We present a complete analysis for the deformation profiles of lipid membranes induced by their interactions with solid elliptical cylinder substrates (e.g., proteins). The theoretical framework for the mechanics of lipid membranes is described in terms of the classical Helfrich model, and the resulting shape equation is formulated in general curvilinear coordinates to accommodate the elliptical shape of the contour surrounding the contact area. Admissible boundary conditions for the contact region are taken from the existing literature but reformulated and adapted to the current framework. A complete semi-analytic solution (in terms of Mathieu functions) is obtained within the limitation of superposed incremental deformations and the Monge representation in the deformed configuration functions. The results predict smooth morphological transitions over the domain of interest when a lipid membrane interacts with a rigid substrate through an elliptical contact region.