Systems and methods include obtaining observation points of a lane line using a sensor of a vehicle. Each observation point indicates a location of a point on the lane line. A method includes generating or updating a lane model with the observation points. The lane model indicates a path of the lane line and the lane model is expressed in a lane-specific coordinate system that differs from a vehicle coordinate system that is defined by an orientation of the vehicle. The method also includes transforming the lane-specific coordinate system to maintain a correspondence between the lane-specific coordinate system and the vehicle coordinate system based on a change in orientation of the vehicle resulting in a change in the vehicle coordinate system.