Current haptic surfaces performing friction modulation still do not convey realistic shape sensations. In this study, we designed three elongated shapes that were rendered by ultrasonic lubrication within their inner area: a straight line, a sine and a second-degree polynomial. For these three shapes, we investigated how design properties such as the shape's complexity and thickness can influence users recognition of the displayed 2D shape and their strategy for haptic exploration on the display. A first experiment focused on recognition of shapes with varying complexity and on whether the increased or decreased friction within the shape impacts its perception. A second experiment focused on studying the influence of the shape thickness on the recognition rate of two shapes: the straight line and the sine. In experiment 1, participants recognized the line and sine quite well while performance with the polynomial was at chance level. In addition, whether friction was increased or decreased compared to the surrounding area did not impact shape recognition. In experiment 2, participants' recognition performance decreased for larger thickness. Analysis of the finger trajectory in experiment 1 showed that participants who followed closely the shapes performed better than participants who broadly scanned the surface. Taken together, our results suggest that friction recognition alone does not enable accurate recognition of straight or curved 2D shapes. However, parameters related to the friction contrast or the width of the stimulation area can enhance the user capacity to perceive shapes on the screen by touch.