Hyperbolic Distance Calculator

Introduction: Measuring hyperbolic distance inside the Poincaré disk

This hyperbolic distance calculator compares two points in the Poincaré disk model, where the open unit disk stands in for the entire hyperbolic plane. Enter the Cartesian coordinates of each point and keep both points strictly inside the unit circle. If a point lands on the boundary or outside it, the metric no longer applies, because the rim of the disk represents infinitely distant space in this geometry. That is why the page asks for x and y values separately: the calculator needs each point’s position, not just a single radius, before it can measure the hyperbolic separation between them.

The figure below shows the intuition behind that rule. Near the center of the disk, the hyperbolic metric behaves more gently, so small Euclidean moves do not always look dramatic. Closer to the edge, however, the same coordinate change can produce a much larger distance than an ordinary ruler would suggest. This calculator is most useful when you want to compare two candidate points, check whether a construction stays valid, or see how quickly distance expands as a point drifts toward the boundary. It also helps when you are reading coordinates from a sketch, because the same point can feel visually close while being much farther away in the hyperbolic sense.

Formula for hyperbolic distance in the Poincaré disk

For points u = (x1, y1) and v = (x2, y2), the calculator uses the standard Poincaré disk distance formula. The expression is d(u, v) = arcosh(1 + 2 |u - v|^2 / ((1 - |u|^2)(1 - |v|^2))). In other words, it starts with the ordinary squared separation between the two points, then rescales that separation by how close each point is to the unit circle. Because the denominator gets smaller as either point approaches the edge, the same Euclidean gap can correspond to a much larger hyperbolic distance near the rim than near the center. The formula is symmetric, so swapping the two points does not change the answer, and that symmetry is a useful check when you are verifying a hand calculation.

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 a dimensionless hyperbolic distance, not a Euclidean length in pixels, meters, or any other physical unit. The formula assumes both points are strictly inside the disk so the inverse hyperbolic cosine stays real. If you are checking hand calculations, it also helps to remember that the same radius from the origin gives the same distance from the center, even when the angle changes. That means the calculator is sensitive to both position and direction, but the center-to-point special case still has a simple radial interpretation.

Example of hyperbolic distance from the center to the boundary

This hyperbolic distance example uses one point at the center and another point halfway to the boundary along the x-axis. Let u = (0, 0) and v = (1/2, 0). Then |u - v|^2 = 1/4, |u|^2 = 0, and |v|^2 = 1/4, so the formula becomes d(u, v) = arcosh(1 + 2(1/4)/(1 1/4)) = arcosh(5/3). That simplifies to ln 3, which is about 1.098612. The Euclidean distance between the same points is only 0.5, but the hyperbolic value is already larger because the disk metric stretches the outer region. If the second point moves farther toward radius 1 on the same ray, the distance rises quickly even though the visible coordinate change looks modest. If you moved that second point to (0, 1/2) instead, the center-to-point distance would be the same, which is a good reminder that rotation around the origin preserves the radial part of this special case.

Poincaré disk input checks

For this hyperbolic distance calculator, each coordinate can be any real number on its own, but the combined radius of each point must stay below 1. After entry, the page checks x1^2 + y1^2 and x2^2 + y2^2; if either sum is 1 or greater, that point lies on or outside the unit circle and the calculation is rejected. This guard matters because the boundary is not a finite location in the Poincaré model. If a coordinate pair is almost valid but slightly outside the disk, it is better to correct the values than to force a distance from a point the model does not allow. The same rule applies whether the point is near the center or very close to the edge, because the formula only behaves as intended in the open disk.

How to use this hyperbolic distance calculator

1. Enter x1 for the first point in the Poincaré disk, and make sure it belongs to the same coordinate system as the other values.
2. Enter y1 for the first point in the Poincaré disk so the calculator can combine both coordinates into one valid interior point.
3. Enter x2 for the second point in the Poincaré disk, then check that you are comparing the intended pair of locations rather than two unrelated coordinates.
4. Enter y2, then compute the distance. If you are comparing two geometric constructions, try a second pair of interior points and see how the hyperbolic distance changes as the points move closer to or farther from the boundary.

Limitations and assumptions for the Poincaré disk metric

This hyperbolic distance calculator follows the idealized Poincaré disk formula exactly as written above, so it assumes clean Cartesian inputs and an exact unit circle. It does not model measurement noise, rounded coordinates from another tool, or a different normalization of the hyperbolic metric. Results depend on entering matching x/y pairs for both points and on keeping each point strictly inside the disk. If you are reading coordinates from a textbook, a proof sketch, or software that uses another convention, double-check the source formula before you rely on the number. The calculator is meant for quick geometry checks, not for resolving alternate definitions or boundary cases, and the best way to use it is to verify the coordinates first, then trust the output as a direct evaluation of the disk metric.