In the last decade, many algorithms have been proposed for sensor-based navigation, which keep convergence of a mobile robot to its goal in a 2D uncertain environment. From the convergence viewpoint, all the previous algorithms are categorized into three types of metric, topologic, and geometric algorithms. If position and orientation errors occur, a mobile robot unfortunately gets lost in each type of algorithm. Especially if a mobile robot should trace the boundary of an obstacle faithfully in a dense 2D environment, the metric algorithm is damaged with a small position error, and also the topologic algorithm is destroyed with even the smallest position error. In such a case, we are obliged to rely on the geometric algorithm. This paper proposes a geometric algorithm for sensor-based navigation when a mobile robot is ready for position and orientation errors in a dense 2D uncertain environment. In the proposed algorithm, a robot never joins deadlock (cyclic motion) and consequently arrives near its goal if and only if position and orientation errors are of finite. The wider a free region exists around a goal, the larger bounds of the errors become. Finally it is confirmed theoretically and experimentally that the proposed algorithm makes a robot converge into the neighborhood close to a goal.