Hyperbolic Distance Calculator

Stephanie Ben-Joseph headshotEdited by: Stephanie Ben-Joseph

Introduction: Measure distance inside the Poincare disk

This calculator finds the hyperbolic distance between two points in the Poincare disk model. The inputs are the Cartesian coordinates of two points, and both points must lie strictly inside the unit circle. A point on or outside the boundary is invalid because the boundary represents infinity in this model.

Curved geometry diagram with arcs, points, and grid notes.
In the Poincare disk, points near the boundary are much farther apart than Euclidean distance alone suggests.

Euclidean distance and hyperbolic distance behave very differently here. Near the center of the disk they can feel similar. Near the edge, a small-looking Euclidean move can represent a large hyperbolic distance because the metric stretches space toward the boundary.

Formula

For points u = (x1, y1) and v = (x2, y2), the calculator uses:

d(u, v) = arcosh(1 + 2 |u - v|^2 / ((1 - |u|^2)(1 - |v|^2))).

Here, |u - v|^2 = (x1 - x2)^2 + (y1 - y2)^2, |u|^2 = x1^2 + y1^2, and |v|^2 = x2^2 + y2^2. The result is dimensionless hyperbolic distance.

Example

If one point is at the center and the other is halfway to the boundary, the Euclidean distance is 0.5, but the hyperbolic distance is larger. Move that second point closer to radius 1 and the distance grows quickly. That is the main idea behind the disk model: the edge is not a wall, it is infinitely far away.

Input checks

Enter values between -1 and 1, then check that each point's squared radius is less than 1. If either point has x^2 + y^2 >= 1, the calculator rejects it because the Poincare metric would no longer describe a valid point inside the disk.

How to use this calculator

  1. Enter x1 using the unit or time period shown by the field.
  2. Enter y1 using the unit or time period shown by the field.
  3. Enter x2 using the unit or time period shown by the field.
  4. Run the calculation and compare the output with a second scenario before acting on it.

Limitations and assumptions

This tool is a planning estimate, not a complete model of every edge case. Results depend on accurate inputs, current rates or rules, and consistent units. It does not replace local policy, professional review, or source data that may change over time.

Enter points inside the unit disk.

Arcade Mini-Game: Hyperbolic Distance Calculator Calibration Run

Use this quick arcade run to practice separating useful scenario inputs from common planning mistakes before you rely on the calculator output.

Score: 0 Timer: 30s Best: 0

Start the game, then use your pointer or arrow keys to catch useful inputs and avoid bad assumptions.