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The mobile platform of a parallel cable-driven mechanism is connected in parallel to a base by lightweight links, such as cables. Since the cables can only work in tension, the set of poses of the mobile platform for which the cables can balance any external wrench, i.e., for which the platform of the mechanism is fully constrained, is often limited or even nonexistent. Thus, the study and determination of this set of poses, called the wrench-closure workspace (WCW), is an important issue for parallel cable-driven mechanisms. In this paper, the case of planar parallel cable-driven mechanisms is addressed. Theorems that characterize the poses of the WCW are proposed. Then, these theorems are used to disclose the parts of the reachable workspace which belong to the WCW. Finally, an efficient algorithm that determines the constant-orientation cross-sections of these parts is introduced.