Existing approaches to visual Simultaneous Localization and Mapping (SLAM) typically utilize points as visual feature primitives to represent landmarks in the environment. Since these techniques mostly use image points from a standard feature point detector, they do not explicitly map objects or regions of interest. Our work is motivated by the need for different SLAM techniques in path and riverine settings, where feature points can be scarce or may not adequately represent the environment. Accordingly, the proposed approach uses cubic Bézier curves as stereo vision primitives and offers a novel SLAM formulation to update the curve parameters and vehicle pose. This method eliminates the need for point-based stereo matching, with an optimization procedure to directly extract the curve information in the world frame from noisy edge measurements. Further, the proposed algorithm enables navigation with fewer feature states than most point-based techniques, and is able to produce a map which only provides detail in key areas. Results in simulation and with vision data validate that the proposed method can be effective in estimating the 6DOF pose of the stereo camera, and can produce structured, uncluttered maps.