Intersection numbers and the hyperbolicity of the curve conplex

We give another proof of the result of Masur and Minsky that the complex of curves associated to a compact orientable surface is hyperbolic. Our proof is more combinatorial in nature and can be expressed mostly in terms of intersection numbers. We show that the hyperbolicity constant is bounded above by a logarithmic function of the complexity of the surface, for example the genus plus the number of boundary components. The geodesics in the complex can be described, up to bounded Hausdorff distance, by a simple relation of intersection numbers. We can also give a similar criterion for recognising if two geodesic segments are close.