Superior mobility of droplets on lubricant-infused surfaces (LIS) has recently attracted significant attention for designing liquid-repellent surfaces. Unlike sessile droplets on flat surfaces wherein the contact line is easily visible in experiments, the contact line on LIS is masked by the lubricant meniscus, and special imaging techniques are required to visualize the hidden droplet-lubricant interface. Moreover, the overall shape deviates significantly from the spherical cap geometry even at very low droplet volumes. These difficulties necessitate the need to model interfaces in order to assess the effect of surface and fluid properties on LIS. In this work, we first numerically simulate the droplet shapes to show that at very small volumes, droplet-air and droplet-lubricant interfaces are constant mean curvature (CMC) interfaces. Moreover, we elucidate that these mean curvatures are related by the ratio of interfacial tensions of the droplet-air and the droplet-lubricant interfaces. These insights reduce the modeling of LIS interfacial profiles to a simplified geometric problem, which is solved using the parametric equations of CMC surfaces along with the angles of the Neumann triangle as the boundary conditions. Predicted profiles of the droplet-air interface as a spherical cap, the droplet-lubricant interface as a nodoid, and the lubricant-air interface as a catenoid/nodoid show good agreement with experimental results in the literature. Importantly, we for the first time provide a framework, which accurately predicts the true contact angle at the hidden solid contact line by just using the information of the top spherical cap portion visible in experiments.