This definition of “horocycle”


“In hyperbolic geometry, a horocycle, sometimes called an oricycle, oricircle, or limit circle, is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horosphere.”

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Hyperbolical geometry describes a weird but possible space that cannot quite be ELI5’d but in that space directions are more “tightly packed” than usual: if you start drawing two parallel straight lines, they will diverge at a distance; area of a square is greater than pi*R^(2) etc. Turns out, in that space there’s a special kind of circles possible: they curve infinitely and have infinite length without ever reaching the starting point, their center lies infinitely far away. That’s what is called a horocycle.

I recommend trying a game called Hyperrogue to get a feel for hyperbolic geometry (it also has a TVTropes page). It can be downloaded for free on its website, and it specifically illustrates weird hyperbolic things like horocycles, some areas are based on them.

Horocycle is just a horosphere of a specific dimension.

In the upper half-space model of hyperbolic geometry any vertical line is a geodesic. All of those lines converge asymptotically to the unique “point at infinity” of this model. Any horizontal plane is perpendicular to any vertical line, therefore this plane along with “the point at infinity” is a horosphere.

In Poincaré model, for example, horospheres actually look like Euclidean spheres. If you pick a “point on the boundary” of this model, the circular arcs that are perpendicular to the boundary at that point are the geodesics of this model and the horospheres are spheres that are tangent to the boundary at that point.

Horocycles occur when you do geometry in the hyperbolic plane. The hyperbolic plane is a strange world that has many descriptions, but let say that it’s a curved surface with curvature equal to -1 at every point. (A sphere has curvature +1 at everypoint, A doughnut has variable curvature between +1 and -1, etc …). It’s not easy to make a drawing of an actual surface with constant curvature -1, so instead we draw maps of this hyperbolic plane to make sense of it (Imagine the equivalent of a world map : a curved surface that we project on a flat piece of paper).

Now, to illustrate the concept of horocycle on a [picture]( :

The disk is our map of the entire hyperbolic plane. The hyperbolic plane itself is infinite, but we can visualize the entirety of it in that finite disk. So points that are represented on the outer circle of the map are actually not part of the hyperbolic plane, and are in a sense infinitely far away.

The red lines are the so-called geodesics. Geodesics are the shortest paths between points. In usual geometry, the geodesics are just straight lines and that’s how you should think about them. It’s the paths that you would follow at the surface of the object if you are not turning left on right. But when we visualize geodesics on a map of a curved surface, they may appear curved (the same is true for any world map. The trajectory of an airplane from Paris to New-York will not be reepresented by the straight line on the map). So the red lines appear curved in that particular map of the hyperbolic plane, but they are actually the “straight lines” of hyperbolic geometry.

In that picture, all the drawn red lines are “converging asymptotically”. What that mean is that all the red lines appear to meet at a single point which is on the outer circle. But remember, the outer circle is not actually a part of the hyperbolic plane, but represent a point at infinity. So these geodesics actually never meet in the hyperbolic plane.

Now, the blue circle represent a curve in the hyperbolic plane. This curve is neither a geodesic nor a circle in the hyperbolic geometry. It’s something else, something intermediate, that has the following property : at any point of this blue circle, if you start perpendicular to that circle (the notion of perpendicular makes sense in the hyperbolic plane) and follow the geodesic in that direction, you will always arrive at the same point. Any curve with that specific property will look similar, and will be represented on the map as a circle that touches the outer circle of the map in just one point. That’s what a horocyle is.