
# 13.5: Conformal interpretation


Let us give another interpretation of the h-distance.

Lemma $$\PageIndex{1}$$

Consider the h-plane with the unit circle centered at $$O$$ as the absolute. Fix a point $$P$$ and let $$Q$$ be another point in the h-plane. Set $$x = PQ$$ and $$y = PQ_h$$. Then

$\lim_{x\to 0} \dfrac{y}{x} = \dfrac{2}{1-OP^2}.$

The above formula tells us that the h-distance from $$P$$ to a nearby point $$Q$$ is almost proportional to the Euclidean distance with the coefficient $$\dfrac{2}{1-OP^2}$$. The value $$\lambda(P)=\dfrac{2}{1-OP^2}$$ is called the conformal factor of the h-metric.

The value $$\dfrac{1}{\lambda(P)}=\dfrac{1}{2} \cdot (1-OP^2)$$ can be interpreted as the speed limit at the given point $$P$$. In this case the h-distance is the minimal time needed to travel from one point of the h-plane to another point.

Proof

If $$P=O$$, then according to Lemma 12.3.2

$\dfrac{y}{x}=\dfrac{\ln \dfrac{1+x}{1-x}}{x}\to 2$

as $$x\to0$$.

If $$P\ne O$$, let $$Z$$ denotes the inverse of $$P$$ in the absolute. Suppose that $$\Gamma$$ denotes the circle with the center $$Z$$ perpendicular to the absolute.

According to the main observation (Theorem 12.3.1) and Lemma 12.3.1, the inversion in $$\Gamma$$ is a motion of the h-plane which sends $$P$$ to $$O$$. In particular, if $$Q'$$ denotes the inverse of $$Q$$ in $$\Gamma$$, then $$OQ'_h=PQ_h$$.

Set $$x'=OQ'$$. According to Lemma 10.1.1,

$$\dfrac{x'}{x}=\dfrac{OZ}{ZQ}.$$

Since $$Z$$ is the inverse of $$P$$ in the absolute, we have that $$PO\cdot OZ=1$$. Therefore,

$$\dfrac{x'}{x} \to \dfrac{OZ}{ZP}=\dfrac{1}{1-OP^2}$$

as $$x \to 0$$.

According to 13.5.1, $$\dfrac{y}{x'} \to 2$$ as $$x' \to 0$$. Therefore

$$\dfrac{y}{x}=\dfrac{y}{x'} \cdot \dfrac{x'}{x} \to \dfrac{2}{1-OP^2}$$

as $$x \to 0$$.

Here is an application of the lemma above.

Proposition $$\PageIndex{1}$$

The circumference of an h-circle of the h-radius $$r$$ is

$$2 \cdot \pi \cdot \sinh r,$$

where $$\sinh r$$ denotes the hyperbolic sine of $$r$$; that is,

$$\sinh r := \dfrac{e^r-e^{-r}}{2}.$$

Before we proceed with the proof, let us discuss the same problem in the Euclidean plane.

The circumference of a circle in the Euclidean plane can be defined as the limit of perimeters of regular $$n$$-gons inscribed in the circle as $$n \to \infty$$.

Namely, let us fix $$r>0$$. Given a positive integer $$n$$, consider $$\triangle AOB$$ such that $$\measuredangle AOB=\dfrac{2\cdot\pi}{n}$$ and $$OA=OB=r$$. Set $$x_n=AB$$. Note that $$x_n$$ is the side of a regular $$n$$-gon inscribed in the circle of radius $$r$$. Therefore, the perimeter of the $$n$$-gon is $$n\cdot x_n$$.

The circumference of the circle with the radius $$r$$ might be defined as the limit

$\lim_{n\to\infty} n\cdot x_n=2\cdot\pi\cdot r.$

(This limit can be taken as the definition of $$\pi$$.)

In the following proof, we repeat the same construction in the h-plane.

Proof

Without loss of generality, we can assume that the center $$O$$ of the circle is the center of the absolute.

By Lemma 12.3.2, the h-circle with the h-radius $$r$$ is the Euclidean circle with the center $$O$$ and the radius

$$a=\dfrac{e^r-1}{e^r+1}.$$

Let $$x_n$$ and $$y_n$$ denote the side lengths of the regular $$n$$-gons inscribed in the circle in the Euclidean and hyperbolic plane respectively.

Note that $$x_n\to0$$ as $$n\to\infty$$. By Lemma $$\PageIndecx{1}$$,

$$\lim_{n\to\infty} \dfrac{y_n}{x_n} = \dfrac{2}{1-a^2}.$$

Applying 13.5.2, we get that the circumference of the h-circle can be found the following way:

$$\begin{array} {rcl} {\lim_{n \to \infty} n \cdot y_n} & = & {\dfrac{2}{1 - a^2} \cdot \lim_{n \to \infty} n \cdot x_n =} \\ {} & = & {\dfrac{4 \cdot \pi \cdot a}{1 - a^2} =} \\ {} & = & {\dfrac{4 \cdot \pi \cdot (\dfrac{e^r - 1}{e^r + 1})}{1 - (\dfrac{e^r - 1}{e^r + 1})^2} =} \\ {} & = & {2 \cdot \pi \cdot \dfrac{e^r - e^{-r}}{2} =} \\ {} & = & {2 \cdot \pi \cdot \sinh r.} \end{array}$$

Exercise $$\PageIndex{1}$$

Let $$\circum_h(r)$$ denote the circumference of the h-circle of the h-radius $$r$$. Show that

$$\\text{circum}_h(r+1)>2\cdot \text{circum}_h(r)\]) for all \(r>0$$.

Hint

Apply Proposition $$\PageIndex{1}$$. Use that $$e > 2$$ and in particular the function $$r \mapsto e^{-r}$$ is decreasing.