How to Find Distance in the Poincaré and Klein Disk Models

The Poincaré disk and the Klein disk both represent the hyperbolic plane inside the Euclidean unit disk, but they use different coordinates and different-looking distance formulas. The formulas below use curvature −1. They give hyperbolic distance, not ordinary straight-line Euclidean distance.

Distance in the Poincaré disk

Represent points by complex numbers $z,w$ with $|z|<1$ and $|w|<1$. The Poincaré disk distance is

$$d_P(z,w)=2\operatorname{artanh}\left(\frac{|z-w|}{|1-\overline{w}z|}\right).$$

An equivalent form, often convenient for calculation, is

$$d_P(z,w)=\operatorname{arcosh}\left(1+\frac{2|z-w|^2}{(1-|z|^2)(1-|w|^2)}\right).$$

Here $\overline{w}$ is the complex conjugate of $w$. In two-dimensional vector notation, the denominator inside $\operatorname{artanh}$ is $|1-z\cdot w-i\det(z,w)|$; using complex coordinates avoids needing that expanded expression.

These formulas come from the Poincaré metric

$$ds=\frac{2|dz|}{1-|z|^2}.$$

The metric makes lengths grow near the boundary, so points that are close in Euclidean coordinates can still be far apart hyperbolically. Geodesics are diameters or circular arcs meeting the boundary at right angles.

Distance in the Klein disk

Represent points by vectors $x,y\in\mathbb{R}^2$ with Euclidean norms less than $1$. The Klein distance satisfies

$$\cosh d_K(x,y)=\frac{1-x\cdot y}{\sqrt{(1-|x|^2)(1-|y|^2)}}.$$

Thus you can calculate the distance as

$$d_K(x,y)=\operatorname{arcosh}\left(\frac{1-x\cdot y}{\sqrt{(1-|x|^2)(1-|y|^2)}}\right).$$

In this model, hyperbolic geodesics are straight Euclidean chords. That does not make their hyperbolic lengths equal to their ordinary Euclidean lengths: the Klein disk has its own position-dependent metric.

Worked example: from the center

Take the Poincaré disk points $z=0$ and $w=1/2$. The distance formula gives

$$\begin{aligned} d_P(0,1/2) &=2\operatorname{artanh}(1/2)\\ &=\ln 3\\ &\approx 1.099. \end{aligned}$$

You can also get this result by integrating the Poincaré metric along the radius:

$$\int_0^{1/2}\frac{2\,dr}{1-r^2}=\ln 3.$$

To represent the same nonzero point in the Klein disk, convert its coordinate. The map from Poincaré coordinates $p$ to Klein coordinates $x$ is

$$x=\frac{2p}{1+|p|^2}.$$

For $p=(1/2,0)$, this gives $x=(4/5,0)$. The origin remains the origin. Applying the Klein formula,

$$\cosh d_K(0,(4/5,0))=\frac{1}{\sqrt{1-16/25}}=\frac{5}{3},$$

so $d_K=\operatorname{arcosh}(5/3)=\ln 3$. The two calculations agree because the points correspond under the coordinate map.

Converting between the models

For a Poincaré coordinate $p$ and its Klein coordinate $x$,

$$x=\frac{2p}{1+|p|^2}.$$

In the other direction,

$$p=\frac{x}{1+\sqrt{1-|x|^2}}.$$

Apply these conversions to each point when moving between formulas. A pair of identical-looking coordinates—for example, $(0,0)$ and $(1/2,0)$—does not generally identify corresponding points if you simply reuse the numbers in both models.

Common mistakes

• Using Euclidean distance: $|z-w|$ or $|x-y|$ alone is not hyperbolic distance.
• Mixing coordinate systems: the Poincaré and Klein formulas take points expressed in their own model.
• Including the boundary: points must be strictly inside the unit disk. The boundary represents points at infinity, not finite points of the hyperbolic plane.
• Using a different curvature without scaling: these formulas assume curvature $-1$. If the model has curvature $-1/R^2$, multiply the resulting distance by $R$.
• Confusing geodesics with lengths: Klein geodesics are straight chords and Poincaré geodesics are orthogonal arcs or diameters, but neither fact makes hyperbolic distance ordinary Euclidean length.