Beam-to-beam contact can be dually described depending on the scale as contact between curves or contact between surfaces. The surface-to-surface algorithm requires a computationally expensive continuum description of the beams. The curve-to-curve contact algorithm is generated from the corresponding closest point projection procedure determining a minimal distance between curves and involves less expensive beam finite elements.The existence and uniqueness criterion of the closest point projection procedure formulated already in earlier publications is analyzed in detail, especially in the case of multiple solutions for "parallel curves." It is shown that the curve-to-curve contact algorithm cannot be applied for general parallel curves, which can be generated in an arbitrary range of angles between the tangent lines.A new curve-to-solid beam contact algorithm is here developed. This algorithm requires a special solid-beam finite element and is combining the advantages of surface contact together with the flexibility of the application to beam-to-beam contact.Further, a special case with "parallel tangent" vectors is considered from the classical Hertzian contact problem. It is found numerically that the corresponding length of a contact zone is well correlated with the Hertz theory. A possible application of the developed strategy for wire ropes is analyzed numerically.