Cayley transform

Example

Let $\mathbb{H} = \{ z \in \mathbb{C} : \operatorname{Im} z > 0 \}$ be the upper half-plane and $\mathbb{D} = \{ w \in \mathbb{C} : |w| < 1 \}$ the open unit disc. The Cayley transform is $$W(z) = \frac{z - i}{z + i}.$$ It is a Möbius transformation with $a = 1$, $b = -i$, $c = 1$, $d = i$, and $ad - bc = 2i \neq 0$. For $z \in \mathbb{C}$, $|W(z)| < 1$ holds exactly when $z$ is nearer to $i$ than to $-i$, that is, when $\operatorname{Im} z > 0$. The map $W$ is a bijection from $\mathbb{H}$ onto $\mathbb{D}$, with inverse $w \mapsto i \frac{1 + w}{1 - w}$.

Remarks

Some values are $W(i) = 0$, $W(0) = -1$, $W(1) = -i$, $W(-1) = i$ and $W(\infty) = 1$. The map $W$ sends the real line together with $\infty$ onto the unit circle. It is an explicit case of the Riemann mapping theorem for $\mathbb{H}$. In hyperbolic geometry, it connects the half-plane model and the disc model.

Explanations

L1 human

The Cayley transform turns the upper half-plane into the unit disc. A point of $\mathbb{H}$ is nearer to $i$ than to $-i$, so the quotient of the two distances is less than $1$. It lets you move a problem on the half-plane to the disc, where it can be easier.