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I thought on a hyperbolic plane all geodesics are lines/arcs across the plane whose endpoints are perpendicular to the boundary of the plane.

I've heard that all geodesics on a hyperbolic plane either

  • intersect ("meet") once (somewhere in the middle of the plane)
  • meet once at the boundary
  • don't meet

How can two different geodesics meet at the boundary?

Drawing a line perpendicular to the boundary of a hyperbolic plane... why doesn't this lead to only a single unique geodesic for every point on the boundary?

minseong
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2 Answers2

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The boundary points of the hyperbolic plane are NOT in the plane. The plane two lines that meet only on the boundary are parallel or non-intersecting. (in Euclidean geometry there exist only one line through a given point parallel to a given line. In hyperbolic geometry there can are infinitely many lines through a given point that do not intersect a given line. Typically we only call the two "bounding" such lines "parallels".)

user247327
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Take a look at pictures of the hyperbolic plane, such as one of Escher's drawings. Perhaps, from that picture, you will be able to get an intuition for the following fact:

The boundary of the hyperbolic plane is infinitely far away.

To say that two points "meet at boundary" does not mean that they meet at a point of the hyperbolic plane itself, because the points at the boundary are infinitely far away. However, what is true is that two lines meet at the boundary if and only if the distance between them approaches zero, as those lines get farther and farther away. This is quite unlike the Euclidean situation, where two lines that do not meet keep the same distance from each other, as those lines get farther and farther away.

Lee Mosher
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  • So they never actually touch, they just get infinitely close to touching. But this distinction is why it's possible to have both these lines go in different directions across the plane? – minseong Oct 19 '19 at 09:35
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    For your first sentence, that's roughly speaking correct, although just keep in mind that there is no actual point of the hyperbolic plane AT WHICH they get infinitely close to touching. For your second sentence, I would also say that two lines which do not meet --- neither infinitely far away at the boundary, nor at an actual point of the hyperbolic plane --- can also be described as going "in different directions across the plane". – Lee Mosher Oct 19 '19 at 20:06