We consider Jordan curves passing through a given set of n points with the property such that each edge is the hyperbolic geodesic in the complement of the rest of the curve. Those curves are related to the minimizers of the Loewner energy and give a specific complex projective structure on the n-punctured sphere with real and parabolic holonomy around the punctures. Similar to a result of Takhtajan-Zograf, we also obtain a new type of accessory parameters and show that they can be expressed as differentials of the Loewner energy. This is based on the joint work with Don Marshall and Steffen Rohde.