Hyperbolic geometry where the metric stretches to infinity at the horizon
Here all of hyperbolic space lives above the x-axis. The closer you get to that axis, the more each step stretches, so you never reach it.
Click pairs of points to draw geodesics. Line two points up vertically to get a straight line; otherwise you get a semicircle on the x-axis.
The upper half-plane model places all of hyperbolic space in the region y > 0. The metric ds = |dz|/y means that near the x-axis (the "boundary at infinity"), distances stretch without bound — a finite Euclidean step near y = 0 corresponds to a vast hyperbolic journey. Geodesics are vertical lines and semicircles centered on the x-axis.
This model is the natural home for the group PSL(2, R) of Mobius transformations with real coefficients, and is central to the theory of modular forms, Fuchsian groups, and Teichmuller spaces.
Click pairs of points to draw geodesics. When two points share the same x-coordinate, the geodesic is a vertical line. Otherwise, it is a semicircle centered on the x-axis. The background shading shows the hyperbolic metric distortion: darker regions near the boundary represent the stretching of distances where ds = |dz|/y.
Horocycles are "circles of infinite radius". Watch them appear as horizontal lines and as circles touching the x-axis, then toggle to equidistant curves to compare.
Horocycles are curves that are orthogonal to every geodesic converging to the ideal point. They can be thought of as "circles centered at infinity." In the upper half-plane, horocycles at a finite boundary point ξ are Euclidean circles tangent to the x-axis at ξ. Horocycles at ∞ (dashed amber) are horizontal lines y = const.
Horocycles are "circles of infinite radius" — curves equidistant from an ideal point on the boundary. In the upper half-plane, horocycles through the point at infinity are horizontal lines y = const, while horocycles at finite boundary points are circles tangent to the x-axis. Toggle to see equidistant curves (loci at constant distance from a geodesic) for comparison.
Horocycles — Horizontal lines and circles tangent to the boundary are the "circles of infinite radius," crucial for understanding parabolic isometries
Click in either view to place points and draw geodesics. The same figure appears in the other model: angles match, distances reshape.
The Poincare disk and the upper half-plane are two windows into the same hyperbolic geometry, connected by the Cayley transform. Click in either view to place points and draw geodesics — the corresponding figure appears simultaneously in the other model, demonstrating how the conformal map preserves angles but reshapes distances.
Adjust the four matrix entries and watch the map move a grid of geodesics. See whether it comes out elliptic, parabolic, or hyperbolic.
Every orientation-preserving isometry of the upper half-plane is a Mobius transformation T(z) = (az + b) / (cz + d) with real coefficients and ad − bc = 1. Adjust the four matrix entries to see the transformation act on a grid of geodesics in real time. The classification — elliptic, parabolic, or hyperbolic — determines the qualitative dynamics.
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