Does the property "any three non-collinear points lie on a unique circle" hold true for hyperbolic circle?

Question

In Euclidean geometry, there is a property about circles that any three non-collinear points lie on a unique circle. Is this true for hyperbolic circles?

Answer 1

It depends on what you consider a circle. I would think about this in the Poincaré disk model (but half plane works just as well, with some tweaks to my formulations). Here are the three possible interpretations I can think of:

1. A hyperbolic circle is a Euclidean circle that doesn't intersect the unit circle. This corresponds to a circle as the set of points that are the same real hyperbolic distance away from a hyperboloic center point. This is the strictest of views.

   Here you can see how the Euclidean circle through three given points may end up intersecting the unit circle. So some combinations of three hyperboloic points won't have a common circle in the above sense.

   There is actually a sight distinction of this case into two sub-cases, depending on whether you require the circle to lie within the closed or open unit disk. In the former case the definition of a circle includes a horocycle, which would not have a hyperbolic center. In the latter case horocycles are excluded as well.

2. A hyperbolic circle is the intersection of a Euclidean circle and the unit disk. This is equivalent to seeing a circle as a curve of constant curvature.

   Place a car on the plane, lock the steering wheel in place and drive as long as you can, forward and backward. In the Euclidean plane you get a circle, except if you locked the steering wheel in the center and end up with a straight line. In the hyperboloic plane you also might get curved lines of infinite length, at constant distance to a given geodesic.

   Here your statement is true. Given three points in the unit disk you can always get a Euclidean circle for them. And you can intersect that Euclidean circle with the unit disk and the result will still coincide with the three points.

3. A hyperbolic circle is a Euclidean circle combined with its inversion in the unit circle. If you intersect this with the unit disk, you get a moon shape there. This takes into account the fact that the outside of the Poincaré disk is just a mirror image of the inside, and it makes sense to identify the points outside with their inverse on the inside.

   Where the previous definition was treating a curve a fixed distance to a geodesic as a circle, this definition here combines the curves at that distance on both sides of the geodesic into a single circle.

   If you use the Beltrami-Klein model then this definition of a circle is the most reasonable one because both the lines of the moon shape in the Poincaré model correspond to a single conic tangent to the fundamental conic in the Beltrami-Klein model.

   In this definition, you can construct a circle through three given points by first inverting any number of them in the unit circle, then constructing the Euclidean circle through them, then adding the inverse of that circle as well. There are $2^3=8$ choices which of the defining points you invert. But if you flip them all, i.e. invert all three points again, you end up with the same pair of Euclidean circles. So you count every circle twice, and end up with $4$ distinct circles thorough every three points. At least three of these will appear moon-shaped, perhaps even all four. Thus the uniqueness of your statement will be violated.

Just like the simple concept of “parallel line” from Euclidean geometry breaks down in hyperboloic geometry into different concepts of limit parallel, ultraparallel, both of these together or non-line constant distance curve depending on what definition you use, the concept of a circle breaks down in a similar way. All of the above can be considered valid counterparts, and you need to distinguish them when discussing their properties.

Note that my arguments above essentially discuss the existence of a hyperbolic circle based on the existence of a Euclidean circle. So you need three points defining a Euclidean circle for that. There is a catch here in that the statement assumes the three point to not lie on a common hyperbolic line, but they might still be collinear in the Euclidean sense. To deal with that you should consider generalized circles, which include straight lines as a special case. Then three distinct points always define a unique Euclidean generalized circle, and all the above will work for that. In particular inversion in the unit circle may turn lines into circles and vice versa, so that too requires generalized circles to avoid dealing with special cases.