A simple method is presented for eliminating most pixel-positioning errors when rendering lines and polygons with J.E. Bresenham's algorithm (see IBM Syst. J., vol.4, no.1, p.25-30 (1965)). The method affects only the calculation of the initial values for Bresenham's vector-generating algorithm. It does not alter the actual vector-generating algorithm, requiring only inter arithmetic to find the next pixel in a vector. The method eliminates all dropouts and virtually all overlaps between adjacent polygons whose edges lie on the same line. This eliminates the need to grow polygons to avoid dropout and opens the possibility of drawing surfaces composed of adjacent polygons with read modify/write pixel operations such as add or alpha buffering. It is shown that most rendering artifacts of today's display controllers ultimately result from pixel-positioning errors, not insufficient z-buffer resolution.