A convertible lenticular liquid crystal (LC) lens architecture is demonstrated using an index-matched planarization layer on a periodically undulated electrode for the homogeneous alignment of an LC. It is found that the in-plane component of the electric field by the undulated electrode plays a primary role in the flat-to-lens effect while the out-of-plane component contributes to the anchoring enhancement of the LC molecules in the surface layer. Our LC device having an index-matched planarization layer on the undulated electrode is capable of achieving the electrical tunability from the flat surface to the lenticular lens suitable for 2D/3D convertible displays.