Let $P = (0, q)$ be the hyperbolic midpoint of $A = (0, a)$ and $B = (0, a + b)$. Compute the limit of $q$ as $a \rightarrow \infty$

Question

Let $P = (0, q)$ be the hyperbolic midpoint of $A = (0, a)$ and $B = (0, a + b)$. Compute the limit of $q$ as $a \rightarrow \infty$, as a function of the fixed number $b$.

I know this is a vertical line and that to compute the distance of a vertical line in the hyperbolic model. $AB*=|\ln(\frac{AM}{BM})|$ where $M$ is the endpoint of the line so $M=(0,0)$(and $AM$ and $BM$ are computed using the standard distance formula). Now computing the distance as $a \rightarrow \infty$ I see that the distance becomes $0$. Now I imagine as the distance goes to $0$ as $a$ gets big then the midpoint should go approach $B$ or $(0, a+b)$ yet not sure how to justify that with the limit. As of now we know that $Q=\frac{a+a+b}{2}=\frac{2a+b}{2}$ as $a$ gets bigger then the gap between $A$ and $B$ goes to $0$.

This is all taking place in the Hyperbolic/Half-Plane Model.

Answer 1

Independently of the model you are using, this is a question related to absolute geometry. It is a big mistake to think that the model counts!

In absolute geometry the axioms of congruence hold. And the facts in the question are facts of absolute geometry in which the axiom of parallelism is not fixed -- it can be the Euclidean axiom or it can be the hyperbolic axiom.

What is the answer to your question in Euclidean geometry? Let's ask:

We are in Euclidean geometry and we have a ray. Let's take a point $A$ on this ray $a$ units away from the beginning of the ray. Then we have a distance $b$ and a point $B$ on this ray $b$ units away form $A$. The midpoint's distance form the beginning of the ray is $\frac{a+b}2$. What happens to this length if $a$ tends to $\infty$? The answer is intuitive: This limit tends to the infinity because the distance of the midpoint remains the same: $\frac{a+b}2$.

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The same question in hyperbolic geometry: ... The answer is also the same.

This is all independent of the models we use to intuit the geometries we examine.

(Yes, the models / the geometries. The Euclidean geometry in the usual intuition is also only a model of the axioms of Euclidean geometry.)