In the analysis of multibody system (MBS) dynamics, contact between two arbitrary rigid bodies is a fundamental feature in a variety of models. Many procedures have been proposed to solve the rigid body contact problem, most of which belong to one of the two categories: offline and online contact search methods. This investigation will focus on the development of a contact surface model for the rigid body contact problem in the case where an online three-dimensional nonconformal contact evaluation procedure, such as the elastic contact formulation—algebraic equations (ECF-A), is used. It is shown that the contact surface must have continuity in the second-order spatial derivatives when used in conjunction with ECF-A. Many of the existing surface models rely on direct linear interpolation of profile curves, which leads to first-order spatial derivative discontinuities. This, in turn, leads to erroneous spikes in the prediction of contact forces. To this end, an absolute nodal coordinate formulation (ANCF) thin plate surface model is developed in order to ensure second-order spatial derivative continuity to satisfy the requirements of the contact formulation used. A simple example of a railroad vehicle negotiating a turnout, which includes a variable cross-section rail, is tested for the cases of the new ANCF thin plate element surface, an existing ANCF thin plate element surface with first-order spatial derivative continuity, and the direct linear profile interpolation method. A comparison of the numerical results reveals the benefits of using the new ANCF surface geometry developed in this investigation.