In this study, the shapes of the wheel and rail are represented by using 3-dimensional surface functions with surface parameters and a 3-dimensional wheel-rail contact analysis is presented. A whole numerical solution of wheel-rail contact at tread and flange including 2-point contacts can be achieved with the proposed numerical algorithm. Kinematic characteristics such as variances of vertical displacement and roll angle, and variance of wheel radius difference for arbitrary yaw and lateral displacement of wheelset, are determined for the KTX wheel-rail pair as an example. The condition of yaw and lateral displacement occurring 2-point contacts to analyze derailment are compared between standard and worn wheels. Differences of contact characteristics between curved and straight rails are also analyzed.