Horocycle

Remarks

Another ray to $\omega$ changes $b$ by an additive constant, so the set of horocycles centred at $\omega$ depends only on $\omega$. Example: in the Poincaré half-plane model, take $\omega = \infty$ and $\gamma(t) = i e^t$. For $x = u + iy$, the distance formula $\cosh d(z, w) = 1 + \abs{z - w}^2 / (2 \operatorname{Im} z \operatorname{Im} w)$ gives $d(x, i e^t) = t - \ln y + o(1)$ as $t \to \infty$. So $b(x) = -\ln y$, and the horocycles centred at $\infty$ are the horizontal lines $y = \text{const}$. Their normal geodesics are the vertical half-lines, which all end at $\infty$. Horocycles centred at a real point $\omega$ are Euclidean circles tangent to the real axis at $\omega$, without $\omega$. Non-example: the ray $\set{r e^{i\theta} : r > 0}$ with $0 < \theta < \pi/2$ is not a horocycle. Its normal geodesics are the semicircles centred at $0$, and they end at different real points, so no single ideal point $\omega$ works. This ray is a curve at constant distance from the imaginary axis, called a hypercycle.