Abstract: We present a Hybrid Rendering Algorithm (HRA) for rendering parametric curves and surfaces. The algorithm uses a series of Direct Rendering Criteria (DRC) for determining whether the curve surface can be directly rendered by forward differencing with a constant step size. The DRCS test the geometric flatness of the curve/surface, its parametric uniformity, and the ability to use only integer arithmetic in the forward differencing algorithm. If any of the DRCS is not fulfilled, the curve, surface is subdivided, The location of the subdivision in parameter space is chosen to increase the chances that the new segments will satisfy the DRCS, For the integer arithmetic DRC we introduce a general method for determining an alignment of tbe forward differences. We show that for cubic [quartic) curves whose control points lie in a 128K × 128K space this alignment enables up to 2¹³(2¹¹) forward steps. The method is applicable to curves of any order.