea applies in the Poincaré half-plane model. As the distance of a hypercycle from its axis tends to infinity, the hypercycle approaches a horocycle, making hypercycles an important intermediate class of curves between hyperbolic geodesics and horocycles. They also arise in the study and classification of conics in hyperbolic geometry.
Key takeaways
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Hypercycles combine properties reminiscent of both Euclidean lines and circles. They are symmetric about every line perpendicular to them, a straight line can intersect a hypercycle in at most two points, and two distinct hypercycles can also meet in at most two points. Their axis and distance from it uniquely determine them, and two hypercycles are congruent exactly when they have the same distance from their axes. In a hyperbolic plane of curvature $-1$, if the hypercycle has radius $r$ and the corresponding points on its axis are separated by distance $d$, the length of the hypercycle arc is $l=dcoshr.l=d\cosh r.$
Thus the farther the curve lies from its axis, the more rapidly its length grows compared with the corresponding segment of the axis.
In common models of hyperbolic geometry, hypercycles have characteristic representations. In the Poincaré disk, they appear as lines or circular arcs meeting the boundary circle at angles other than $90^\circ$, whereas their axes meet the boundary orthogonally. The same idea applies in the Poincaré half-plane model. As the distance of a hypercycle from its axis tends to infinity, the hypercycle approaches a horocycle, making hypercycles an important intermediate class of curves between hyperbolic geodesics and horocycles. They also arise in the study and classification of conics in hyperbolic geometry.
Key takeaways
- A hypercycle is the hyperbolic analogue of a curve at a constant distance from a straight line.
- It is completely determined by its axis and its distance $r$ from that axis.
- Its arc length satisfies $l=d\cosh r$ in curvature $-1$.
- As $r\to\infty$, hypercycles approach horocycles.
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