A system and method operate an autonomous vehicle. A sensor senses a road and an object. A processor determines, in a Cartesian reference frame, a representation of the road and a source point representative of the object, samples a first waypoint and a second waypoint from the representation of the road, determines a linear projection of the source point to a line connecting the first waypoint and the second waypoint, determines a first estimate of a longitudinal component of the source point in a road-based reference frame based on the linear projection, the first estimate being on a curve representing the road between the first waypoint and the second waypoint, determines a second estimate of the longitudinal component from the first estimate, determines a coordinate of the source point in the road-based reference frame from the second estimate and operates the vehicle with respect to the object using the coordinate.