Based on the attitude angle of asperities, a multi-stage model is established for the curved contact between fractal surfaces. The asperity exhibits deviation between its axis and the global normal, thereby the attitude angle of asperity occurs. Asperity deformation depending on attitude angle is derived at length scale. Attitude angle also leads to a two-dimensional load of asperity, presenting double origins of plastic mode and corresponding critical length scales. With increasing load, the critical length scale varies, and the contact processes four stages incorporating the effects of attitude angle. The contact characteristics of each stage are calculated, and verification by experimental data is provided. This work illustrates the substantive characteristics of curved contact due to the asperity attitude.