Mobius Transformations & Isometries

The symmetries of hyperbolic space — rotations, translations, and limit rotations

Symmetries of Hyperbolic Space

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.

Möbius Transform Playground

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.

Handle: (0.000, 0.000)Drag the red handle to apply a hyperbolic translation

Try it: Drag the handle toward the boundary and watch tiles crowd together on one side while expanding on the other. The fixed points of the transformation are marked with diamonds.

Isometry Classifier

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.

Isometry Info

Place 3 source–target pairs by clicking in the disk:

  • S1 → T1
  • S2 → T2
  • S3 → T3

(0/6 points placed)

Click alternating source/target points: S1, T1, S2, T2, S3, T3

Hyperbolic Translation

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.

Step 0/20 — Hyp. dist per step: 0.5

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.

Cross-Ratio Invariance

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.

z₁
(0.300, 0.400)
z₂
(-0.200, 0.300)
z₃
(-0.400, -0.200)
z₄
(0.200, -0.500)
CR = 1.6838+0.1564i|CR| = 1.691019Drag points or sweep the rotation slider — the cross-ratio stays constant

Key Takeaways

  • Möbius transformations — Every orientation-preserving isometry of the Poincaré disk is a Möbius transformation in SU(1,1)
  • Three families — Elliptic (|tr| < 2, rotation), parabolic (|tr| = 2, boundary-fixing), hyperbolic (|tr| > 2, axis translation)
  • Exponential metric growth — Equal hyperbolic steps correspond to exponentially shrinking Euclidean steps near the boundary
  • Cross-ratio invariance — The cross-ratio of four points is preserved under all Möbius transformations, linking hyperbolic geometry to projective geometry