The Geometry of the Poincaré Disk

The Poincaré disk model describes a two-dimensional universe where lines are arcs that meet the unit circle at right angles. It visualizes hyperbolic geometry, a non-Euclidean system in which the parallel postulate does not hold. To measure distances in this space, we use a modified metric. Two points in the open unit disk, represented as $\left(\mathrm{x\_1},,,\mathrm{y\_1}\right)$ and $\left(\mathrm{x\_2},,,\mathrm{y\_2}\right)$, have hyperbolic separation determined by the arcosh formula.

Specifically, if we denote the Euclidean distance squared as ${\mathrm{\backslash Delta}}^2=(\mathrm{x\_1}-\mathrm{x\_2})2^{}+(\mathrm{y\_1}-\mathrm{y\_2})2^{}$ and the squared radii $r_1{2}^{}={\mathrm{x\_1}}^2+{\mathrm{y\_1}}^2$, $r_2{2}^{}={\mathrm{x\_2}}^2+{\mathrm{y\_2}}^2$, then the hyperbolic distance $\mathrm{d\_H}$ is determined by

$\mathrm{\backslash cosh}\left(\mathrm{d\_H}\right)=1+\frac{2{\mathrm{\backslash Delta}}^2}{(1-r_1{2}^{})(1-r_2{2}^{})}$.

This expression ensures that points near the boundary of the unit disk appear infinitely far away. The result is a geometry with constant negative curvature, where triangles have angle sums less than $180$ degrees and circles grow exponentially in area as their radius increases. The formula above captures these exotic features in a compact numerical computation.

Why Hyperbolic Distance Matters

Hyperbolic spaces arise in several areas of mathematics and physics. They provide models for certain kinds of large-scale geometric structures, appear in the study of complex analysis via the Poincaré metric, and underlie algorithms in modern network science. Computing distance in this setting is crucial when analyzing tilings of the disk, exploring Fuchsian groups, or understanding the geometry of moduli spaces. The negative curvature shapes the behavior of geodesics and makes hyperbolic distance a natural measurement for non-Euclidean growth.

Using This Calculator

Enter coordinates for two points strictly inside the unit circle. The script verifies that each radius is less than one; otherwise, it warns you that the coordinates lie outside the hyperbolic plane. When valid, it computes the hyperbolic distance by evaluating the formula above with $\mathrm{\backslash cosh^\{-1\}}$, which is implemented by the JavaScript Math.acosh function. The result appears in the text field, giving an exact measure of separation under the Poincaré metric. You can experiment with points near the boundary to observe how distances grow rapidly as you approach the circle, reflecting the space's negative curvature.

Exploring Hyperbolic Geometry

The Poincaré disk is just one representation of hyperbolic space. Another common model is the upper half-plane, where circles and vertical lines represent geodesics. Möbius transformations connect the two models. Because distances in hyperbolic space behave differently than Euclidean ones, the disk model reveals surprising phenomena. For instance, it is possible to arrange infinitely many non-overlapping disks of equal Euclidean size within the unit disk. These properties have deep connections to the theory of tessellations, Kleinian groups, and the uniformization theorem in complex analysis.

Studying hyperbolic distance helps mathematicians understand growth rates in groups and surfaces, classify Riemann surfaces, and analyze algorithms for network embeddings. The Poincaré disk also provides insights into relativity and cosmology, where negative curvature plays a role in theoretical models of the universe.