In this investigation, a numerical procedure that can be used for the analysis of a wheel and rail contact geometry is developed using the constraint contact formulation. The locations of contact points are determined for given lateral and yaw displacements of a wheelset when one-point contact is considered for each wheel, while these two displacements are no longer independent when the two-point contact occurs. A systematic procedure for predicting the flange, as well as the back-of-flange contact points, is developed and used for the two-point contact geometry analysis of a wheel and rail. Numerical results that involve tread, flange, and back-of-flange contacts are presented in order to demonstrate the use of the contact algorithm developed in this investigation. In particular, the back-of-flange contact is discussed for assessing contact configurations of a wheel and a grooved rail in light rail vehicle applications.