Extended Klein model and a bound on curves with negative self-intersection

Let $S$ be an irreducible smooth projective surface and $\mathcal{F}$ a collection of curves with negative self-intersection on $S$ such that no positive combination $aC_1 + bC_2$ is connected nef. In this paper, we provide an alternate proof that $\mathcal{F}$ is bounded by an exponential function of the Picard number $\rho(S)$ of $S$ using an extended version of the Klein disc model for hyperbolic space.