In this paper, a three dimensional catenary model was developed using a cable element based on the absolute nodal coordinate formulation (ANCF). The formulation of the lower order cable element was derived and verified using a simple example. A three-dimensional catenary model was created using the cable element which includes a contact, messenger wire, droppers and steady arms. Detailed characteristics including slackening of the dropper, pre-sag and stagger were considered. In order to verify the reliability of the developed catenary model, the static and dynamic behavior such as the static uplift, pre-sag, wave propagation speed, natural frequency and deflection by side wind were evaluated by comparing the results of the analysis model with the theoretical values, step by step. Finally, by presenting a curved track catenary model, the possibilities of a differentiated analysis extended from the existing two-dimensional catenary model were proposed.