A translation surface is a singular geometric structure on a surface modeled on the plane where transition maps are translations. Some recent research has focused on extending results known for translation surfaces to dilation surfaces, where we broaden allowable transition maps to include dilations of the plane. Such surfaces do not have natural metrics; however, one can ask: “Are natural analogs of geodesic representatives in this context?” Relatedly, translation surfaces which are not closed (e.g., infinite genus surfaces) may or may not have geodesic representatives in every homotopy class. We will describe conditions on surfaces that guarantee that canonical representatives of homotopy classes of curves exist. In doing so, we realize that even less structure is needed: we describe a class of geometric structures on surfaces that are not modeled on the plane at all, but still have canonical curve representatives. This is joint work with Ferrán Valdez and Barak Weiss (arXiv:2301.03727).