$\textbf{Context:}$ Let $K$ be a connected graph endowed with a distance $d$ given by the length of the shortest path between two points. A path $\alpha$ between two vertices $x$ and $y$ is called a geodesic if its length equals $d(x,y)$. More generally, for $r\geq 0$, a path $\alpha$ is an $r$-quasi-geodesic if all its vertices are distinct and if for any finite subpath $\beta=(x_{0},...,x_{n})$ of $\alpha$, the length of $\beta$ is smaller than $d(x,y)+r$. We also consider infinite paths $(x_{0}, x_{1}, . . .)$ or bi-infinite paths $(. . . , x_{1}, x_{0}, x_{1}, . . .)$. For $r\geq 0$, such an infinite or bi-infinite path is called $r$-quasi-geodesic if all its finite subpaths are $r$-quasi-geodesics. Two infinite quasi-geodesics are equivalent if their Hausdorff distance is finite. The Gromov boundary $\partial K$ of $K$ is the set of equivalence classes of infinite quasi-geodesics. For a hyperbolic group $G$ with Cayley graph $\Gamma$, define $\Delta\Gamma=\partial\Gamma\cup\Gamma$.
$\textbf{My Question:}$ In the paper I am reading the authors write: "Consider $x,y,z\in\Delta\Gamma$ and let $\alpha$ be a geodesic between $x$ and $y$. A point $z_{0}\in\alpha$ which minimizes the distance from $z$ to $\alpha$, is called a projection of $z$ on $\alpha$". If $z$ is a vertex of $\Gamma$, then I understand why such a minimizing point exists. But I don't know how to interpret this in the case when $z\in\partial\Gamma$. I know that for any $z\in\partial\Gamma$, there is a $0$-quasi-geodesic between $z$ and every point of $\alpha$, but I cannot figure out what is meant by the "distance" between $z$ and a vertex in $\alpha$.
If you consider a geodesic ray $ r: [0,\infty[\to K$, say $r(t)$ such that $r(\infty)=z$, the functionn $b(x,t)= d(x,r(t))-t$ is eventually constant. So the distance to the point $z$ is not defined, but $b(x)$ is, and is is not hard to see that it has a minimum. It is called the Busemann function. Unfortunately it is depends on the choice of $r$, but nit too much in the case of hyperbolic spaces. Anyhow, in hypernolic spaces triangles with points in the boundary also are thin (perhaps $2\delta$ thin), so you can see which points minimizes this function by considering such a triangle and looking at his "center".