Good way to describe "converging parallel lines"?

Question

Not sure if this question is on topic.

I am looking for a nice correct and succinct way to describe "2 lines are limiting parallel " that is:

• Understandable for newbies to hyperbolic geometry.
• Geometrically correct
• not mentions "ideal points". (because they don't really exist. "going in the same direction" is also not allowable, lines don't move.)

background:

I was editing wikipedia>hyperbolic triangles > Triangles with ideal vertices:

https://en.wikipedia.org/wiki/Hyperbolic_triangle#Triangles_with_ideal_vertices but could not find a nice way to describe an omega triangle.

And also found problems in the descriptions at:

https://en.wikipedia.org/wiki/Hyperbolic_geometry#Non-intersecting_.2F_parallel_lines

• uses ideal points

https://en.wikipedia.org/wiki/Limiting_parallel :

A ray $Aa$ Aa is a limiting parallel to a ray $ Bb$ if they are coterminal or if they lie on distinct lines not equal to the line $AB$, they do not meet, and every ray in the interior of the angle $ \angle BAa$ meets the ray $Bb$

• Do not like the $Aa$ and $Bb$ and the "they" in "they do not meet"?

also

• lines get closer together not really geomatrically correct (lines don't move)

Are not really good.

ps this is a soft question, maybe more good and creative answers are possible :)

Suggestions welcome

Answer 1

Unless I made a mistcake, two distinct and non-intersecting lines are limit-parallel iff any of the following equivalent conditions holds:

• they don't have a common perpendicular
• they come arbitrary close to one another ($\forall\varepsilon>0\;\exists…$)
• the can be transformed into one another using a limit rotation

I'll leave it to you to decide whether you consider any of this easier than what you quoted in your question.

Personally I'd disagree with “ideal points don't really exist”. They are not points in the aximatization of your plane. But the are things which newbies in particular can easily visualize, which can be defined even without referring to a model (although doing so would likely be as complicated as defining limit parallel), and which are useful for many considerations. But I come from a background of projective geometry, so I'm very much used to the idea of treating a point at infinity as something which exists and has a very well-defined and useful meaning.

Nevertheless, for Wikipedia I'd go with the concept of an ideal point, giving that name as a link plus a sentence explaining it in (short but slightly imprecise) layman's terms:

Two limiting parallel lines have an ideal point in common; i.e. the have a common “endpoint” on the boundary of the model.

Answer 2

First of all, "two lines are limiting parallels" makes sense only when they both go through a point not on a third line.

A geodesic is non-secant with an infinity of other geodesics going through a point outside the given geodesics, these geodesics are ultraparallel to the given geodesic.

A geodesic has (at least and at most) two geodesics that are its limiting parallels (horoparallel) and that go through a point not on the given geodesic.

You made two assertions on what is easily understandable and what is truth :

• Ideal points do exist : consider the Poincaré Disk Model, Ideal Points are on the Disk so that $|z| = 1$. Their hyperbolic distance from any other point is infinity.

Take a point A on a geodesic for example, and take another point B on the same geodesic. You can make the distance from A to B as great as you want, and when this distance is big enough (infinity) you will eventually reach an Ideal Point.

So, every geodesic that does through A and B can also be expressed as a geodesic between two Ideal Points.

A geodesic that goes through a point not on a second geodesic and with which an Ideal Point is shared is called a limiting parallel (Horoparallel).

Every geodesic having two Ideal Points, it is easy to see why there are at least and at most two Horoparallels of a geodesic going through a point not on the given geodesic.

• Lines don't move but points can : fore sure lines don't move. But if you take an arbitrary point on a geodesic, you can only move it two ways : forward or backward. And as you can evaluate a distance between two points, you can know if two geodesics are asymptotic or not.

"Going in the same direction" is a non-sense in hyperbolic geometry, because there is no notion of collinearity.

But a geodesic IS a direction, and has two ways. Given two geodesics, we can know if they are asymptotic by checking in four ways if they converge.

Given a point $P$ on a geodesic $G$, you can move it forward or backward. Consider the point $P_1$, that lies on $G$ and the hyperbolic distance between $P$ and $P_1$ is 1 and the way is forward. Consider the point $P_{-1}$, that lies on $G$ and the hyperbolic distance between $P$ and $P_1$ is 1 and the way is backward. Now consider a point $Q$ on a geodesic $L$, with $Q_1$ and $Q_{-1}$ built the same as $P_1$ and $P_{-1}$.

1. $dist(P_n, Q_n) = F_n$
2. $dist(P_n, Q_{-n}) = F_{-n}$
3. $dist(P_{-n}, Q_n) = B_n$
4. $dist(P_{-n}, Q_{-n}) = B_{-n}$

There are three cases :

• G and L are non-secant, there exist n such as when n goes greater, every distances go greater too.

• G and L are secant, there exist n such as when n goes greater, every distances go greater too.

• G and L reach the same Ideal Point, there exist n such as when n goes greater, one of the four distance will converge to 0.

Answer 3

The answer by MvG to this similar question uses the same terms as planetmath concerning parallel lines in hyperbolic geometry. So there remains the question how to geometrically differentiate limiting parallel lines from ultraparallel lines without talking about "ideal points".

Two parallel lines $g$ and $h$ are limiting parallel if $$d(g,h):=\inf\bigl\{d(x,y)\>|\>x\in g, \ y\in h\bigr\}=0\ ,$$ and ultraparallel, if $d(g,h)>0$.