The contribution at hand aims at a numerical method to represent contact between two or more different solid bodies. The proposed computational contact algorithm is based on a displacement-driven approach to describe the geometrical constraint resulting from the impenetrability of the solid bodies in contact. A solution method of contact between one deformable and one rigid body is assumed for the initial computationally fast implementation of the underlying equations. Elastic or inelastic material behavior can be considered for the deformable body. Furthermore, the proposed approach is capable of finite deformations. Moreover, a node-to-segment (NTS) formulation is utilized for contact element discretization. Various numerical examples are presented to demonstrate the numerical method's applicability for different contact mechanics tasks. First, the presented examples verify the displacement-driven approach by investigating its convergence behavior and consistency. Then, comparisons of solutions by the proposed formulation to the analytical solution and the penalty method are provided to validate its accuracy. Moreover, the stick and slip condition of large sliding motion in a frictional contact problem is successfully depicted.