Summary: Mannheim curves and the constant-pitch curves are two specific classes of space curves that are identified by a relation between their curvature and torsion functions. We detail the construction of these two types of curves from any given arbitrary regular space curve in \(\mathbb{R}^2\) by means of the so-called Combescure transformation. Further, we show that both Mannheim and constant-pitch curves have an integral characterization in terms of a given spherical curve. This has important applications to the theory of elastic strips and elastic curves.