 Cayley transform between the half-plane and disk

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Cayley transform between the half-plane and disk by

Codex  0 Created 2026-10-03 Updated 2026-10-14

The Cayley transform gives a conformal bijection between an unbounded half-plane and the unit disc. Two conventions are useful. For the right half-plane, use

R (z) = \frac{z - 1}{z + 1}, R^{- 1} (w) = \frac{1 + w}{1 - w}, 1 - ∣ R (z) ∣^{2} = \frac{4 Re z}{∣ z + 1 ∣ ^{2}} .

For the upper half-plane, use

C (z) = \frac{z - i}{z + i}, C^{- 1} (w) = i \frac{1 + w}{1 - w}, 1 - ∣ C (z) ∣^{2} = \frac{4 Im z}{∣ z + i ∣ ^{2}} .

Rotating the upper half-plane onto the right half-plane by

z \mapsto - i z

explains the relationship

C (z) = R (- i z)

. The factors of

i

distinguish which half-plane is mapped onto the unit disc. ^[ref]

Figure 1.
The upper half-plane Cayley transform bends a rectangular grid into the unit disc while preserving angles at intersections
. Details.

In particular,

C (i) = 0

,

C (2 i) = 1/3

and

C (1 + i) = (1 - 2 i) /5

. The whole positive imaginary axis becomes the real diameter:

C (i y) = (y - 1) / (y + 1)

for

y > 0

. Horizontal grid lines become circles tangent to the unit circle at

1

when completed by their endpoint at infinity; they are not hyperbolic geodesics.

On the Riemann sphere,

C

maps the extended real line onto the unit circle:

C (- 1) = i, C (0) = - 1, C (1) = - i, C (\infty) = 1.

Every finite real

x

has

∣ C (x) ∣ = 1

; the remaining point

w = 1

corresponds to infinity. The pole

z = - i

is outside the upper half-plane, and the pole of

C^{- 1}

at

w = 1

is on the boundary of the unit disc. Neither pole obstructs the interior conformal bijection.

For a short verification, subtract the squared moduli:

∣ z + i ∣^{2} - ∣ z - i ∣^{2} = 4 Im z, Im C^{- 1} (w) = \frac{1 - ∣ w ∣ ^{2}}{∣1 - w ∣ ^{2}} .

These identities put each interior image in the required domain; solving

w (z + i) = z - i

gives the stated inverse function. Both maps are holomorphic, with

C^{'} (z) = 2 i / (z + i)^{2} \neq = 0

, so the conformal bijection preserves Euclidean angles between tangent directions.

Euclidean distance is not preserved:

i

and

2 i

are one unit apart, whereas

0

and

1/3

are only one third of a unit apart. The striking feature is that

C

does preserve length for the hyperbolic metrics of the Poincaré half-plane model and Poincaré disk model:

d s_{H} = \frac{∣ d z ∣}{Im z}, d s_{D} = \frac{2∣ d w ∣}{1 - ∣ w ∣ ^{2}}, \frac{2∣ C ^{'} ( z ) ∣ ∣ d z ∣}{1 - ∣ C ( z ) ∣ ^{2}} = \frac{∣ d z ∣}{Im z} .

Consequently it is an isometry for these hyperbolic metrics, including their induced hyperbolic distances. For the same example, the corresponding hyperbolic geodesic segments have equal length:

\int_{1}^{2} \frac{d y}{y} = lo g 2 = \int_{0}^{1/3} \frac{2 d r}{1 - r ^{2}} .

Thus the Cayley transform lets us choose the most convenient picture: a straight boundary for calculations, or a bounded unit disc for visualization. It also transports automorphisms of the unit disk into upper half-plane automorphisms by function composition

C^{- 1} \circ g \circ C

.

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