For a projective curve ${\Bbb P}_k^1$, we have a very ample line bundle ${\cal O}_{{\Bbb P}_k^1}(1)$. Geometrically, I prefer to consider ${\cal O}_{{\Bbb P}_k^1}(1)$ as the fibration
\begin{equation*} \qquad \qquad \pi \colon {\Bbb P}_k^2 \setminus \infty \to {\Bbb P}_k^1 \phantom{AA} \cdots\cdots \phantom{A} (\lozenge) \end{equation*} where ${\Bbb P}_k^1 = (0\,\colon k\,\colon k) \hookrightarrow {\Bbb P}_k^2 = (k\, \colon k \,\colon k)$ and $\infty = (1 \, \colon 0 \,\colon 0)$. Moreover for each point $p = (0 \, \colon \alpha \, \colon \beta) \in {\Bbb P}_k^1$, the fibre $\pi^{-1}(p) = (\lambda \,\colon \alpha\, \colon \beta) = k$. This is a purely geometric definition of ${\cal O}_{{\Bbb P}_k^1}(1)$.
However, experts often choose a point $x \in {\Bbb P}_k^1$ and consider the sheaf ${\cal O}_{{\Bbb P}_k^1}(x)$ defined as $\Gamma(U,{\cal O}_{{\Bbb P}_k^1}(x)) = \{f \in k(X) \,|\, ((f) + x) |_{U} \geq 0 \}$, i.e. the divisor $(f) + x$ is effective on $U$. Thus if $U(x)$ is the defining equation of a point $x$ on $U$, we have
\begin{equation*} \Gamma(U,{\cal O}_{{\Bbb P}_k^1}(x)) = {\cal O}(U)[\frac{1}{U(x)}] = {\cal O}(U)e_U, \end{equation*} where $e_U$ is very often referred to as a local frame of ${\cal O}_{{\Bbb P}_k^1}(x)$. This yields the following transition function for two open $U, V$ of ${\Bbb P}_k^1$$\colon$
\begin{equation*} \qquad \qquad \theta_{VU} \colon= \frac{V(x)}{U(x)} \phantom{AA} \cdots \cdots \phantom{A} (\blacklozenge) \end{equation*}
Q. Very often is quoted that ${\cal O}_{{\Bbb P}_k^1}(1) = {\cal O}_{{\Bbb P}_k^1}(x)$. Then through this equality how can I interpret the transition function $\theta_{VU}$ at $(\blacklozenge)$ in the framework of $\pi$ at $(\lozenge)$?