The symmetries of hyperbolic space — rotations, translations, and limit rotations
Möbius transformations are the rigid motions of the Poincaré disk. They warp how things look while keeping every hyperbolic distance and angle exactly the same.
Drag the handle toward the edge and watch the tiling slide: tiles crowd together on one side and spread out on the other. Diamonds mark the fixed points.
Möbius transformations are the isometries of the Poincaré disk — the distance-preserving maps that play the role rotations and translations play in Euclidean geometry. Every orientation-preserving isometry of the hyperbolic plane can be written as a Möbius transformation z → (az + b) / (cz + d), where the matrix lies in SU(1,1). These transformations fall into three families — elliptic (rotations), parabolic (horocyclic translations), and hyperbolic (axis translations) — classified by the trace of the matrix.
Drag the handle point inside the Poincaré disk to define a hyperbolic translation that moves the origin to that point. The entire tiling warps in real time, giving a visceral sense of how rigid motions in hyperbolic space distort Euclidean appearances while preserving hyperbolic distances and angles.
Click inside the disk to place three source–target point pairs. The panel names the map that sends one triple to the other and shows its fixed points.
Place 3 source–target pairs by clicking in the disk:
(0/6 points placed)
Click inside the disk to place three source–target point pairs. The unique Möbius transformation mapping one triple to the other is computed automatically, and the info panel shows its matrix entries, trace, classification type, and geometric invariants. Fixed points are drawn directly on the disk.
Three families — Elliptic (|tr| < 2, rotation), parabolic (|tr| = 2, boundary-fixing), hyperbolic (|tr| > 2, axis translation)
Step or animate the polygon along its axis. Every step is the same hyperbolic distance, yet the polygon seems to shrink and slow near the edge.
A polygon is translated step by step along a geodesic axis. Each step covers the same hyperbolic distance, yet in Euclidean terms the polygon appears to shrink and slow as it approaches the boundary — a direct consequence of the exponential growth of the metric near the ideal boundary.
Key insight: The side graph shows that while hyperbolic displacement grows linearly with each step, the Euclidean displacement from the origin decays exponentially — the conformal factor compresses everything near the boundary.
Drag the four points, then sweep the rotation slider. All four points move, but the cross-ratio holds perfectly steady.
The cross-ratio of four points is a fundamental projective invariant: it does not change under any Möbius transformation. Drag the four points freely to see the cross-ratio update, then sweep the rotation slider to apply a smooth Möbius transform — all four points move, but the cross-ratio holds perfectly steady.
Cross-ratio invariance — The cross-ratio of four points is preserved under all Möbius transformations, linking hyperbolic geometry to projective geometry
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